Holt McDougal Florida Larson Algebra 1 - rjssolutions.com · Holt McDougal Florida Larson Algebra 1 Notetaking Teacher’s Guide Volume 1: Chapters 1–6 LAH_A1_11_FL_NTG_FM_Vol1.indd

Holt McDougal Florida

Larson Algebra 1

Notetaking Teacher’s GuideVolume 1: Chapters 1–6

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ISBN 13: 978-0-547-24991-9ISBN 10: 0-547-24991-9

1 2 3 4 5 6 7 8 9 XXX 15 14 13 12 11 10 09

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iiiAlgebra 1

Notetaking Guide

ContentsChapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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Goal p Evaluate algebraic expressions and use exponents.

ALGEBRAIC EXPRESSIONS

Algebraic Expression Meaning Operation

7t 7 times t

x } 20 Division

y 2 8

12 1 a

VOCABULARY

Variable

Algebraic expression

Evaluating an expression

Power

Base

Exponent

An algebraic expression is also called a variable expression.

Your Notes

Evaluate Expressions1.1

Copyright © Holt McDougal. All rights reserved. Lesson 1.1 • Algebra 1 Notetaking Guide 1

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Goal p Evaluate algebraic expressions and use exponents.

ALGEBRAIC EXPRESSIONS

Algebraic Expression Meaning Operation

7t 7 times t Multiplication

x } 20 x divided by 20 Division

y 2 8 y minus 8 Subtraction

12 1 a 12 plus a Addition

VOCABULARY

Variable A letter used to represent one or more numbers

Algebraic expression A collection of numbers, variables, and operations

Evaluating an expression Substituting a number for the variable, performing the operation(s), and simplifying the result if necessary

Power An expression that represents repeated multiplication of the same factor

Base The number or expression that is used as a factor in repeated multiplication

Exponent A number or expression that represents the number of times the base is used as a factor

An algebraic expression is also called a variable expression.

Your Notes

Evaluate Expressions1.1



Your Notes

Evaluate the expression when n 5 4.

a. 11 3 n 5 11 3 Substitute for n.

5 .

b. 12 } n 5 12 } Substitute for n.

5 .

c. n 2 3 5 2 3 Substitute for n.

5 .

Example 1 Evaluate algebraic expressions

To evaluate an expression, substitute a number for the variable, perform the operation(s), and simplify.

Write the power in words and as a product.

Power Words Product

a. 121 twelve to the power

b. 23 two to the power, or two

c. 1 1 } 4 2 2 one fourth to the power, or one fourth

d. a4 a to the power

Example 2 Read and write powers

1. 7y 2. y 4 2 3. 10 2 y 4. y 1 6

Checkpoint Evaluate the expression when y 5 8.

2 Lesson 1.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.


Your Notes

4

Evaluate the expression when n 5 4.

a. 11 3 n 5 11 3 4 Substitute 4 for n.

5 44 Multiply .

b. 12 } n 5 12 } Substitute 4 for n.

5 3 Divide .

c. n 2 3 5 4 2 3 Substitute 4 for n.

5 1 Subtract .

Example 1 Evaluate algebraic expressions

To evaluate an expression, substitute a number for the variable, perform the operation(s), and simplify.

Write the power in words and as a product.

Power Words Product

a. 121 twelve to the 12 first power

b. 23 two to the 2 p 2 p 2 third power, or two cubed

c. 1 1 } 4 2 2 one fourth to the second power, or one fourth squared

d. a4 a to the fourth a p a p a p a power

1 } 4 p 1 } 4

Example 2 Read and write powers

1. 7y 2. y 4 2 3. 10 2 y 4. y 1 6

56 4 2 14

Checkpoint Evaluate the expression when y 5 8.



Your Notes

Evaluate the expression.

a. y3 when y 5 3 b. a5 when a 5 1.2

Solution

a. y3 5 3 Substitute for y.

5 .

5 .

b. a5 5 5 Substitute for a.

5 .

5 .

Example 3 Evaluate powers

Homework

5. 75 6. 1 1 } 3 2 2 7. (1.4)3

Checkpoint Write the power in words and as a product.

8. t2 when t 5 3 9. m5 when 10. x3 when

m 5 1 } 2 x 5 4

Checkpoint Evaluate the expression.



Your Notes


a. y3 when y 5 3 b. a5 when a 5 1.2

Solution

a. y3 5 3 3 Substitute 3 for y.

5 3 p 3 p 3 Write factors .

5 27 Multiply .

b. a5 5 1.2 5 Substitute 1.2 for a.

5 (1.2)(1.2)(1.2)(1.2)(1.2) Write factors .

5 2.48832 Multiply .

Example 3 Evaluate powers

Homework

5. 75 6. 1 1 } 3 2 2 7. (1.4)3

seven to the one third to the one and four fifth power second power, tenths to the

or one third third power, or7 p 7 p 7 p 7 p 7 squared one and four tenths cubed 1 }

3 p 1 }

3

(1.4)(1.4)(1.4)

Checkpoint Write the power in words and as a product.

8. t2 when t 5 3 9. m5 when 10. x3 when

9 m 5 1 } 2 x 5 4

1 } 32

64




1.2 Apply Order of OperationsGoal p Use the order of operations to evaluate expressions.

VOCABULARY

Order of Operations

ORDER OF OPERATIONS

To evaluate an expression involving more than one operation, use the following steps.

Step 1 Evaluate expressions inside .

Step 2 Evaluate .

Step 3 and divide from left to right.

Step 4 Add and from left to right.

Evaluate the expression 30 3 2 4 22 2 5.

SolutionStep 1

There are no grouping symbols, so go to Step 2.

Step 2

30 3 2 4 22 2 5 5 30 3 2 4 2 5 power.

Step 3

30 3 2 4 2 5 5 4 2 5 .

5 2 5 .

Step 4

2 5 5 .

Example 1 Evaluate Expressions


Your Notes


1.2 Apply Order of OperationsGoal p Use the order of operations to evaluate expressions.

VOCABULARY

Order of Operations The rules established to evaluate an expression involving more than one operation

ORDER OF OPERATIONS

To evaluate an expression involving more than one operation, use the following steps.

Step 1 Evaluate expressions inside grouping symbols .

Step 2 Evaluate powers .

Step 3 Multiply and divide from left to right.

Step 4 Add and subtract from left to right.

Evaluate the expression 30 3 2 4 22 2 5.

SolutionStep 1

There are no grouping symbols, so go to Step 2.

Step 2

30 3 2 4 22 2 5 5 30 3 2 4 4 2 5 Evaluate power.

Step 3

30 3 2 4 4 2 5 5 60 4 4 2 5 Multiply .

5 15 2 5 Divide .

Step 4

15 2 5 5 10 Subtract .

Example 1 Evaluate Expressions


Your Notes



a. 6(9 1 3) 5 6( ) within parentheses.

5 .

b. 50 2 (32 1 1) 5 50 2 ( 1 1) power.

5 50 2 ( ) within parentheses.

5 .

c. 3[5 1 (52 1 5)] 5 3[5 1 ( 1 5)] power.

5 3[5 1 ( )] within parentheses.

5 3[ ] within brackets.

5 .

Example 2 Evaluate expressions with grouping symbols

Grouping symbols such as parentheses ( ) and brackets [ ] indicate that operations inside the grouping symbols should be performed first.

1. 10 1 32 2. 16 2 23 1 4

3. 28 4 22 1 1 4. 4 p 52 1 4



Your Notes



a. 6(9 1 3) 5 6( 12 ) Add within parentheses.

5 72 Multiply .

b. 50 2 (32 1 1) 5 50 2 ( 9 1 1) Evaluate power.

5 50 2 ( 10 ) Add within parentheses.

5 40 Subtract .

c. 3[5 1 (52 1 5)] 5 3[5 1 ( 25 1 5)] Evaluate power.

5 3[5 1 ( 30 )] Add within parentheses.

5 3[ 35 ] Add within brackets.

5 105 Multiply .

Example 2 Evaluate expressions with grouping symbols

Grouping symbols such as parentheses ( ) and brackets [ ] indicate that operations inside the grouping symbols should be performed first.

1. 10 1 32 2. 16 2 23 1 4

19 12

3. 28 4 22 1 1 4. 4 p 52 1 4

8 104



Your Notes


5. 6(3 1 32) 6. 2[(10 2 4) 4 3]


Evaluate the expression 12k } 3(k2 1 4)

when k 5 2.

Solution

12k } 3(k2 1 4)

5 Substitute for k.

5 power.

5 within parentheses.

5 .

5 .

Example 3 Evaluate an algebraic expression

12( )

3( 2 1 4)

12( )

3( 1 4)

12( )

3( )

7. x3 2 5 8. 6x 1 2 } x 1 7

Checkpoint Evaluate the expression when x 5 3.

Homework

A fraction bar can act as a grouping symbol. Evaluate the numerator and denominator before dividing.


Your Notes


5. 6(3 1 32) 6. 2[(10 2 4) 4 3]

72 4


Evaluate the expression 12k } 3(k2 1 4)

when k 5 2.

Solution

12k } 3(k2 1 4)

5 Substitute 2 for k.

5 Evaluate power.

5 Add within parentheses.

5 Multiply .

5 1 Divide .

Example 3 Evaluate an algebraic expression

212( )

23( 2 1 4)

212( )

43( 1 4)

212( )

83( )

24

24

7. x3 2 5 8. 6x 1 2 } x 1 7

22 2

Checkpoint Evaluate the expression when x 5 3.

Homework

A fraction bar can act as a grouping symbol. Evaluate the numerator and denominator before dividing.


Your Notes


Example 1 Determine the truth of a statement

Analyze Always, Sometimes, Never Statements Goal p Determine whether statements are always,

sometimes, or never true.

Tell whether the statement is always, sometimes, or never true.

For a given whole number, n, the product 3n represents an odd whole number.

SolutionTest the expression for some values for n. When n is 1, 3n 5 , an number. When n is 2, 3n 5 , an number. The statement is true.



For a given whole number, n, the expression 4n 1 2 represents an even whole number.

SolutionTest the expression for some values of n. When n is 1, , an number. When n is 2,

, an number. Will this happen for any value of n?

Check whether dividing the expression by results in a whole number.

(4n 1 2) 4 5 (4n 4 ) 1 (2 4 )

5

Because n is a whole number, will be a whole number. So, (4n 1 2) divisible by , and the original statement is true.

Your Notes

Copyright © Holt McDougal. All rights reserved. 1.2 Focus On Reasoning • Algebra 1 Notetaking Guide 7

Focus On ReasoningUse after Lesson 1.2



Analyze Always, Sometimes, Never Statements Goal p Determine whether statements are always,

sometimes, or never true.


For a given whole number, n, the product 3n represents an odd whole number.

SolutionTest the expression for some values for n. When n is 1, 3n 5 3 , an odd number. When n is 2, 3n 5 6 , an even number. The statement is sometimes true.



For a given whole number, n, the expression 4n 1 2 represents an even whole number.

SolutionTest the expression for some values of n. When n is 1, 4n 1 2 5 6 , an even number. When n is 2, 4n 1 2 5 10 , an even number. Will this happen for any value of n?

Check whether dividing the expression by 2 results in a whole number.

(4n 1 2) 4 2 5 (4n 4 2 ) 1 (2 4 2 )

5 2n 1 1

Because n is a whole number, 2n 1 1 will always be a whole number. So, (4n 1 2) is divisible by 2 , and the original statement is always true.

Your Notes






For a given whole number, n, the expression (2n)2 1 1 represents an even whole number.

SolutionTest the expression for some values of n. When n is 1,

, an number. When n is 2, , . Will this

happen for any value of n?

Check whether results in a .

((2n)2 1 1) 4 5 ((2n)2 4 ) 1 (1 4 )

5 ( 4 ) 1 (1 4 )

5 1 1 }

Because n is a whole number, the first term

will be a whole number. This means that

the sum

be a whole number. So,

(2n)2 1 1 divisible by , and the original

statement is true.

Homework

Checkpoint Tell whether the statement is always, sometimes, or never true..

1. For a given whole number, n, the expression represents an even whole number.

2. For a given odd whole number, n, the expression n2 1 2 represents an even whole number.

8 1.2 Focus On Reasoning • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Your Notes




For a given whole number, n, the expression (2n)2 1 1 represents an even whole number.

SolutionTest the expression for some values of n. When n is 1, (2n)2 1 1 5 5 , an odd number. When n is 2, (2n)2 1 1 5 17 , an odd number . Will this happen for any value of n?

Check whether dividing the expression by 2 results in a whole number .

((2n)2 1 1) 4 2 5 ((2n)2 4 2 ) 1 (1 4 2 )

5 ( 4n2 4 2 ) 1 (1 4 2 )

5 2n2 1 1 }

2

Because n is a whole number, the first term 2n2

will always be a whole number. This means that

the sum 2n2 1 1 } 2

will not be a whole number. So,

(2n)2 1 1 is not divisible by 2 , and the original

statement is never true.

Homework

Checkpoint Tell whether the statement is always, sometimes, or never true..

1. For a given whole number, n, the expression represents an even whole number.

sometimes

2. For a given odd whole number, n, the expression n2 1 2 represents an even whole number.

never


Your Notes


1.3 Write ExpressionsGoal p Translate verbal phrases into expressions.

VOCABULARY

Verbal model

Rate

Unit rate

TRANSLATING VERBAL PHRASES

Operation Verbal Phrase Expression

Addition The of 3 and a number n

A number x 10

Subtraction The of 7

and a number a

Twelve than a number x

Multiplication Five a number y

The of 2 and a number n

Division The of a number a and 6

Eight intoa number y

Order is important when writing subtraction and division expressions.


Your Notes


1.3 Write ExpressionsGoal p Translate verbal phrases into expressions.

VOCABULARY

Verbal model An expression that describes a problem using words as labels and using math symbols to relate the words

Rate A fraction that compares two quantities measured in different units

Unit rate A rate per one given unit

TRANSLATING VERBAL PHRASES

Operation Verbal Phrase Expression

Addition The sum of 3 and 3 1 n a number n

A number x plus 10 x 1 10

Subtraction The difference of 7 7 2 a

and a number a

Twelve less than x 2 12 a number x

Multiplication Five times a number y 5y

The product of 2 2n and a number n

Division The quotient of a a } 6

number a and 6

Eight divided into y }

8

a number y

Order is important when writing subtraction and division expressions.


Your Notes


Translate the verbal phrase into an expression.

Verbal Phrase Expression

a. 6 less than the quantity 8 times a number x

b. 2 times the sum of 5 and a number a

c. The difference of 17 and the cube of a number n

Example 1 Translate verbal phrases into expressions

1. The product of 5 and the quantity 12 plus a number n

2. The quotient of 10 and the quantity a number x minus 3

Checkpoint Translate the verbal phrase into an expression.

Food Drive You and three friends are collecting canned food for a food drive. You each collect the same number of cans. Write an expression for the total number of cans collected.

SolutionStep 1 Write a verbal model. Amount 3 Number of

of cans

Step 2 Translate the verbal 3 model into an algebraic expression.

An expression that represents the total number of cans is .

Example 2 Use a verbal model to write an expression

The words “the quantity” tell you what to group when translating verbal phrases.


Your Notes


Translate the verbal phrase into an expression.

Verbal Phrase Expression

a. 6 less than the quantity 8x 2 6 8 times a number x

b. 2 times the sum of 5 2(5 1 a) and a number a

c. The difference of 17 and 17 2 n3 the cube of a number n

Example 1 Translate verbal phrases into expressions

1. The product of 5 and the quantity 12 plus a number n

5(12 1 n)

2. The quotient of 10 and the quantity a number x minus 3

10 } x 2 3

Checkpoint Translate the verbal phrase into an expression.

Food Drive You and three friends are collecting canned food for a food drive. You each collect the same number of cans. Write an expression for the total number of cans collected.

SolutionStep 1 Write a verbal model. Amount 3 Number of of cans people

Step 2 Translate the verbal c 3 4 model into an algebraic expression.

An expression that represents the total number of cans is c 3 4 .

Example 2 Use a verbal model to write an expression

The words “the quantity” tell you what to group when translating verbal phrases.


Your Notes


3. In Example 2, suppose that the total number of cans collected are distributed equally to 2 food banks. Write an expression that represents the number of cans each food bank receives.

Checkpoint Complete the following exercise.

Example 3 Find a unit rate

Homework

A construction crew digs a tunnel 430 feet long in 3 days. Find their unit rate in miles per week.

Solution

} days

5 } days

• 1 mile }

5280 • days }

1

5 mile }

1

Check that the expression produces the units you expect.

} days

• mile } •

days } 5

mile }

Their unit rate is mile per .

Use unit analysis, also called dimensional analysis, to check that the expression produces an answer in miles per week.

Checkpoint Find the unit rate.

4. A worker earns a nickel every eight seconds. What is his rate in dollars per hour?


Your Notes


3. In Example 2, suppose that the total number of cans collected are distributed equally to 2 food banks. Write an expression that represents the number of cans each food bank receives.

c 3 4 } 2


Example 3 Find a unit rate

Homework

A construction crew digs a tunnel 430 feet long in 3 days. Find their unit rate in miles per week.

Solution

430

feet }

3 days

5 430

feet }

3 days • 1 mile }

5280 feet • 7 days }

1 week

5 0.19 mile } 1 week

Check that the expression produces the units you expect.

feet }

days • mile

} feet

• days

} week

5 mile

} week

Their unit rate is 0.19 mile per week .

Use unit analysis, also called dimensional analysis, to check that the expression produces an answer in miles per week.

Checkpoint Find the unit rate.

4. A worker earns a nickel every eight seconds. What is his rate in dollars per hour?

$22.50/hr


Your Notes


Write Equations and Inequalities

1.4Goal p Translate verbal sentences into equations or

inequalities.

VOCABULARY

Open sentence

Equation

Inequality

Solution of an equation

Solution of an inequality

EXPRESSING OPEN SENTENCES

Symbol Meaning Associated Words

a 5 b a is to b a is the as b

a b a is b a is than b

a ≥ b a is than a is b, or to b a is than b

Your Notes



Write Equations and Inequalities

1.4Goal p Translate verbal sentences into equations or

inequalities.

VOCABULARY

Open sentence A mathematical statement that contains two expressions and a symbol that compares them

Equation An open sentence that contains the symbol 5

Inequality An open sentence that contains one of the symbols <, ≤, >, or ≥

Solution of an equation A number that when substituted for the variable in an equation results in a true statement

Solution of an inequality A number that when substituted for the variable in an inequality results in a true statement

EXPRESSING OPEN SENTENCES

Symbol Meaning Associated Words

a 5 b a is equal to b a is the same as b

a b a is greater than b a is more than b

a ≥ b a is greater than a is at least b, or equal to b a is no less than b

Your Notes



Write an equation or an inequality.

Verbal Sentence Equation or Inequality

a. The sum of three times a number a and 4 is 25.

b. The quotient of a number x and 4 is fewer than 10.

c. A number n is greater than 6 and less than 12.

Example 1 Write equations and inequalitiesYour Notes

Sometimes two inequalities are combined. For example, the inequalities a < b and b < c can be combined to form the inequality a < b < c.

Check whether 2 is a solution of the equation or inequality.

Equation or Substitute ConclusionInequality

a. 7x 2 8 5 9 7(2) 2 8 0 9

a solution.

b. 4 1 5y < 18 4 1 5(2) <? 18

a solution.

c. 6n 2 9 ≥ 2 6(2) 2 9 ≥? 2

a solution.

Example 2 Check possible solutions

1. 6r 1 1 5 25 2. x2 2 5 > 10 3. 7a ≤ 21

r 5 4 x 5 5 a 5 6

Checkpoint Check whether the given number is a solution of the equation or inequality.



Write an equation or an inequality.

Verbal Sentence Equation or Inequality

a. The sum of three times a 3a 1 4 5 25 number a and 4 is 25.

b. The quotient of a number x } 4 < 10x and 4 is fewer than 10.

c. A number n is greater 6 < n < 12 than 6 and less than 12.

Example 1 Write equations and inequalitiesYour Notes

Sometimes two inequalities are combined. For example, the inequalities a < b and b < c can be combined to form the inequality a < b < c.

Check whether 2 is a solution of the equation or inequality.

Equation or Substitute ConclusionInequality

a. 7x 2 8 5 9 7(2) 2 8 0 9 6 Þ 9

2 is not a solution.

b. 4 1 5y < 18 4 1 5(2) <? 18 14 < 18

2 is a solution.

c. 6n 2 9 ≥ 2 6(2) 2 9 ≥? 2 3 ≥ 2

2 is a solution.

Example 2 Check possible solutions

1. 6r 1 1 5 25 2. x2 2 5 > 10 3. 7a ≤ 21

r 5 4 x 5 5 a 5 6

solution solution not a solution

Checkpoint Check whether the given number is a solution of the equation or inequality.



Homework

Your Notes Example 3 Use mental math to solve an equation

Solve the equation using mental math.

a. n 1 6 5 11 b. 18 2 x 5 10

c. 7a 5 56 d. b } 11 5 3

Solution Equation Think Solution Check

a. n 1 6 5 11 What number 1 6 5 11 plus 6 equals

11?

b. 18 2 x 5 10 18 2 5 10

c. 7a 5 56 7( ) 5 56

d. b } 11 5 3 } 11 5 3

4. x 1 9 5 14 5. 5t 2 4 5 11 6. y } 4 5 15

Checkpoint Solve the equation using mental math.

Think of an equation as a question when solving using mental math.



Homework

Your Notes Example 3 Use mental math to solve an equation

Solve the equation using mental math.

a. n 1 6 5 11 b. 18 2 x 5 10

c. 7a 5 56 d. b } 11 5 3

Solution Equation Think Solution Check

a. n 1 6 5 11 What number 5 5 1 6 5 11 plus 6 equals

11?

b. 18 2 x 5 10 18 minus what 8 18 2 8 5 10 number equals

10?

c. 7a 5 56 7 times 8 7( 8 ) 5 56

what number

equals 56?

d. b } 11 5 3 What number 33 } 11 5 3

divided by 11

equals 3?

33

4. x 1 9 5 14 5. 5t 2 4 5 11 6. y } 4 5 15

5 3 60

Checkpoint Solve the equation using mental math.

Think of an equation as a question when solving using mental math.



1.5 Use a Problem Solving PlanGoal p Use a problem solving plan to solve problems.

VOCABULARY

Formula

A PROBLEM SOLVING PLAN

Use the following four-step plan to solve a problem.

Step 1 Read the problem carefully. Identify what you want to know and what you want to find out.

Step 2 Decide on an approach to solving the problem.

Step 3 Carry out your plan. Try a new approach if the first one isn’t successful.

Step 4 Check that your answer is reasonable.

You have $7 to buy orange juice and bagels at the store. A container of juice costs $1.25 and a bagel costs $.75. If you buy two containers of juice, how many bagels can you buy?

Solution

Step 1 What do you know? You know how much money you have and the price of a and a container of juice.

What do you want to find out? You want to find out the number of you can buy.

Step 2 Use what you know to write a that represents what you want to find out. Then write an and solve it.

Example 1 Read a problem and make a plan


Your Notes


1.5 Use a Problem Solving PlanGoal p Use a problem solving plan to solve problems.

VOCABULARY

Formula An equation that relates two or more quantities

A PROBLEM SOLVING PLAN

Use the following four-step plan to solve a problem.

Step 1 Read and Understand Read the problem carefully. Identify what you want to know and what you want to find out.

Step 2 Make a Plan Decide on an approach to solving the problem.

Step 3 Solve the Problem Carry out your plan. Try a new approach if the first one isn’t successful.

Step 4 Look Back Check that your answer is reasonable.

You have $7 to buy orange juice and bagels at the store. A container of juice costs $1.25 and a bagel costs $.75. If you buy two containers of juice, how many bagels can you buy?

Solution

Step 1 Read and Understand What do you know? You know how much money you have and the price of a bagel and a container of juice.

What do you want to find out? You want to find out the number of bagels you can buy.

Step 2 Make a Plan Use what you know to write a verbal model that represents what you want to find out. Then write an equation and solve it.

Example 1 Read a problem and make a plan


Your Notes


Solve the problem in Example 1 by carrying out the plan. Then check your answer.

Solution

Step 3 Write a verbal model. Then write an equation. Let b be the number of bagels you buy.

Price of Number Price of Number Costjuice of bagel of (in dollars)

(in dollars) containers (in dollars) bagels

p 1 p b 5

The equation is 1 b 5 . One way to solve the equation is to use the strategy guess, check, and revise.

Guess an even number that is easily multiplied by . Try 4.

1 b 5 Write equation.

1 (4) 0 Substitute 4 for b.

Simplify; 4 check.

Because , try an even number 4. Try 6.

1 b 5 Write equation.

1 (6) 0 Substitute 6 for b.

Simplify.

For you can buy bagels and containers of juice.

Step 4 Each additional bagel you buy adds to the you pay for the juice. Make a table.

Bagels 0 1 2 3 4 5 6

Total Cost

The total cost is when you buy bagels and containers of juice. The answer in step 3 is .

Example 2 Solve a problem and look backYour Notes



Solve the problem in Example 1 by carrying out the plan. Then check your answer.

Solution

Step 3 Solve the Problem Write a verbal model. Then write an equation. Let b be the number of bagels you buy.

Price of Number Price of Number Costjuice of bagel of (in dollars)

(in dollars) containers (in dollars) bagels

1.25 p 2 1 0.75 p b 5 7

The equation is 2.5 1 0.75 b 5 7 . One way to solve the equation is to use the strategy guess, check, and revise.

Guess an even number that is easily multiplied by 0.75 . Try 4.

2.5 1 0.75 b 5 7 Write equation.

2.5 1 0.75 (4) 0 7 Substitute 4 for b.

5.5 Þ 7 Simplify; 4 does not check.

Because 5.5 < 7 , try an even number greater than 4. Try 6.

2.5 1 0.75 b 5 7 Write equation.

2.5 1 0.75 (6) 0 7 Substitute 6 for b.

7 5 7 Simplify.

For $7 you can buy 6 bagels and 2 containers of juice.

Step 4 Look Back Each additional bagel you buy adds $.75 to the $2.50 you pay for the juice. Make a table.

Bagels 0 1 2 3 4 5 6

Total Cost 2.50 3.25 4 4.75 5.5 6.25 7

The total cost is $7 when you buy 6 bagels and 2 containers of juice. The answer in step 3 is correct .

Example 2 Solve a problem and look backYour Notes



1. Suppose in Example 1 that you have $12 and you decide to buy three containers of juice. How many bagels can you buy?


FORMULA REVIEW

Temperature

C 5 5 } 9 (F 2 32), where F 5

and C 5

Simple interest

I 5 Prt, where I 5 , P 5 ,r 5 (as a decimal), and t 5

Distance traveled

d 5 rt, where d 5 , r 5 , and t 5

Profit

P 5 I 2 E, where P 5 , I 5 , and E 5

2. In Example 1, the store where you bought the juice and bagels had an income of $7 from your purchase. The profit the store made from your purchase is $2.50. Find the store’s expense for the juice and bagels.


Homework


Your Notes


1. Suppose in Example 1 that you have $12 and you decide to buy three containers of juice. How many bagels can you buy?

11 bagels


FORMULA REVIEW

Temperature

C 5 5 } 9 (F 2 32), where F 5 degrees Fahrenheit

and C 5 degrees Celsius

Simple interest

I 5 Prt, where I 5 interest , P 5 principal ,r 5 interest rate (as a decimal), and t 5 time

Distance traveled

d 5 rt, where d 5 distance traveled , r 5 rate , and t 5 time

Profit

P 5 I 2 E, where P 5 profit , I 5 income , and E 5 expenses

2. In Example 1, the store where you bought the juice and bagels had an income of $7 from your purchase. The profit the store made from your purchase is $2.50. Find the store’s expense for the juice and bagels.

$4.50


Homework


Your Notes


1.6 Represent Functionsas Rules and TablesGoal p Represent functions as rules and as tables.

VOCABULARY

Function

Domain

Range

Independent variable

Dependent variable

The input-output table shows temperatures over various increments of time. Identify the domain and range of the function.

Input (hours) 0 2 4 6

Output (°C) 24 27 30 33

Solution

Domain:

Range:

Example 1 Identify the domain and range of a function


Your Notes


1.6 Represent Functionsas Rules and TablesGoal p Represent functions as rules and as tables.

VOCABULARY

Function A rule that establishes a relationship between two quantities, called the input and the output. For each input, there is exactly one output

Domain The collection of all input values

Range The collection of all output values

Independent variable The input variable

Dependent variable The output variable

The input-output table shows temperatures over various increments of time. Identify the domain and range of the function.

Input (hours) 0 2 4 6

Output (°C) 24 27 30 33

Solution

Domain: 0, 2, 4, 6

Range: 24, 27, 30, 33

Example 1 Identify the domain and range of a function


Your Notes


1. Input 4 7 11 13

Output 10 20 35 45

Checkpoint Identify the domain and range of the function.

Tell whether the pairing is a function. Explain your reasoning.

Solutiona.

482

Input Output

123

b. Input Output

2 2

2 4

3 6

4 8

Example 2 Identify a function

Mapping diagrams are often used to represent functions. Take note of the pairings to make your decision.

2. Input 5 5 10 15

Output 3 4 6 8

3. Input 0 4 12 20

Output 3 5 9 13

Checkpoint Tell whether the pairing is a function.


Your Notes


1. Input 4 7 11 13

Output 10 20 35 45

Domain: 4, 7, 11, 13

Range: 10, 20, 35, 45

Checkpoint Identify the domain and range of the function.

Tell whether the pairing is a function. Explain your reasoning.

Solutiona.

482

Input Output

123

b. Input Output

2 2

2 4

3 6

4 8

The pairing is a function The pairing is not a because each input is function because thepaired with exactly one input 2 is paired with output. 2 and 4.

Example 2 Identify a function

Mapping diagrams are often used to represent functions. Take note of the pairings to make your decision.

2. Input 5 5 10 15

Output 3 4 6 8

No, the pairing is not a function.

3. Input 0 4 12 20

Output 3 5 9 13

Yes, the pairing is a function.

Checkpoint Tell whether the pairing is a function.


Your Notes


FUNCTIONS

Verbal Rule Equation Table

The output is Input 2 4 6 8 10

Output2 less thanthe input.

The domain of the function y 5 3x is 0, 1, 2, and 3. Make a table for the function, then identify the range of the function.

Solution

x

y 5 3x

The range of the function is .

Example 3 Make a table for a function

Write a rule for the function.

Input 3 5 7 9 11

Output 6 10 14 18 22

SolutionLet x be the input and let y be the output. Notice that each output is the corresponding input. So, a rule for the function is .

Example 4 Write a function rule

A function may be represented using a rule that relates one variable to another.

Homework 4. Yarn (yd) 1 2 3 4

Total Cost ($) 1.5 3 4.5 6

Checkpoint Write a rule for the function. Identify thedomain and the range.


Your Notes


FUNCTIONS

Verbal Rule Equation Table

The output is y 5 x 2 2 Input 2 4 6 8 10

Output 0 2 4 6 82 less thanthe input.

The domain of the function y 5 3x is 0, 1, 2, and 3. Make a table for the function, then identify the range of the function.

Solution

x 0 1 2 3

y 5 3x 3(0) 5 0 3(1) 5 3 3(2) 5 6 3(3) 5 9

The range of the function is 0, 3, 6, and 9 .

Example 3 Make a table for a function

Write a rule for the function.

Input 3 5 7 9 11

Output 6 10 14 18 22

SolutionLet x be the input and let y be the output. Notice that each output is twice the corresponding input. So, a rule for the function is y 5 2x .

Example 4 Write a function rule

A function may be represented using a rule that relates one variable to another.

Homework 4. Yarn (yd) 1 2 3 4

Total Cost ($) 1.5 3 4.5 6

Function Rule: y 5 1.5x

Domain: 1, 2, 3, 4 Range: 1.5, 3, 4.5, 6

Checkpoint Write a rule for the function. Identify thedomain and the range.


Your Notes


1.7 Represent Functions as GraphsGoal p Represent functions as graphs.

GRAPHING A FUNCTION

• You can use a graph to represent a .

• In a given table, each corresponding pair of input and output values forms an .

• An ordered pair of numbers can be plotted as a.

• The x-coordinate is the .

• The y-coordinate is the .

• The horizontal axis of the graph is labeled with the.

• The vertical axis is labeled with the the .

Graph the function y 5 x 1 1 with domain 1, 2, 3, 4, and 5.

Solution

Step 1 Make an table.

x

y

Step 2 Plot a point for each (x, y).

y

xO 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1

Example 1 Graph a function


Your Notes


1.7 Represent Functions as GraphsGoal p Represent functions as graphs.

GRAPHING A FUNCTION

• You can use a graph to represent a function .

• In a given table, each corresponding pair of input and output values forms an ordered pair .

• An ordered pair of numbers can be plotted as a point .

• The x-coordinate is the input .

• The y-coordinate is the output .

• The horizontal axis of the graph is labeled with the input variable .

• The vertical axis is labeled with the the output variable .

Graph the function y 5 x 1 1 with domain 1, 2, 3, 4, and 5.

Solution

Step 1 Make an input-output table.

x 1 2 3 4 5

y 2 3 4 5 6

Step 2 Plot a point for each ordered pair (x, y).

y

xO 1 2 3 4 5 6 7 8

7

6

5

4

3

2

1



Your Notes


Write a function rule for the function represented by the graph. Identify the domain and the range of the funtion.

y

xO 2 4 6 8 10 12 14 16

7

6

5

4

3

2

1

Solution

Step 1 Make a for the graph.

x

y

Step 2 Find a between the input and output values.

Step 3 Write a that describes the relationship.

y 5

A rule for the function is y 5 . The domain of the function is .

The range is .

Example 2 Write a function rule for a graph


Your Notes


Write a function rule for the function represented by the graph. Identify the domain and the range of the funtion.

y

xO 2 4 6 8 10 12 14 16

7

6

5

4

3

2

1

Solution

Step 1 Make a table for the graph.

x 2 4 6 8 10

y 0 1 2 3 4

Step 2 Find a relationship between the input and output values.

From the table, each output value is 1 less than half the corresponding input value.

Step 3 Write a function rule that describes the relationship.

y 5 1 } 2 x 2 1

A rule for the function is y 5 1 } 2 x 2 1 . The domain of the function is 2, 4, 6, 8, and 10 .

The range is 0, 1, 2, 3, and 4 .

Example 2 Write a function rule for a graph


Your Notes


Homework

1. Graph the function y 5 1 } 3 x 1 1 with domain 0, 3, 6, 9, and 12.

y

xO 2 4 6 8 10 12 14

7

6

5

4

3

2

1


Checkpoint Write a rule for the function represented by the graph. Identify the domain and the range of the function.

2. y

xO 122 2 3 4 5 6

14

12

10

8

6

4

2

3. y

xO 224 4 6 8 10 12

14

12

10

8

6

4

2


Your Notes


Homework

1. Graph the function y 5 1 } 3 x 1 1 with domain 0, 3, 6, 9, and 12.

x 0 3 6 9 12

y 1 2 3 4 5

y

xO 2 4 6 8 10 12 14

7

6

5

4

3

2

1


Checkpoint Write a rule for the function represented by the graph. Identify the domain and the range of the function.

2. y

xO 122 2 3 4 5 6

14

12

10

8

6

4

2

3. y

xO 224 4 6 8 10 12

14

12

10

8

6

4

2

y 5 3x 2 3 y 5 1 } 2 x 1 5

Domain: 1, 2, 3, 4 Domain: 2, 4, 6, 8

Range: 0, 3, 6, 9 Range: 6, 7, 8, 9


Your Notes


Determine Whether a Relation Is a Function Goal p Determine whether a relation is a function when

the relation is represented by a table or a graph.

VOCABULARY

Relation

Determine whether the relation is a function. Explain your reasoning.

a. b.

Input 3 4 2 0 Input 1 2 1 3 4

Output 4 4 5 5 Output 4 7 5 0 3

Solution

a. Every has one . The relation a function.

b. The input has different . So, .

Example 1 Determine whether a relation is a function

Checkpoint Determine whether the relation is a function.

1. Input 4 3 5 4

Output 6 5 7 2

2. Input 1 2 5 7

Output 4 5 7 6

24 1.7 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Focus On FunctionsUse after Lesson 1.7

Your Notes


Determine Whether a Relation Is a Function Goal p Determine whether a relation is a function when

the relation is represented by a table or a graph.

VOCABULARY

Relation Any pairing of a set of inputs with a set of outputs.

Determine whether the relation is a function. Explain your reasoning.

a. b.

Input 3 4 2 0 Input 1 2 1 3 4

Output 4 4 5 5 Output 4 7 5 0 3

Solution

a. Every input has one output . The relation is a function.

b. The input 1 has two different outputs . So, the relation is not a function .

Example 1 Determine whether a relation is a function

Checkpoint Determine whether the relation is a function.

1. Input 4 3 5 4

Output 6 5 7 2

The relation is not a function.

2. Input 1 2 5 7

Output 4 5 7 6

The relation is a function.



Your Notes


VERTICAL LINE TEST

A graph of a relation is a if no vertical line passes through more than one on the graph.

Copyright © Holt McDougal. All rights reserved. 1.7 Focus On Functions • Algebra 1 Notetaking Guide 25

x

y

x

y

x

y

2 4

2

4

6Ox

y

Determine whether the graph represents a function.

a. b.

Example 2 Use the vertical line test

A vertical line be drawn through more than one point. The graph a function.

A vertical line be drawn through more than one point. The graph

a function.

62 4

2

4

Ox

y

2 4

2

4

6Ox

y

Checkpoint Determine whether the graph represents a function.

3. 4.

2 4

2

4

6Ox

yHomework

Your Notes


VERTICAL LINE TEST

A graph of a relation is a function if no vertical line passes through more than one point on the graph.

Function Not a function Not a function


x

y

x

y

x

y

2 4

2

4

6Ox

y

Determine whether the graph represents a function.

a. b.

Example 2 Use the vertical line test

A vertical line cannot be drawn through more than one point. The graph represents a function.

A vertical line can be drawn through more than one point. The graph does not represent a function.

62 4

2

4

Ox

y

2 4

2

4

6Ox

y

Checkpoint Determine whether the graph represents a function.

3. 4.

function not a function

2 4

2

4

6Ox

yHomework

Your Notes


Variable

Power, Base, Exponent

Rate

Equation

Formula

Domain

Relation


Verbal model

Unit rate

Inequality

Function

Range

Review your notes and Chapter 1 by using the Chapter Review on pages 54–57 of your textbook.

26 Words to Review • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Words to ReviewGive an example of the vocabulary word.


Variable

x, where x is any real number

Power, Base, Exponent

23

2 is the base, 3 is the exponent

Rate

80 miles } 4 hours

Equation

2a 1 4 5 10

Formula

d 5 rt

Domain

Input 0 1 2

Output 2 4 6

The domain is 0, 1, 2 for this input-output table.

Relation

Input 1 1 2

Output 2 3 4


3x

Verbal model

number of CDs 4 number of customers

Unit rate

20 mi/h

Inequality

5n ≤ 20

Function

y 5 5x

Range

Input 0 1 2

Output 2 4 6

The range is 2, 4, 6 for this input-output table.





VOCABULARY

Whole number

Integer

Rational number

Opposite

Absolute value

Conditional statement

Graph 22 and 25 on a number line. Then tell which number is less.

Solution

25 24 23 22 21 0 1 2 3 4 5

On the number line, is to the left of .

So, < .

Example 1 Graph and compare integers

Negative integers are integers less than 0 and positive integers are integers greater than 0. The integer 0 is neither negative nor positive.

Goal p Graph and compare positive and negative numbers.

Use Integers and Rational Numbers

2.1


Your Notes


VOCABULARY

Whole number The numbers 0, 1, 2, 3, . . . are whole numbers.

Integer Any of the numbers . . . 23, 22, 21, 0, 1, 2, 3, . . . are integers.

Rational number A number that can be written as a quotient of two integers

Opposite Two numbers that are the same distance from 0 on a number line but are on opposite sides of 0

Absolute value The distance between a number a and 0 on a number line; represented by ⏐a⏐

Conditional statement A statement that has a hypothesis and a conclusion

Graph 22 and 25 on a number line. Then tell which number is less.

Solution

25

25 22

24 23 22 21 0 1 2 3 4 5

On the number line, 25 is to the left of 22 .

So, 25 < 22 .

Example 1 Graph and compare integers

Negative integers are integers less than 0 and positive integers are integers greater than 0. The integer 0 is neither negative nor positive.

Goal p Graph and compare positive and negative numbers.

Use Integers and Rational Numbers

2.1


Your Notes



Tell whether each of the following numbers is a whole number, an integer, or a rational number: 3, 1.7, 214,

and 2 1 } 2 .

Solution

Number Whole Number ? Integer ? Rational Number ?

3

1.7

214

2 1 } 2

Example 2 Classify numbers

Temperature The table shows the low daily temperatures for a town over a five-day period. Order the days from warmest to coldest.

Day 1 2 3 4 5

Temperature 08C 108C 228C 58C 278C

SolutionStep 1

Graph the numbers on a number line.

210 28 26 24 22 0 2 4 6 8 10

Step 2

Read the numbers from left to right:

.

From warmest to coldest the days are .

Example 3 Order rational numbers

Your Notes



Tell whether each of the following numbers is a whole number, an integer, or a rational number: 3, 1.7, 214,

and 2 1 } 2 .

Solution

Number Whole Number ? Integer ? Rational Number ?

3 Yes Yes Yes

1.7 No No Yes

214 No Yes Yes

2 1 } 2 No No Yes


Temperature The table shows the low daily temperatures for a town over a five-day period. Order the days from warmest to coldest.

Day 1 2 3 4 5

Temperature 08C 108C 228C 58C 278C

SolutionStep 1

Graph the numbers on a number line.

210

27 22 0 5 10

28 26 24 22 0 2 4 6 8 10

Step 2

Read the numbers from left to right:

27, 22, 0, 5, 10 .

From warmest to coldest the days are 2, 4, 1, 3, and 5 .

Example 3 Order rational numbers

Your Notes



Your Notes

1. Tell whether each of the following numbers is a whole number, an integer, or a rational number:

0.8, 217, 25 3 } 4 , and 2. Then order the numbers from

least to greatest.


ABSOLUTE VALUE OF A NUMBER

Words Numbers

If x is a positive number, ⏐5⏐ 5 then ⏐x⏐ 5 .

If x is 0, then ⏐x⏐ 5 . ⏐0⏐ 5

If x is a number, ⏐24⏐ 5 then ⏐x⏐5 2x. 5

a. If a 5 24.8, then 2a 5 5 .

b. If a 5 5 } 6 , then 2a 5 .

Example 4 Find opposites of numbers

a. If a 5 2 3 } 7 , then ⏐a⏐ 5 5 5 .

b. If a 5 2.9, then ⏐a⏐ 5 5 .

Example 5 Find absoute values of numbers



Your Notes

1. Tell whether each of the following numbers is a whole number, an integer, or a rational number:

0.8, 217, 25 3 } 4 , and 2. Then order the numbers from

least to greatest.

whole number: 2; integer: 217, 2; rational number:

0.8, 217, 25 3 } 4 , 2; order: 217, 25 3 } 4 , 0.8, 2


ABSOLUTE VALUE OF A NUMBER

Words Numbers

If x is a positive number, ⏐5⏐ 5 5 then ⏐x⏐ 5 x .

If x is 0, then ⏐x⏐ 5 0 . ⏐0⏐ 5 0

If x is a negative number, ⏐24⏐ 5 2(24) then ⏐x⏐5 2x. 5 4

a. If a 5 24.8, then 2a 5 2(24.8) 5 4.8 .

b. If a 5 5 } 6 , then 2a 5 2 5 } 6 .

Example 4 Find opposites of numbers

a. If a 5 2 3 } 7 , then ⏐a⏐ 5 ⏐2

3 } 7 ⏐ 5 2 1 2 3 } 7 2 5 3 } 7 .

b. If a 5 2.9, then ⏐a⏐ 5 ⏐2.9⏐ 5 2.9 .

Example 5 Find absoute values of numbers


Your Notes

5. If a number is negative, then the absolute value of the number is negative.

Checkpoint Identify the hypothesis and conclusion of the statement. Tell whether the statement is true or false. If it is false, give a counterexample.

Identify the hypothesis and the conclusion of the statement “If a number is an integer, then the number is positive.” Tell whether the statement is true or false. If it is false, give a counterexample.

Solution

Hypothesis:

Conclusion:

The statement is .

Example 6 Analyze a conditional statement

2. a 5 6 3. a 5 29.5 4. a 5 2 3 } 8

Checkpoint For the given value of a, find 2a and ⏐a⏐.

Homework



Your Notes

5. If a number is negative, then the absolute value of the number is negative.

Hypothesis: a number is negative; Conclusion: the absolute value of the number is negative; false; 23 is negative, but the absolute value of 23 is 3.

Checkpoint Identify the hypothesis and conclusion of the statement. Tell whether the statement is true or false. If it is false, give a counterexample.

Identify the hypothesis and the conclusion of the statement “If a number is an integer, then the number is positive.” Tell whether the statement is true or false. If it is false, give a counterexample.

Solution

Hypothesis: a number is an integer

Conclusion: the number is positive

The statement is false. For example, 25 is an integer, but it is not positive .

Example 6 Analyze a conditional statement

2. a 5 6 3. a 5 29.5 4. a 5 2 3 } 8

26; 6 9.5; 9.5 3 } 8 ; 3 }

8

Checkpoint For the given value of a, find 2a and ⏐a⏐.

Homework



Goal p Apply set theory to numbers and functions

Apply Sets to Numbers and Functions

VOCABULARY

Set

Element

Empty set

Universal set

Union

Intersection

Complement

Cross product

Your Notes

Let U be the set of integers from 0 to 7. Let A 5 {1, 2, 6} and B 5 {0, 2, 3, 6}. Find (a) A : B and (b) A " B.

Example 1 Find the union and intersection of two sets



a. The consists of the elements in

.

A BU

10

b. The consists of elements in

.

A BU

10

A ø B = {0, 1, , , } A ù B =


Goal p Apply set theory to numbers and functions

Apply Sets to Numbers and Functions

VOCABULARY

Set A collection of distinct objects

Element An object in a set

Empty set Set with no elements, [

Universal set Set of all elements under consideration, U

Union The union of two sets, A U B, is the set of all elements in either set A or set B.

Intersection The intersection of two sets, A ∩ B, is the set of all elements in both set A or set B.

Complement The complement of a set, A’, is all the elements in U that are not in A.

Cross product A 3 B is the set of all possible ordered pairs (x, y) where x is an element of A and y is an element of B.

Your Notes

Let U be the set of integers from 0 to 7. Let A 5 {1, 2, 6} and B 5 {0, 2, 3, 6}. Find (a) A : B and (b) A " B.

Example 1 Find the union and intersection of two sets



a. The union consists of the elements in either set.

A BU

12

4 5 7

6 30

b. The intersection consists of elements in both sets.

A BU

12

4 5 7

6 30

A ø B = {0, 1, 2, 3, 6} A ù B = {2, 6}


Your Notes

1. Let U be the set of integers from 0 to 6. Let A 5 (0, 3, 5) and B 5 (2, 4, 6). Find A : B, A " B, A', B', and A 3 B.


Example 2 Find complements and cross products of sets

Let U be the set of integers from 1 to 6. Let A 5 {2, 6} and B 5 {1, 3, 4, 6}. Find (a) A', (b) B', (c) A 3 B, and (d) A' 3 B'a. The complement of set B

consists of the elements in that are

A BU

13

B' 5

b. Note A and B are always ital. Note spaces after all commas:

A 3 B’ consists of all the with a

first coordinate from and a second coordinate from .

A 3 B’ 5 {}

Example 3 Write a function and its range as sets

Consider the function y 1 4x with domain D 5 {1,2,3}. Write the range and the function using set notation.

x 1 2

4( ) = 4 4( ) =

The range is R 5 {4, , }.f 5 {(1, 4), (2, ), (3, )


2. Consider the function y 5 x 2 3 with domain D 5 {0, 2, 4}. Write the range and the function using set notation.

Checkpoint Complete the following exercise.Homework


Your Notes

1. Let U be the set of integers from 0 to 6. Let A 5 (0, 3, 5) and B 5 (2, 4, 6). Find A : B, A " B, A', B', and A 3 B.

A : B 5 {0, 2, 3, 4, 5, 6,}, A " B 5 \, A' 5 {1, 2, 4, 6}, B' 5 {0, 1, 3, 5},

A 3 B 5 {(0, 2), (0, 4), (0, 6), (3, 2), (3, 4), (3, 6), (5, 2), (5, 4), (5, 6)}


Example 2 Find complements and cross products of sets

Let U be the set of integers from 1 to 6. Let A 5 {2, 6} and B 5 {1, 3, 4, 6}. Find (a) A', (b) B', (c) A 3 B, and (d) A' 3 B'a. The complement of set B

consists of the elements in U that are not in B.

A BU

13

4

26

5

B' 5 {2,5}

b. Note A and B are always ital. Note spaces after all commas:

A 3 B’ consists of all the ordered pairs with a first coordinate from A and a second coordinate from B’.

A 3 B’ 5 {(2, 2), (2, 5), (6, 2), (6, 5)}

Example 3 Write a function and its range as sets

Consider the function y 1 4x with domain D 5 {1,2,3}. Write the range and the function using set notation.

x 1 2 3

y 4(1) = 4 4(2) = 8 4(3) =12The range is R 5 {4, 8, 12}.f 5 {(1, 4), (2, 8), (3, 12)


2. Consider the function y 5 x 2 3 with domain D 5 {0, 2, 4}. Write the range and the function using set notation.

R 5 {23, 21, 1}, f 5 {(0, 23), (2, 21), (4, 1)}

Checkpoint Complete the following exercise.Homework



Goal p Add positive and negative numbers.

Add Real Numbers2.2

VOCABULARY

Additive identity

Additive inverse

Your Notes

Use the number line to find the sum.

a. 25 1 7

28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8

Start at .

To add, move units to the .

End at .

Answer: 25 1 7 5 .

b. 23 1 (24)

28 27 26 25 24 23 22 21 0 1 2 3 4

Start at .

To add, move units to the .

End at .

Answer: 23 1 (24) 5 .

Example 1 Add two integers using a number line

Remember: To add a positive number, move to the right on a number line. To add a negative number, move to the left.



Goal p Add positive and negative numbers.

Add Real Numbers2.2

VOCABULARY

Additive identity The number 0 in the identity property; the sum of a number a and 0 is a.

Additive inverse The opposite of a as stated in the inverse property; the sum of a number a and its opposite is 0.

Your Notes

Use the number line to find the sum.

a. 25 1 7

|7| units right

28 27 26 25 24 23 22 21 0 1 2 3 4 5 6 7 8

Start at 25 .

To add, move ⏐7⏐ units to the right .

End at 2 .

Answer: 25 1 7 5 2 .

b. 23 1 (24)

|24| units left

28 27 26 25 24 23 22 21 0 1 2 3 4

Start at 23 .

To add, move ⏐24⏐ units to the left .

End at 27 .

Answer: 23 1 (24) 5 27 .

Example 1 Add two integers using a number line

Remember: To add a positive number, move to the right on a number line. To add a negative number, move to the left.


RULES OF ADDITION

To add two numbers with the same sign:

1. Add their .

2. The sum has the as the numbers added.

Example: 25 1 (27) 5

To add two numbers with different signs:

1. Subtract the absolute value.

2. The sum has the as the number with the absolute value.

Example: 210 1 4 5

Find the sum.

a. 22.5 1 (24.2) 5 2( 1 ) Rule of same signs

5 2( 1 ) Take absolute values.

5 Add.

b. 10.5 1 (215.0) 5 2 Rule of different signs

5 2 Take absolute values.

5 Subtract and take sign from greater absolute value.

Example 2 Add real numbers

Your Notes

1. 27 1 (23) 2. 9.6 1 (22.1)

Checkpoint Find the sum.



RULES OF ADDITION

To add two numbers with the same sign:

1. Add their absolute values .

2. The sum has the same sign as the numbers added.

Example: 25 1 (27) 5 ]12

To add two numbers with different signs:

1. Subtract the lesser absolute value.

2. The sum has the same sign as the number with the greater absolute value.

Example: 210 1 4 5 ]6

Find the sum.

a. 22.5 1 (24.2) 5 2(⏐22.5⏐ 1 ⏐24.2⏐) Rule of same signs

5 2( 2.5 1 4.2 ) Take absolute values.

5 26.7 Add.

b. 10.5 1 (215.0) 5 ⏐215.0⏐ 2 ⏐10.5⏐ Rule of different signs

5 15.0 2 10.5 Take absolute values.

5 24.5 Subtract and take sign from greater absolute value.

Example 2 Add real numbers

Your Notes

1. 27 1 (23) 2. 9.6 1 (22.1)

]10 7.5

Checkpoint Find the sum.




Homework

Your Notes PROPERTIES OF ADDITION

Commutative Property The order in which you add two numbers does not change the sum.

a 1 b 5 1

Example: 21 1 3 5 1

Associative Property The way you group three numbers in a sum does not change the sum.

(a 1 b) 1 c 5 1 ( 1 )

Example: (1 1 2) 1 3 5 1 ( 1 )

Identity Property The sum of a number and 0 is the number.

a 1 0 5 1 5

Example: 4 1 0 5

Inverse Property The sum of a number and its opposite is 0.

a 1 (2a) 5 1 5

Example: 29 1 5 0

Identify the property illustrated by the statement.

Statement Property Illustrated

a. x 1 5 5 5 1 x of addition

b. y 1 0 5 y of addition

Example 3 Identify properties of addition

3. 25 1 5 5 0

4. (25 1 2) 1 3 5 25 1 (2 1 3)

Checkpoint Identify the property being illustrated.



Homework

Your Notes PROPERTIES OF ADDITION

Commutative Property The order in which you add two numbers does not change the sum.

a 1 b 5 b 1 a

Example: 21 1 3 5 3 1 (]1)

Associative Property The way you group three numbers in a sum does not change the sum.

(a 1 b) 1 c 5 a 1 ( b 1 c )

Example: (1 1 2) 1 3 5 1 1 ( 2 1 3 )

Identity Property The sum of a number and 0 is the number.

a 1 0 5 0 1 a 5 a

Example: 4 1 0 5 4

Inverse Property The sum of a number and its opposite is 0.

a 1 (2a) 5 2a 1 a 5 0

Example: 29 1 9 5 0

Identify the property illustrated by the statement.

Statement Property Illustrated

a. x 1 5 5 5 1 x Commutative property of addition

b. y 1 0 5 y Identity property of addition

Example 3 Identify properties of addition

3. 25 1 5 5 0

Inverse property of addition

4. (25 1 2) 1 3 5 25 1 (2 1 3)

Associative property of addition

Checkpoint Identify the property being illustrated.


2.3 Subtract Real Numbers Goal p Subtract real numbers.

Your Notes

1. 24 2 8 2. 9 2 18

Checkpoint Find the difference.

Find the difference.

a. 210 2 4 5 210 1 5

b. 13 2 (211) 5 13 1 5

Example 1 Subtract real numbers

SUBTRACTION RULE

Words: To subtract b from a, add the of b to a.

Algebra: a 2 b 5 1

Numbers: 15 2 7 5 1

Evaluate the expression a 2 b 1 5.3 when a 5 6.5 and b 5 23.

Solution

a 2 b 1 5.3 5 2 1 5.3 Substitute values.

5 1 1 5.3 Add the opposite of .

5 Add.

Example 2 Evaluate a variable expression



2.3 Subtract Real Numbers Goal p Subtract real numbers.

Your Notes

1. 24 2 8 2. 9 2 18

212 29

Checkpoint Find the difference.

Find the difference.

a. 210 2 4 5 210 1 (24) 5 214

b. 13 2 (211) 5 13 1 11 5 24

Example 1 Subtract real numbers

SUBTRACTION RULE

Words: To subtract b from a, add the opposite of b to a.

Algebra: a 2 b 5 a 1 (2b)

Numbers: 15 2 7 5 15 1 (27)

Evaluate the expression a 2 b 1 5.3 when a 5 6.5 and b 5 23.

Solution

a 2 b 1 5.3 5 6.5 2 (23) 1 5.3 Substitute values.

5 6.5 1 3 1 5.3 Add the opposite of 23 .

5 14.8 Add.

Example 2 Evaluate a variable expression




Homework

Your Notes

3. m 2 t 1 2 4. (m 2 3) 2 t

Checkpoint Evaluate the expression when m 5 3.2 and t 5 24.

Hiking Trail A sign at the start of a hiking trail states you are 320 feet below sea level. At the end of the trail another sign states you are 880 feet above sea level. Find the change in elevation of the trail.

SolutionStep 1 Write a verbal model of the situation.

Change in elevation 5

Elevation at

of trail 2

Elevation at

of trail

Step 2 Find the change in elevation.

Change in elevation 5 2 Substitute values.

5 1 Add the opposite of .

5 Add and .

The change in elevation is feet.

Example 3 Evaluate change

5. In the morning, the temperature was 238F. In the afternoon, the temperature was 218F. What was the change in temperature?




Homework

Your Notes

3. m 2 t 1 2 4. (m 2 3) 2 t

9.2 4.2

Checkpoint Evaluate the expression when m 5 3.2 and t 5 24.

Hiking Trail A sign at the start of a hiking trail states you are 320 feet below sea level. At the end of the trail another sign states you are 880 feet above sea level. Find the change in elevation of the trail.

SolutionStep 1 Write a verbal model of the situation.

Change in elevation 5

Elevation at

end of trail 2

Elevation at

start of trail

Step 2 Find the change in elevation.

Change in elevation 5 880 2 (2320) Substitute values.

5 880 1 320 Add the opposite of 2320 .

5 1200 Add 880 and 320 .

The change in elevation is 1200 feet.

Example 3 Evaluate change

5. In the morning, the temperature was 238F. In the afternoon, the temperature was 218F. What was the change in temperature?

248F



2.4 Multiply Real Numbers Goal p Multiply real numbers.

VOCABULARY

Multiplicative identity Your Notes

THE SIGN OF A PRODUCT

The product of two real numbers with the same sign is .

Examples: 5(2) 5

24(25) 5

The product of two real numbers with different signs is .

Examples: 5(23) 5

28(4) 5

Find the product.

Solution

a. 27(23) 5 Same signs: product is .

b. 3(4)(22) 5 (22) Multiply 3 and 4.

5 Different signs: product is .

c. 1 } 4 (216)(23) 5 (23) Multiply 1 } 4 and 216.

5 Same signs: product is .

Example 1 Multiply real numbers



2.4 Multiply Real Numbers Goal p Multiply real numbers.

VOCABULARY

Multiplicative identity The number 1; The identity property states the product of a and 1 is a.

Your Notes

THE SIGN OF A PRODUCT

The product of two real numbers with the same sign is positive .

Examples: 5(2) 5 10

24(25) 5 20

The product of two real numbers with different signs is negative .

Examples: 5(23) 5 215

28(4) 5 232

Find the product.

Solution

a. 27(23) 5 21 Same signs: product is positive .

b. 3(4)(22) 5 12 (22) Multiply 3 and 4.

5 224 Different signs: product is negative .

c. 1 } 4 (216)(23) 5 24 (23) Multiply 1 } 4 and 216.

5 12 Same signs: product is positive .

Example 1 Multiply real numbers




Your Notes

1. 24(26) 2. 23(22)(27)

Checkpoint Find the product.

PROPERTIES OF MULTIPLICATION

Commutative Property The order in which two numbers are multiplied does not change the product.

a p b 5 p

Example: 3 p 4 5 p

Associative Property The way you group three numbers when multiplying does not change the product.

(a p b) p c 5 p ( p )

Example: (2 p 3) p 4 5 p ( p )

Identity Property The product of a number and 1 is that number.

a p 1 5 p 5

Example: (22) p 1 5

Property of Zero The product of a number and 0 is 0.

a p 0 5 p 5

Example: 4 p 5 0

Property of 21 The product of a number and 21 is the opposite of the number.

a p (21) 5 p 5

Example: 25 p (21) 5



Your Notes

1. 24(26) 2. 23(22)(27)

24 242

Checkpoint Find the product.

PROPERTIES OF MULTIPLICATION

Commutative Property The order in which two numbers are multiplied does not change the product.

a p b 5 b p a

Example: 3 p 4 5 4 p 3

Associative Property The way you group three numbers when multiplying does not change the product.

(a p b) p c 5 a p ( b p c )

Example: (2 p 3) p 4 5 2 p ( 3 p 4 )

Identity Property The product of a number and 1 is that number.

a p 1 5 1 p a 5 a

Example: (22) p 1 5 22

Property of Zero The product of a number and 0 is 0.

a p 0 5 0 p a 5 0

Example: 4 p 0 5 0

Property of 21 The product of a number and 21 is the opposite of the number.

a p (21) 5 21 p a 5 2a

Example: 25 p (21) 5 5


Your Notes

Identify the property illustrated by each expression.

SolutionStatement Property Illustrated

a. 3 p 0 5 0

b. t p 1 5 t of multiplication

c. a p 3 5 3 p a of multiplication

d. n p (3 p 5) 5 (n p 3) p 5 of multiplication

e. 27(21) 5 7

Example 2 Identify properties of multiplication

3. 24 p 0 5 0

4. 6 p 2 5 2 p 6

5. (4 p 5) p 6 5 4 p (5 p 6)

6. 4 p (21) 5 24

Checkpoint Identify the property illustrated.



Your Notes

Identify the property illustrated by each expression.

SolutionStatement Property Illustrated

a. 3 p 0 5 0 Multiplicative property of zero

b. t p 1 5 t Identity property of multiplication

c. a p 3 5 3 p a Commutative property of multiplication

d. n p (3 p 5) 5 (n p 3) p 5 Associative property of multiplication

e. 27(21) 5 7 Multiplicative property of 21

Example 2 Identify properties of multiplication

3. 24 p 0 5 0

Multiplicative property of zero

4. 6 p 2 5 2 p 6

Commutative property of multiplication

5. (4 p 5) p 6 5 4 p (5 p 6)

Associative property of multiplication

6. 4 p (21) 5 24

Multiplicative property of ]1

Checkpoint Identify the property illustrated.




Homework

Your Notes

7. 2 1 } 2 (2)(3y)

8. (22)(a)(25)

Checkpoint Find the product. Justify your steps.

Find the product (0.5)(22x)(6). Justify your steps.

Solution

(0.5)(22x)(6) 5 (22x)(0.5)(6)

5 (22x)(0.5 p 6)

5 (22x)(3)

5 3 p (22x)

5 [3 p (22)]x

5 26x

Example 3 Use properties of multiplication



Homework

Your Notes

7. 2 1 } 2 (2)(3y)

2 1 } 2 (2)(3y) 5 (21)(3y) Product of 2 1 }

2 and 2 is

21.

5 [(21) p 3]y Associative property of multiplication

5 23y Multiplicative property of 21

8. (22)(a)(25)

(22)(a)(25) 5 (a)(22)(25) Commutative property of multiplication

5 a p 10 Product of 22 and 25 is 10.

5 10a Commutative property of multiplication

Checkpoint Find the product. Justify your steps.

Find the product (0.5)(22x)(6). Justify your steps.

Solution

(0.5)(22x)(6) 5 (22x)(0.5)(6) Commutative property of multiplication

5 (22x)(0.5 p 6) Associative property of multiplication

5 (22x)(3) Product of 0.5 and 6 is 3.

5 3 p (22x) Commutative property multiplication

5 [3 p (22)]x Associative property of multiplication

5 26x Product of 3 and 22 is 26.

Example 3 Use properties of multiplication


Apply the Distributive Property2.5 Goal p Apply the distributive property.

VOCABULARY

Equivalent expressions

Distributive property

Terms

Coefficient

Constant term

Like terms

THE DISTRIBUTIVE PROPERTY

Let a, b, and c be real numbers.

Algebra Numbers

a(b 1 c) 5 ab 1 4(2 1 3) 5 1

(b 1 c)a 5 ba 1 (3 1 5)2 5 1

a(b 2 c) 5 ab 2 7(5 2 3) 5 2

(b 2 c)a 5 ba 2 (6 2 4)9 5 2

Your Notes



Apply the Distributive Property2.5 Goal p Apply the distributive property.

VOCABULARY

Equivalent expressions Two expressions that have the same value for all values of the variable

Distributive property A property used to find the product of a number and a sum or difference

Terms The parts of an expression that are added together

Coefficient The number part of a term with a variable part

Constant term A term that has a number but no variable part

Like terms Terms that have the same variable parts

THE DISTRIBUTIVE PROPERTY

Let a, b, and c be real numbers.

Algebra Numbers

a(b 1 c) 5 ab 1 ac 4(2 1 3) 5 4(2) 1 4(3)

(b 1 c)a 5 ba 1 ca (3 1 5)2 5 3(2) 1 5(2)

a(b 2 c) 5 ab 2 ac 7(5 2 3) 5 7(5) 2 7(3)

(b 2 c)a 5 ba 2 ca (6 2 4)9 5 6(9) 2 4(9)

Your Notes



Use the distributive property to write an equivalent equation.

Solution

a. 4(a 1 3) 5

b. (a 1 5)6 5

c. 3(x 2 8) 5

d. (4 2 x)(x) 5

Example 1 Apply the distributive property


Solutiona. 23(7 1 x)

5 (7) 1 (x) Distribute .

5 .

b. (6 2 a)(22a)

5 6( ) 2 a( ) Distribute .

5 .

Example 2 Distribute a negative number

Use the distributive property to combine like terms with variable parts. Your expression is simplified if there are no grouping symbols and all like terms are combined.


Your Notes

Checkpoint Use the distributive property to write an equivalent equation.

1. 5(n 1 4) 2. 2a(3 1 a)

Example 3

Be sure to distribute the factor outside of the parentheses to all of the numbers inside the parentheses.



Solution

a. 4(a 1 3) 5 4a 1 12

b. (a 1 5)6 5 6a 1 30

c. 3(x 2 8) 5 3x 2 24

d. (4 2 x)(x) 5 4x 2 x2

Example 1 Apply the distributive property


Solutiona. 23(7 1 x)

5 23 (7) 1 (23) (x) Distribute 23 .

5 221 2 3x Simplify .

b. (6 2 a)(22a)

5 6( 22a ) 2 a( 22a ) Distribute 22a .

5 212a 1 2a2 Simplify .

Example 2 Distribute a negative number

Use the distributive property to combine like terms with variable parts. Your expression is simplified if there are no grouping symbols and all like terms are combined.


Your Notes

Checkpoint Use the distributive property to write an equivalent equation.

1. 5(n 1 4) 2. 2a(3 1 a)

5n 1 20 23a 2 a2

Example 3

Be sure to distribute the factor outside of the parentheses to all of the numbers inside the parentheses.


Homework

Your Notes

Identify the terms, like terms, coefficients, and constant terms of the expression 2x 2 5 1 8x 2 3.

Solution

Write the expression as a sum.

Terms: Like terms:

Coefficients: Constant terms:

Example 3 Identify parts of an expression

3. 10 1 3a 2 4 2 6a

4. 7y 2 11 2 4y 2 1

Checkpoint Identify the terms, like terms, coefficients, and constant terms of the expressions.



Homework

Your Notes

Identify the terms, like terms, coefficients, and constant terms of the expression 2x 2 5 1 8x 2 3.

Solution

2x 1 (25) 1 8x 1 (23) Write the expression as a sum.

Terms: Like terms:

2x, 25, 8x, 23 2x, 8x; 25, 23

Coefficients: Constant terms:

2, 8 25, 23

Example 3 Identify parts of an expression

3. 10 1 3a 2 4 2 6a

Terms: 10, 3a, 24, 26a

Like terms: 10, 24; 3a, 26a

Coefficients: 3, 26

Constant terms: 10, 24

4. 7y 2 11 2 4y 2 1

Terms: 7y, 211, 24y, 21

Like terms: 7y, 24y; 211, 21

Coefficients: 7, 24

Constant terms: 211, 21

Checkpoint Identify the terms, like terms, coefficients, and constant terms of the expressions.



2.6 Goal p Divide real numbers.

Divide Real Numbers


VOCABULARY

Multiplicative inverse

Your Notes

INVERSE PROPERTY OF MULTIPLICATION

Words

The of a nonzero number and its multiplicative inverse is .

Algebra

a p 1 } a 5 , a Þ

Numbers

4 p 1 } 4 5

Identify the multiplicative inverse and justify your answer.

Solution Number Multiplicative Reason

inverse

a. 9

b. 2 5 } 6

Example 1 Find multiplicative inverses of numbers


2.6 Goal p Divide real numbers.

Divide Real Numbers


VOCABULARY

Multiplicative inverse The reciprocal of a nonzero

number a, written 1 } a ; 1 } a is the multiplicative inverse

of a.

Your Notes

INVERSE PROPERTY OF MULTIPLICATION

Words

The product of a nonzero number and its multiplicative inverse is 1 .

Algebra

a p 1 } a 5 1 , a Þ 0

Numbers

4 p 1 } 4 5 1

Identify the multiplicative inverse and justify your answer.

Solution Number Multiplicative Reason

inverse

a. 9 1 } 9 9 p 1 }

9 5 1

b. 2 5 } 6 2 6 } 5 2

5 } 6 p 1 2 6 } 5 2 5 1

Example 1 Find multiplicative inverses of numbers


Your Notes

1. 2 2 } 3 2. 3

Checkpoint Find the multiplicative inverse.

DIVISION RULE

Words

To divide a number a by a nonzero number b, multiply by the multiplicative inverse of .

Algebra

a 4 b 5 a p , b Þ

Numbers

7 4 3 5

THE SIGN OF A QUOTIENT

The quotient of two real numbers with the same sign is .

The quotient of two real numbers with different signs is .

The quotient of 0 and any nonzero real number is .

You cannot divide a real number by 0, because 0 does not have a multiplicative inverse.

Find the quotient.

Solution

a. 25 4 5 5 25 p 5

b. 240 4 2 } 3 5 240 p 5

Example 2 Divide real numbers



Your Notes

1. 2 2 } 3 2. 3

2 3 } 2 1 }

3

Checkpoint Find the multiplicative inverse.

DIVISION RULE

Words

To divide a number a by a nonzero number b, multiply a by the multiplicative inverse of b .

Algebra

a 4 b 5 a p 1 } b , b Þ 0

Numbers

7 4 3 5 7 p 1 } 3

THE SIGN OF A QUOTIENT

The quotient of two real numbers with the same sign is positive .

The quotient of two real numbers with different signs is negative .

The quotient of 0 and any nonzero real number is 0 .

You cannot divide a real number by 0, because 0 does not have a multiplicative inverse.

Find the quotient.

Solution

a. 25 4 5 5 25 p 1 } 5 5 5

b. 240 4 2 } 3 5 240 p 3 } 2 5 260

Example 2 Divide real numbers




Homework

Your Notes

3. 1 } 2 4 3 } 4 4. 16 4 1 2

1 } 4 2

Checkpoint Find the quotient.

Simplify the expression 48y 2 32

} 8 .

Solution

48y 2 32

} 8 5 (48y 2 32) 4 Rewrite fraction as

division.

5 (48y 2 32) p Division rule

5 48y p 2 32 p Distributive property

5 Simplify.

Example 3 Simplify an expression

5. 3a 1 4 } 2 6. 12x 2 8 } 4

Checkpoint Simplify the expression.



Homework

Your Notes

3. 1 } 2 4 3 } 4 4. 16 4 1 2

1 } 4 2

2 } 3 264

Checkpoint Find the quotient.

Simplify the expression 48y 2 32

} 8 .

Solution

48y 2 32

} 8 5 (48y 2 32) 4 8 Rewrite fraction as

division.

5 (48y 2 32) p 1 } 8 Division rule

5 48y p 1 } 8 2 32 p 1 } 8 Distributive property

5 6y 2 4 Simplify.

Example 3 Simplify an expression

5. 3a 1 4 } 2 6. 12x 2 8 } 4

3 } 2 a 1 2 3x 2 2

Checkpoint Simplify the expression.


2.7 Find Square Roots and Compare Real Numbers Goal p Find square roots and compare real numbers.

VOCABULARY

Square root

Radicand

Perfect square

Irrational number

Real number

SQUARE ROOT OF A NUMBER

Words

If b2 5 a, then is a square root of .

Numbers

52 5 25 and (25)2 5 25, so and are square roots of 25.


Your Notes


2.7 Find Square Roots and Compare Real Numbers Goal p Find square roots and compare real numbers.

VOCABULARY

Square root If b2 5 a, then b is a square root of a. Square roots are written with the radical symbol Ï

}

.

Radicand The number or expression inside the radical symbol

Perfect square A number whose square roots are integers

Irrational number A number that cannot be written as the quotient of two integers

Real number The set of all rational and irrational numbers

SQUARE ROOT OF A NUMBER

Words

If b2 5 a, then b is a square root of a .

Numbers

52 5 25 and (25)2 5 25, so 5 and 25 are square roots of 25.


Your Notes



Your Notes


Solution

a. 2 Ï}

36 5 The negative square root of 36 is .

b. Ï}

16 5 The positive square root of 16 is .

c. 6 Ï}

64 5 The positive and negative square roots of 64 are and .

Example 1 Find square roots

All positive real numbers have two square roots, a positive and a negative square root. The positive square root is called the principal square root.

1. Ï}

100 2. 2 Ï}

1


Tell whether each number is a real number, a rational number, an irrational number, an integer, or a whole number: Ï

}

144 , 2 Ï}

49 , Ï}

32 .

Solution

Number Real Number ?

Rational Number?

Irrational Number ?

Integer ? Whole Number ?

Ï}

144

2 Ï}

49

Ï}

32




Your Notes


Solution

a. 2 Ï}

36 5 26 The negative square root of 36 is 26 .

b. Ï}

16 5 4 The positive square root of 16 is 4 .

c. 6 Ï}

64 5 68 The positive and negative square roots of 64 are 8 and 28 .

Example 1 Find square roots

All positive real numbers have two square roots, a positive and a negative square root. The positive square root is called the principal square root.

1. Ï}

100 2. 2 Ï}

1

10 ]1


Tell whether each number is a real number, a rational number, an irrational number, an integer, or a whole number: Ï

}

144 , 2 Ï}

49 , Ï}

32 .

Solution

Number Real Number ?

Rational Number?

Irrational Number ?

Integer ? Whole Number ?

Ï}

144 Yes Yes No Yes Yes

2 Ï}

49 Yes Yes No Yes No

Ï}

32 Yes No Yes No No



Homework

Your Notes

3. Tell whether each number is a real number, rational number, irrational number, integer,

or whole number: Ï}

49 , 0, 2 6 } 4 , 22, Ï}

17 .

4. Compare Ï}

3 and 5 } 2 .

43210

Checkpoint Complete the following exercises.


Example 3 Graph and compare real numbers

Compare 2 Ï}

6 and 23.

SolutionGraph the numbers on a number line.

4 0123

Because 2 Ï}

6 is the of 23,

2 Ï}

6 ?


Homework

Your Notes

3. Tell whether each number is a real number, rational number, irrational number, integer,

or whole number: Ï}

49 , 0, 2 6 } 4 , 22, Ï}

17 .

Real number: Ï}

49 , 0, 2 6 } 4 , 22, Ï

}

17

Rational number: Ï}

49 , 0, 2 6 } 4 , 22

Irrational number: Ï}

17

Integer: Ï}

49 , 0, 22

Whole number: Ï}

49 , 0

4. Compare Ï}

3 and 5 } 2 .

1.73 52

3

43210

Ï}

3 , 5 } 2




Compare 2 Ï}

6 and 23.

SolutionGraph the numbers on a number line.

3 2.456

4 0123

Because 2 Ï}

6 is the right of 23,

2 Ï}

6 . 23

?


Goal p Find, compare, and order cube roots.

Find Cube Roots

VOCABULARY

Cube root Your Notes


a. 3

Î}

125 5 The cube root of 125 is .

b. 3

Î}

2216 5 The cube root of 2216 is .

Example 1 Find cube roots

1. 3

Î}

28

2. 3

Î}

8000


A cube-shaped building has a volume of 50,000 cubic feet. Approximate the length of each side of the building to the nearest foot.

Solution

Find the length s such that 5 50,000. Use a table.

Number 34 35 36 37

Cube of number 39,304

From the table, you can see that 50,000 is between 363 5 and 373 5 , so , s , . The average of 36 and 37 is , and 5 48,627.125.

Because 50,000 . 48,627.125, S 5 is closer to than to . The length of each edge of the building is about feet.

Example 1 Approximate a cube root

Copyright © Holt McDougal. All rights reserved. 2.7 Focus on Operations • Algebra 1 Notetaking Guide 51

Focus On OperationsUse after Lesson 2.7


Goal p Find, compare, and order cube roots.

Find Cube Roots

VOCABULARY

Cube root If b3 5 a then b is the cube root of a.Your Notes


a. 3

Î}

125 5 5 The cube root of 125 is 5.

b. 3

Î}

2216 5 26 The cube root of 2216 is 26.

Example 1 Find cube roots

1. 3

Î}

28

22

2. 3

Î}

8000

20


A cube-shaped building has a volume of 50,000 cubic feet. Approximate the length of each side of the building to the nearest foot.

Solution

Find the length s such that s3 5 50,000. Use a table.

Number 34 35 36 37

Cube of number 39,304 42,875 46,656 50,653

From the table, you can see that 50,000 is between 363 5 46,656 and 373 5 50,653 , so 36 , s , 37 . The average of 36 and 37 is 36.5 , and 36.53 5 48,627.125.

Because 50,000 . 48,627.125, S 5 3 Î} 50,000 is closer

to 37 than to 36 . The length of each edge of the building is about 37 feet.

Example 1 Approximate a cube root

Copyright © Holt McDougal. All rights reserved. 2.7 Focus on Operations • Algebra 1 Notetaking Guide 51



3. An ice cube has a volume of 25,000 cubic millimeters. Approximate the length of each side of the ice cube to the nearest millimeter. Assume the ice cube is a perfect cube.


Copy and complete the statement using < or >.

a. 2 7 } 3 ? 3

Î}

250 b. 3

Î}

81 ? Ï}

25

Solution

Graph each pair of numbers on a number line.


You can use a calculator to approximate a cube root.

4. Ï}

7 ? 3

Î}

42

5. 7 } 2 ? 3

Î}

30

Checkpoint Copy and complete the statement using < or >.

52 2.7 Focus on Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Homework

Your Notes

a.

503 3.68 2.33

4 0123

73

Because is to the

of

b.

54

25813 4.33

Because 3

Î}

81 is to the

of Ï}

25, .

LAH_A1_11_FL_NTG_Ch02_27-54.indd 52LAH_A1_11_FL_NTG_Ch02_27-54.indd 52 12/16/09 6:49:36 AM12/16/09 6:49:36 AM

3. An ice cube has a volume of 25,000 cubic millimeters. Approximate the length of each side of the ice cube to the nearest millimeter. Assume the ice cube is a perfect cube.

29 millimeters


Copy and complete the statement using < or >.

a. 2 7 } 3 ? 3

Î}

250 b. 3

Î}

81 ? Ï}

25

Solution

Graph each pair of numbers on a number line.


You can use a calculator to approximate a cube root.

4. Ï}

7 ? 3

Î}

42

Ï}

7 < 3

Î}

42

5. 7 } 2 ? 3

Î}

30

7 } 2 >

3

Î}

30

Checkpoint Copy and complete the statement using < or >.

52 2.7 Focus on Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Homework

Your Notes

a.

503 3.68 2.33

4 0123

73

Because 2 7 } 3 is to the

right of 3 Î}

250 ,

2 7 } 3 >

3

Î}

250 .

b.

54

25813 4.33

Because 3

Î}

81 is to the

left of Ï}

25, 3

Î}

81 < Ï}

25 .


Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 53


Whole number

Rational number

Absolute Value

Set

Empty set

Union

Additive identity/Additive inverse

Integer

Opposite

Conditional Statement

Element

Universal set

Intersection

Multiplicative identity




Whole number

0, 1, 2, 3, . . .

Rational number

3 } 4 , 2, 11 } 2

Absolute Value

⏐22⏐ 5 2

Set

{1, 2, 5, 6}

Empty set

The set of positive numbers less than 0

Union

If A 5 {1, 2, 7} and B 5 {2, 4, 7}, A U B 5 {1, 2, 4, 7}

Additive identity/Additive inverse

0; a 1 (2a) 5 0

Integer

. . ., 23, 22, 21, 0, 1,

2, 3, . . .

Opposite

7 and 27 are opposites.

Conditional Statement

If a number is a rational number, then it is a real number

Element

2 in the set {1, 2, 6}

Universal set

{1, 2, 3, 4, 5, 6} when rolling a number cube.

Intersection

If A 5 {1, 2, 7} and B 5 {2, 4, 7}, A ∩ B 5 {2, 7}

Multiplicative identity

1




Terms

Constant term


Radicand

Irrational number

Cube root


Coefficient

Like terms

Square root

Perfect square

Real number





4(x 1 3) 5 4x 1 12

Terms

5x 2 9 2 2x 1 4Terms: 5x, 29, 22x, 4

Constant term

5x 2 9 2 2x 1 4Constant terms: 29, 4


1 } a is the multiplicative

inverse of a.

Radicand

Ï}

36 Radicand

Irrational number

Ï}

10 , 2 Ï}

23

Cube root

3

Î}

27 5 3


2(x 1 1) 5 2(x) 1 2(1)

Coefficient

5x 2 9 2 2x 1 4Coefficients: 5, 22

Like terms

5x 2 9 2 2x 1 4Like terms: 5x, 22x; 29, 4

Square root

Ï}

36 5 6

Perfect square

The square of an integer.

Real number

2 1 } 5 , Ï

}

25 , 9, 0, 2 Ï}

3



VOCABULARY

Inverse operations

Equivalent equations

Goal p Solve one-step equations using algebra.


3.1

ADDITION PROPERTY OF EQUALITY

Words Adding the same number to each side of an equation produces an .

Algebra If x 2 a 5 b, then x 2 a 1 a 5 1 or x 5 1 .

SUBTRACTION PROPERTY OF EQUALITY

Words Subtracting the same number from each side of an equation produces an

.

Algebra If x 1 a 5 b, then x 1 a 2 a 5 2 or x 5 2 .

Solve One-Step Equations

Your Notes


VOCABULARY

Inverse operations Two operations that undo each other, such as addition and subtraction

Equivalent equations Equations that have the same solution(s)

Goal p Solve one-step equations using algebra.


3.1

ADDITION PROPERTY OF EQUALITY

Words Adding the same number to each side of an equation produces an equivalent equation .

Algebra If x 2 a 5 b, then x 2 a 1 a 5 b 1 a or x 5 b 1 a .

SUBTRACTION PROPERTY OF EQUALITY

Words Subtracting the same number from each side of an equation produces an equivalent equation .

Algebra If x 1 a 5 b, then x 1 a 2 a 5 b 2 a or x 5 b 2 a .

Solve One-Step Equations

Your Notes


Solve y 1 3 5 10.

Solution y 1 3 5 10 Write original equation.

y 1 3 2 5 10 2 Use subtraction property of equality: Subtract from each side.

y 5 Simplify.

The solution is .

CHECK

y 1 3 5 10 Write original equation.

1 3 0 10 Substitute for y.

5 10 ✓ Solution checks.

Example 1 Solve an equation using subtraction

Remember to check your solution in the original equation for accuracy.

Solve t 2 9 5 11.

Solution t 2 9 5 11 Write original equation.

t 2 9 1 5 11 1 Use addition property of equality: Add to each side.

t 5 Simplify.

The solution is .

CHECK

t 2 9 5 11 Write original equation.

2 9 0 11 Substitute for t.


Example 2 Solve an equation using addition


Your Notes


Solve y 1 3 5 10.

Solution y 1 3 5 10 Write original equation.

y 1 3 2 3 5 10 2 3 Use subtraction property of equality: Subtract 3 from each side.

y 5 7 Simplify.

The solution is 7 .

CHECK

y 1 3 5 10 Write original equation.

7 1 3 0 10 Substitute 7 for y.

10 5 10 ✓ Solution checks.

Example 1 Solve an equation using subtraction

Remember to check your solution in the original equation for accuracy.

Solve t 2 9 5 11.

Solution t 2 9 5 11 Write original equation.

t 2 9 1 9 5 11 1 9 Use addition property of equality: Add 9 to each side.

t 5 20 Simplify.

The solution is 20 .

CHECK

t 2 9 5 11 Write original equation.

20 2 9 0 11 Substitute 20 for t.


Example 2 Solve an equation using addition


Your Notes


MULTIPLICATION PROPERTY OF EQUALITY

Words Multiplying each side of an equation by the same non-zero number produces an

.

Algebra If x } a 5 b and a Þ 0, then a p x } a 5 p

or x 5 .

DIVISION PROPERTY OF EQUALITY

Words Dividing each side of an equation by the same non-zero number produces an

.

Algebra If ax 5 b, and a Þ 0, then ax } a 5 or x 5 .


1. a 1 6 5 17 2. b 2 17 5 12

3. 23 5 x 1 2 4. y 2 4 5 26

Checkpoint Solve each equation. Check your solution.Your Notes


MULTIPLICATION PROPERTY OF EQUALITY

Words Multiplying each side of an equation by the same non-zero number produces an equivalent equation .

Algebra If x } a 5 b and a Þ 0, then a p x } a 5 a p b

or x 5 ab .

DIVISION PROPERTY OF EQUALITY

Words Dividing each side of an equation by the same non-zero number produces an equivalent equation .

Algebra If ax 5 b, and a Þ 0, then ax } a 5 a

b or x 5

a

b.


1. a 1 6 5 17 2. b 2 17 5 12

a 5 11 b 5 29

3. 23 5 x 1 2 4. y 2 4 5 26

x 5 25 y 5 22

Checkpoint Solve each equation. Check your solution.Your Notes


Solve 8x 5 56.

Solution 8x 5 56 Write original equation.

5 Use division property of equality: Divide each side by .

x 5 Simplify.

The solution is .

CHECK

8x 5 56 Write original equation.

8( ) 0 56 Substitute for x.


Example 3 Solve an equation using division

8x 56

The division property of equality can be used to solve equations involving multiplication.

Solve a } 5 5 12.

Solution

a } 5 5 12 Write original equation.

p a } 5 5 p 12 Use multiplication property of equality: Multiply each side by .

a 5 Simplify.

The solution is .

CHECK


5 0 12 Substitute for a.


Example 4 Solve an equation using multiplication

The multiplication property of equality can be used to solve equations involving division.


Your Notes


Solve 8x 5 56.

Solution 8x 5 56 Write original equation.

5 Use division property of equality: Divide each side by 8 .

x 5 7 Simplify.

The solution is 7 .

CHECK

8x 5 56 Write original equation.

8( 7 ) 0 56 Substitute 7 for x.


Example 3 Solve an equation using division

8x

8

56

8

The division property of equality can be used to solve equations involving multiplication.

Solve a } 5 5 12.

Solution


5 p a } 5 5 5 p 12 Use multiplication property of equality: Multiply each side by 5 .

a 5 60 Simplify.


CHECK


5

60 0 12 Substitute 60 for a.


Example 4 Solve an equation using multiplication

The multiplication property of equality can be used to solve equations involving division.


Your Notes


Solve 3 } 5 t 5 6.

Solution

The coefficient of t is 3 } 5 . The reciprocal of 3 } 5 is .

3 } 5 t 5 6 Write original equation.

p 3 } 5 t 5 p 6 Multiply each side by the

reciprocal .

t 5 Simplify.

The solution is .

CHECK


3 } 5 ( ) 0 6 Substitute for t.


Example 5 Solve an equation by multiplying by a reciprocal

5. 3x 5 39 6. b } 4 5 13

7. 224 5 4x 8. 2 3 } 8 m 5 21

Checkpoint Solve each equation. Check your solution.


Homework

Your Notes


Solve 3 } 5 t 5 6.

Solution

The coefficient of t is 3 } 5 . The reciprocal of 3 } 5 is 5 } 3 .


1 5 } 3 2 p 3 } 5 t 5 1 5 } 3 2 p 6 Multiply each side by the

reciprocal 5 } 3 .

t 5 10 Simplify.


CHECK


3 } 5 ( 10 ) 0 6 Substitute 10 for t.


Example 5 Solve an equation by multiplying by a reciprocal

5. 3x 5 39 6. b } 4 5 13

x 5 13 b 5 52

7. 224 5 4x 8. 2 3 } 8 m 5 21

x 5 26 m 5 256

Checkpoint Solve each equation. Check your solution.


Homework

Your Notes


3.2 Solve Two-Step Equations Goal p Solve two-step equations.

IDENTIFYING OPERATIONS

Identify the operations involved in the equation 3x 1 7 5 19.

Operations performed on x Operations to isolate x

1. Multiply by .

2. Add .

1. Subtract .

2. Divide by .

Solve 3x 1 7 5 19.

Solution 3x 1 7 5 19 Write original equation.

3x 1 7 2 5 19 2 Subtract from eachside.

3x 5 Simplify.

5 Divide each side by .

x 5 Simplify.

The solution is .

CHECK

3x 1 7 5 19 Write original equation.

3( ) 1 7 0 19 Substitute for x.

1 7 0 19 Multiply 3 by .

5 19 ✓ Simplify. Solution checks.

Example 1 Solve a two-step equation

3x 12

When solving a two-step equation, apply the inverse operations in the reverse order of the order of operations.

Your Notes



3.2 Solve Two-Step Equations Goal p Solve two-step equations.

IDENTIFYING OPERATIONS

Identify the operations involved in the equation 3x 1 7 5 19.

Operations performed on x Operations to isolate x

1. Multiply by 3 .

2. Add 7 .

1. Subtract 7 .

2. Divide by 3 .

Solve 3x 1 7 5 19.

Solution 3x 1 7 5 19 Write original equation.

3x 1 7 2 7 5 19 2 7 Subtract 7 from eachside.

3x 5 12 Simplify.

5 Divide each side by 3 .

x 5 4 Simplify.

The solution is 4 .

CHECK

3x 1 7 5 19 Write original equation.

3( 4 ) 1 7 0 19 Substitute 4 for x.

12 1 7 0 19 Multiply 3 by 4 .

19 5 19 ✓ Simplify. Solution checks.

Example 1 Solve a two-step equation

3x

3

12

3

When solving a two-step equation, apply the inverse operations in the reverse order of the order of operations.

Your Notes




Your Notes

1. r } 4 2 12 5 25 2. 7k 2 14 5 42

Checkpoint Solve the two-step equation. Check your solution.

Solve 4a 1 3a 5 63.

Solution4a 1 3a 5 63 Write original equation.

5 63 Combine like terms.

5 63 Divide each side by .

a 5 Simplify.

The solution is .

CHECK

4a 1 3a 5 63 Write original equation.

4( ) 1 3( ) 0 63 Substitute for a.

1 0 63 Multiply 4 by and 3 by .

5 63 ✓ Add. Solution checks.

Example 2 Solve a two-step equation by combining like terms

3. 5z 1 4z 5 36 4. 5b 2 2b 5 9

Checkpoint Solve the equation. Check your solution.



Your Notes

1. r } 4 2 12 5 25 2. 7k 2 14 5 42

r 5 28 k 5 8

Checkpoint Solve the two-step equation. Check your solution.

Solve 4a 1 3a 5 63.

Solution4a 1 3a 5 63 Write original equation.

7a 5 63 Combine like terms.

7

7a 5 63


a 5 9 Simplify.

The solution is 9 .

CHECK

4a 1 3a 5 63 Write original equation.

4( 9 ) 1 3( 9 ) 0 63 Substitute 9 for a.

36 1 27 0 63 Multiply 4 by 9 and 3 by 9 .

63 5 63 ✓ Add. Solution checks.

Example 2 Solve a two-step equation by combining like terms

3. 5z 1 4z 5 36 4. 5b 2 2b 5 9

z 5 4 b 5 3



Homework

Your Notes

The output of a function is 2 more than 4 times the input. Find the input when the output is 14.

SolutionStep 1 Write an equation for the function. Let x be the

input and y be the output.

y 5 y is 2 more than 4 times x.

Step 2 Solve the equation when y 5 14.

y 5 Write original function.

5 Substitute for y.

5 Subtract from each side.

5 Simplify.


5 x Simplify.

An input of produces an output of .

CHECK

y 5 Write original function.

0 Substitute for y and for x.

0 Multiply and .

5 ✓ Simplify. Solution checks.

Example 3 Find an input of a function

5. The output of a function is 3 less than 6 times the input. Find the input when the output is 15.




Homework

Your Notes

The output of a function is 2 more than 4 times the input. Find the input when the output is 14.

SolutionStep 1 Write an equation for the function. Let x be the

input and y be the output.

y 5 4x 1 2 y is 2 more than 4 times x.

Step 2 Solve the equation when y 5 14.

y 5 4x 1 2 Write original function.

14 5 4x 1 2 Substitute 14 for y.

14 2 2 5 4x 1 2 2 2 Subtract 2 from each side.

12 5 4x Simplify.


3 5 x Simplify.

An input of 3 produces an output of 14 .

CHECK

y 5 4x 1 2 Write original function.

14 0 4(3) 1 2 Substitute 14 for y and 3 for x.

14 0 12 1 2 Multiply 4 and 3 .

14 5 14 ✓ Simplify. Solution checks.

Example 3 Find an input of a function

4

12

4

4x

5. The output of a function is 3 less than 6 times the input. Find the input when the output is 15.

x 5 3




Example 1 Use a real-world context

Check Responableness of Solutions Goal p Check that the solutions of a real-world problem is

reasonable.

An orchestra has 163 musicians. The orchestra rehearses with 120 musicians on stage. The remaining musicians rehearse in practice rooms that hold up to 8 people each. How many practice rooms are needed?

Solution

163 5 120 1 r Write orginal equation.

43 5 r Subtract from each side.

} 5 r each side by .

At least practice rooms are needed because performers would not have a place to rehearseif only practice rooms were available.


You have $60 to spend on party decorations. You spend $7.99 on streamers. With your remaining money, you buy paper lanterns at $3.99 each. How many can you buy?

Solution

7.99 1 5 60 Write original equation.

5

from each side.

L ø each side by .

You can afford lanterns, but it is not possible to

buy part of a lantern. Because 7.99 1 ( )

5 , you cannot afford to buy lanterns.

You can afford to buy lanterns because

7.99 1 ( ) 5 .

Your Notes





Check Responableness of Solutions Goal p Check that the solutions of a real-world problem is

reasonable.

An orchestra has 163 musicians. The orchestra rehearses with 120 musicians on stage. The remaining musicians rehearse in practice rooms that hold up to 8 people each. How many practice rooms are needed?

Solution

163 5 120 1 8 r Write orginal equation.

43 5 8 r Subtract 120 from each side.

5 3 }

8 5 r Divide each side by 8 .

At least 6 practice rooms are needed because 3 performers would not have a place to rehearseif only 5 practice rooms were available.


You have $60 to spend on party decorations. You spend $7.99 on streamers. With your remaining money, you buy paper lanterns at $3.99 each. How many can you buy?

Solution

7.99 1 3.99L 5 60 Write original equation.

3.99L 5 52.01 Subract 7.99 from each side.

L ø 13.04 Divide each side by 3.99 .

You can afford 13.04 lanterns, but it is not possible to

buy part of a lantern. Because 7.99 1 3.99 ( 14 )

5 63.85 , you cannot afford to buy 14 lanterns.

You can afford to buy 13 lanterns because

7.99 1 3.99 ( 13 ) 5 59.86 .

Your Notes




Example 3 Check reasonableness

A student’s answer to a real-world problem is shown below. Check the student’s answer for reasonableness.

Solution

The price of $ seems for a marker. The cost of 6 markers would be 6( ) 5 $ . If m equals the price of all the markers then the price of each marker is ÷ 5 . A check for reasonableness suggests the student’s answer is

. The correct price for each marker is $ .


Problem: You spent $21.75 on 6 markers and a $9.15 sketch pad. How much did each marker cost?

Solution: m 1 9.15 5 21.75

m 1 9.15 2 9.15 5 21.75 2 9.15

m 5 12.60

Answer: The price of each marker is $12.60.


1. You have $20 to buy a pizza. A plain pizza costs $14.50 and each topping costs $1.25. How many toppings can you add to your pizza? Show that your answer is reasonable.

2. Now you have $18 to buy the pizza and toppings from the previous problem. How many toppings can you add? A solution to this problem is shown below. Check the answer for reasonableness. 1.25t = 18, so t 5 14.4. You can have 14 toppings.

Homework

Your Notes


Example 3 Check reasonableness

A student’s answer to a real-world problem is shown below. Check the student’s answer for reasonableness.

Solution

The price of $ 12.60 seems high for a marker. The cost of 6 markers would be 6( 12.60 ) 5 $ 75.60 . If m equals the price of all the markers then the price of each marker is 12.60 ÷ 6 5 2.10 . A check for reasonableness suggests the student’s answer is incorrect. The correct price for each marker is $ 2.10 .


Problem: You spent $21.75 on 6 markers and a $9.15 sketch pad. How much did each marker cost?

Solution: m 1 9.15 5 21.75

m 1 9.15 2 9.15 5 21.75 2 9.15

m 5 12.60

Answer: The price of each marker is $12.60.


1. You have $20 to buy a pizza. A plain pizza costs $14.50 and each topping costs $1.25. How many toppings can you add to your pizza? Show that your answer is reasonable.

14.50 1 1.25t 5 20, so 1.25t 5 5.50, and t 5 4.4.You can have 4 toppings.

2. Now you have $18 to buy the pizza and toppings from the previous problem. How many toppings can you add? A solution to this problem is shown below. Check the answer for reasonableness. 1.25t = 18, so t 5 14.4. You can have 14 toppings.

The answer is not reasonable. 14 toppings would increase the price to more than $18.

14.50 1 1.25t 5 18, so 1.25t 5 3.50, and t 5 2.8.You can have 2 toppings.

Homework

Your Notes



3.3 Goal p Solve multi-step equations.

Solve Multi-Step Equations

Your NotesSolve 3t 1 5t 2 5 5 11.

Solution 3t 1 5t 2 5 5 11 Write original equation.

2 5 5 11 Combine like terms.

2 5 1 5 11 1 Add to each side.

5 Simplify.


t 5 Simplify.

The solution is .

Example 1 Solve an equation by combining like terms

Solve 5a 1 3(a 1 2) 5 22.

SolutionMethod 1 Method 2

Show All Steps Do Some Steps Mentally

5a 1 3(a 1 2) 5 22 5a 1 3(a 1 2) 5 22

5a 1 1 5 22 5a 1 1 5 22

1 5 22 1 5 22

5 22 2 5 16

5 16 a 5

5

a 5

Example 2 Solve an equation using the distributive property

16



3.3 Goal p Solve multi-step equations.

Solve Multi-Step Equations

Your NotesSolve 3t 1 5t 2 5 5 11.

Solution 3t 1 5t 2 5 5 11 Write original equation.

8t 2 5 5 11 Combine like terms.

8t 2 5 1 5 5 11 1 5 Add 5 to each side.

8t 5 16 Simplify.


t 5 2 Simplify.

The solution is 2 .

Example 1 Solve an equation by combining like terms

8

8t

8

16

Solve 5a 1 3(a 1 2) 5 22.

SolutionMethod 1 Method 2

Show All Steps Do Some Steps Mentally

5a 1 3(a 1 2) 5 22 5a 1 3(a 1 2) 5 22

5a 1 3a 1 6 5 22 5a 1 3a 1 6 5 22

8a 1 6 5 22 8a 1 6 5 22

8a 1 6 2 6 5 22 2 6 8a 5 16

8a 5 16 a 5 2

5

a 5 2

Example 2 Solve an equation using the distributive property

8

8a

8

16


Homework

Your Notes

1. 9d 2 4d 2 2 5 18 2. 2x 1 7(x 2 3) 5 6

3. 3w 1 4 1 w 5 36 4. 40 5 2(10 1 4k) 1 2k


Solve 3 } 4 (a 2 5) 5 9.

Solution

3 } 4 (a 2 5) 5 9 Write original equation.

p 3 } 4 (a 2 5) 5 p 9 Multiply each side by .

a 2 5 5 Simplify.

a 2 5 1 5 12 1 Add to each side.

a 5 Simplify.

Example 3 Multiply by a reciprocal to solve an equation

5. 1 } 2 (4x 2 2) 5 7 6. 5 } 6 (2y 1 4) 5 10




Homework

Your Notes

1. 9d 2 4d 2 2 5 18 2. 2x 1 7(x 2 3) 5 6

d 5 4 x 5 3

3. 3w 1 4 1 w 5 36 4. 40 5 2(10 1 4k) 1 2k

w 5 8 k 5 2


Solve 3 } 4 (a 2 5) 5 9.

Solution

3 } 4 (a 2 5) 5 9 Write original equation.

1 4 } 3 2 p 3 } 4 (a 2 5) 5 1 4 } 3 2 p 9 Multiply each side by 4 } 3 .

a 2 5 5 12 Simplify.

a 2 5 1 5 5 12 1 5 Add 5 to each side.

a 5 17 Simplify.

Example 3 Multiply by a reciprocal to solve an equation

5. 1 } 2 (4x 2 2) 5 7 6. 5 } 6 (2y 1 4) 5 10

x 5 4 y 5 4




3.4


Solve Equations with Variables on Both Sides Goal p Solve equations with variables on both sides.

VOCABULARY

Identity

Solve 15 1 4a 5 9a 2 5.

Solution 15 1 4a 5 9a 2 5 Write original

equation.

15 1 4a 2 5 9a 2 2 5 Subtract from each side.

15 5 2 5 Simplify.

15 1 5 2 5 1 Add to each side.

5 Simplify.


5 a Simplify.

The solution is .

CHECK

15 1 4a 5 9a 2 5 Write original equation.

15 1 4( ) 0 9( ) 2 5 Substitute for a.

15 1 0 2 5 Multiply.

5 ✓ Solution checks.

Example 1 Solve an equation with variables on both sides

Collect variables on one side of the equation and constant terms on the other to solve equations with variables on both sides.

Your Notes


3.4


Solve Equations with Variables on Both Sides Goal p Solve equations with variables on both sides.

VOCABULARY

Identity An equation that is true for all values of the variable

Solve 15 1 4a 5 9a 2 5.

Solution 15 1 4a 5 9a 2 5 Write original

equation.

15 1 4a 2 4a 5 9a 2 4a 2 5 Subtract 4a from each side.

15 5 5a 2 5 Simplify.

15 1 5 5 5a 2 5 1 5 Add 5 to each side.

20 5 5a Simplify.


4 5 a Simplify.

The solution is 4 .

CHECK

15 1 4a 5 9a 2 5 Write original equation.

15 1 4( 4 ) 0 9( 4 ) 2 5 Substitute 4 for a.

15 1 16 0 36 2 5 Multiply.


Example 1 Solve an equation with variables on both sides

5

20

5

5a

Collect variables on one side of the equation and constant terms on the other to solve equations with variables on both sides.

Your Notes


Your Notes

Solve 4t 2 12 5 6(t 1 3).

Solution4t 2 12 5 6(t 1 3) Write original equation.

4t 2 12 5 1 Distributive property

212 5 1 Subtract from each side.

5 Subtract from each side.

5 t Divide each side by .

Example 2 Solve an equation with grouping symbols

1. 3b 1 7 5 8b 1 2 2. 6d 2 6 5 3 } 4 (4d 1 8)


Solve the equation, if possible.

a. 4x 1 5 5 4(x 1 5) b. 6x 2 3 5 3(2x 2 1)

Solutiona. 4x 1 5 5 4(x 1 5) Original equation

4x 1 5 5 Distributive property

The equation 4x 1 5 5 is because the number 4x equal to 5 more than itself and more than itself. So, the equation has solution.

b. 6x 2 3 5 3(2x 2 1) Original equation

6x 2 3 5 Distributive property

The statement 6x 2 3 5 is for all values of x. So, the equation is an .

Example 3 Identify the number of solutions of an equation



Your Notes

Solve 4t 2 12 5 6(t 1 3).

Solution4t 2 12 5 6(t 1 3) Write original equation.

4t 2 12 5 6t 1 18 Distributive property

212 5 2t 1 18 Subtract 4t from each side.

230 5 2t Subtract 18 from each side.

215 5 t Divide each side by 2 .

Example 2 Solve an equation with grouping symbols

1. 3b 1 7 5 8b 1 2 2. 6d 2 6 5 3 } 4 (4d 1 8)

b 5 1 d 5 4


Solve the equation, if possible.

a. 4x 1 5 5 4(x 1 5) b. 6x 2 3 5 3(2x 2 1)

Solutiona. 4x 1 5 5 4(x 1 5) Original equation

4x 1 5 5 4x 1 20 Distributive property

The equation 4x 1 5 5 4x 1 20 is not true because the number 4x cannot be equal to 5 more than itself and 20 more than itself. So, the equation has no solution.

b. 6x 2 3 5 3(2x 2 1) Original equation

6x 2 3 5 6x 2 3 Distributive property

The statement 6x 2 3 5 6x 2 3 is true for all values of x. So, the equation is an identity .

Example 3 Identify the number of solutions of an equation



Homework

Your Notes

3. 1 } 2 (4t 2 6) 5 2t

4. 10m 2 4 5 22(2 2 5m)

Checkpoint Solve the equation, if possible.

STEPS FOR SOLVING LINEAR EQUATIONS

Step 1 Use the to remove any grouping symbols.

Step 2 the expression on each side of the equation.

Step 3 Use the properties of equality to collect the terms on one side of the equation

and the terms on the other side of the equation.

Step 4 Use the properties of equality to solve for the .

Step 5 Check your in the original equation.



Homework

Your Notes

3. 1 } 2 (4t 2 6) 5 2t

no solution

4. 10m 2 4 5 22(2 2 5m)

identity

Checkpoint Solve the equation, if possible.

STEPS FOR SOLVING LINEAR EQUATIONS

Step 1 Use the distributive property to remove any grouping symbols.

Step 2 Simplify the expression on each side of the equation.

Step 3 Use the properties of equality to collect the variable terms on one side of the equation and the constant terms on the other side of the equation.

Step 4 Use the properties of equality to solve for the variable .

Step 5 Check your solution in the original equation.



3.5 Write Ratios and ProportionsGoals p Find ratios and write and solve proportions.

VOCABULARY

Ratio

Proportion

RATIOS

1. A ratio uses to compare two quantities.

2. The ratio of two quantities, a and b, where b is not equal to 0, can be written in three ways:

3. Each ratio is read “the of a to b”.

4. Ratios should be written in form.

Cell Phone Use A person makes 6 long distance calls and 15 local calls in 1 month.

a. Find the ratio of long distance calls to local calls.

b. Find the ratio of long distance calls to all calls.

Solution

a. long distance calls

}} local calls 5 5

b. long distance calls

}} all calls 5 5

Example 1 Write a ratio

Your Notes



3.5 Write Ratios and ProportionsGoals p Find ratios and write and solve proportions.

VOCABULARY

Ratio The use of division to compare two quantities

Proportion An equation that states that two ratios are equivalent

RATIOS

1. A ratio uses division to compare two quantities.

2. The ratio of two quantities, a and b, where b is not equal to 0, can be written in three ways:

a to b a :b a } b

3. Each ratio is read “the ratio of a to b”.

4. Ratios should be written in simplest form.

Cell Phone Use A person makes 6 long distance calls and 15 local calls in 1 month.

a. Find the ratio of long distance calls to local calls.

b. Find the ratio of long distance calls to all calls.

Solution

a. long distance calls

}} local calls 5 5

b. long distance calls

}} all calls 5 5

Example 1 Write a ratio

15

6

5

2

21

6

7

2

Your Notes




Your Notes

1. The number of tickets Shawn sold to the number of tickets Myra sold

2. The number of tickets Myra sold to the number of tickets Shawn and Myra sold

Checkpoint Shawn and Myra are selling tickets to their school’s talent show. Shawn sold 36 tickets, and Myra sold 44 tickets. Find the specified ratio.

Example 2 Solve a proportion

Solve the proportion y }

15 5 3 }

5 .

Solution

y } 15 5 3 } 5 Write original proportion.

p y } 15 5 p 3 } 5 Multiply each side by .

y 5 Simplify.

y 5 Divide.

Use the same methods for solving equations to solve proportions with a variable in the numerator.

3. 9 } 4 5 c } 28 4. a } 32 5 7 } 8

Checkpoint Solve the proportion. Check your solution.



Your Notes

1. The number of tickets Shawn sold to the number of tickets Myra sold

9 } 11

2. The number of tickets Myra sold to the number of tickets Shawn and Myra sold

11 } 20

Checkpoint Shawn and Myra are selling tickets to their school’s talent show. Shawn sold 36 tickets, and Myra sold 44 tickets. Find the specified ratio.

Example 2 Solve a proportion

Solve the proportion y }

15 5 3 }

5 .

Solution

y } 15 5 3 } 5 Write original proportion.

15 p y } 15 5 15 p 3 } 5 Multiply each side by 15 .

y 5 Simplify.

y 5 9 Divide.5

45

Use the same methods for solving equations to solve proportions with a variable in the numerator.

3. 9 } 4 5 c } 28 4. a } 32 5 7 } 8

c 5 63 a 5 28



Your Notes

Swimming Pool A empty swimming pool is being filled with water. After 5 minutes the pool has 400 gallons of water. If the pool has a volume of 11,200 gallons, how long does it take to fill the empty pool?

SolutionStep 1 Write a proportion involving two ratios that

compare the amount of water in the pool to the amount of time.

400 } 5 5 x

gallons

minutes

Step 2 Solve the proportion.

400 } 5 5 x

Write proportion.

p 400 } 5 5 p x

Multiply each side by .

5

5 Simplify.

p 5

5 p Multiply each side by .

5 Simplify.

x 5 Divide each side by .

The pool is full after minutes.

Example 3 Solve a multi-step problem

5. An Olympic sized pool has a volume of 810,000 gallons. If it is filled at the same rate as the pool in Example 3, how long will it take to fill the pool?


Homework



Your Notes

Swimming Pool A empty swimming pool is being filled with water. After 5 minutes the pool has 400 gallons of water. If the pool has a volume of 11,200 gallons, how long does it take to fill the empty pool?


compare the amount of water in the pool to the amount of time.

400 } 5 5 11,200

x

gallons

minutes


400 } 5 5 11,200

x Write proportion.

x p 400 } 5 5 x p 11,200

x Multiply each side

by x .

400x

5 5 11,200 Simplify.

5 p 400x

5 5 5 p 11,200 Multiply each side

by 5 .

400x 5 56,000 Simplify.

x 5 140 Divide each side by 400 .

The pool is full after 140 minutes.


5. An Olympic sized pool has a volume of 810,000 gallons. If it is filled at the same rate as the pool in Example 3, how long will it take to fill the pool?

10,125 minutes


Homework



3.6


Solve Proportions Using Cross Products Goal p Solve proportions using cross products.

VOCABULARY

Cross product

Scale drawing

Scale model

Scale

CROSS PRODUCTS PROPERTY

Words The cross products of a proportion are .

Example 5 } 6 5 10 } 12 p 10 5 60 p 12 5 60

Algebra If a } b 5 c } d where b Þ 0 and d Þ 0, then

ad 5 .

Your Notes


3.6


Solve Proportions Using Cross Products Goal p Solve proportions using cross products.

VOCABULARY

Cross product The product of the numerator of one ratio and the denominator of the other ratio

Scale drawing A two-dimensional drawing of an object in which the dimensions of the drawing are in proportion to the dimensions of the object

Scale model A three-dimensional model of an object in which the dimensions of the model are in proportion to the dimensions of the object

Scale A relationship between the drawing’s or model’s dimensions and the actual dimensions; This should be written as scale measure : actual measure.

CROSS PRODUCTS PROPERTY

Words The cross products of a proportion are equal .

Example 5 } 6 5 10 } 12 6 p 10 5 60 5 p 12 5 60

Algebra If a } b 5 c } d where b Þ 0 and d Þ 0, then

ad 5 bc .

Your Notes


Your Notes

Plant Food To feed your plants, you need to mix 3 tablespoons of plant food with 16 ounces of water. If it takes 80 ounces of water to feed all of your plants, how many tablespoons of plant food are needed?


compare the amount of plant food with the amount of water.

3 } 16 5 x amount of plant food

amount of water


3 } 16 5 x Write proportion.

3 p 5 p x Cross product property

5 Simplify.

5 x Divide each side by .

You need tablespoons of plant food for 80 ounces of water.

Example 2 Write and solve a proportion

Solve the proportion 5 } y 5 15 }

75 .

Solution

5 } y 5 15 } 75 Write original proportion.

p 75 5 p 15 Cross products property

5 Simplify.

5 y Divide each side by .

The solution is .

Example 1 Solve a proportion using cross products



Your Notes

Plant Food To feed your plants, you need to mix 3 tablespoons of plant food with 16 ounces of water. If it takes 80 ounces of water to feed all of your plants, how many tablespoons of plant food are needed?


compare the amount of plant food with the amount of water.

3 } 16 5 x

80

amount of plant food

amount of water


3 } 16 5 x

80 Write proportion.

3 p 80 5 16 p x Cross product property

240 5 16x Simplify.

15 5 x Divide each side by 16 .

You need 15 tablespoons of plant food for 80 ounces of water.

Example 2 Write and solve a proportion

Solve the proportion 5 } y 5 15 }

75 .

Solution

5 } y 5 15 } 75 Write original proportion.

5 p 75 5 y p 15 Cross products property

375 5 15y Simplify.

25 5 y Divide each side by 15 .


Example 1 Solve a proportion using cross products




Homework

Your Notes

1. 5 } n 5 25 } 45 2. 6 } b 5 3 } b 2 2

3. In Example 2, suppose it takes 120 ounces to feed all of the plants. How many tablespoons of plant food are needed?


Scale Model An architect creates a scale model of a school. The school is 50 feet high. The ratio of the model to the actual school is 1 foot to 75 feet. Estimate the height of the model.

SolutionWrite and solve a proportion to find the height h of the scale model.

1 5 h height of model (feet)

actual height (feet)

1 p 5 p h Cross products property

5 h Simplify.

The height of the scale model is foot, or inches.

Example 3 Use a scale model

4. In Example 3, suppose the ratio of the model to the actual school is 1 foot to 100 feet. Estimate the height of the model.




Homework

Your Notes

1. 5 } n 5 25 } 45 2. 6 } b 5 3 } b 2 2

n 5 9 b 5 4

3. In Example 2, suppose it takes 120 ounces to feed all of the plants. How many tablespoons of plant food are needed?

22.5 tablespoons


Scale Model An architect creates a scale model of a school. The school is 50 feet high. The ratio of the model to the actual school is 1 foot to 75 feet. Estimate the height of the model.

SolutionWrite and solve a proportion to find the height h of the scale model.

1

75 5 h

50

height of model (feet)

actual height (feet)

1 p 50 5 75 p h Cross products property

2 } 3 5 h Simplify.

The height of the scale model is 2 } 3 foot, or 8 inches.

Example 3 Use a scale model

4. In Example 3, suppose the ratio of the model to the actual school is 1 foot to 100 feet. Estimate the height of the model.

1 } 2 foot, or 6 in.



3.7 Solve Percent Problems Goal p Solve percent problems.

SOLVING PERCENT PROBLEMS USING PROPORTIONS

You can represent “a is p percent of b” by using the proportion

a } b 5 p

where a is a part of the base and p

, or p%, is

the .

What percent of 50 is 33?

SolutionWrite a proportion when 50 is the base and 33 is part of the base.

a } b 5 p } 100 Write proportion.

5 p } 100 Substitute for a and for b.

5 50p Cross products property

5 p Divide each side by .

33 is of 50.

Example 1 Find a percent using a proportion


Your Notes


3.7 Solve Percent Problems Goal p Solve percent problems.

SOLVING PERCENT PROBLEMS USING PROPORTIONS

You can represent “a is p percent of b” by using the proportion

a } b 5 p

100

where a is a part of the base b and p

100, or p%, is

the percent .


SolutionWrite a proportion when 50 is the base and 33 is part of the base.

a } b 5 p } 100 Write proportion.

50

33 5

p } 100 Substitute 33 for a and 50 for b.

3300 5 50p Cross products property

66 5 p Divide each side by 50 .

33 is 66% of 50.

Example 1 Find a percent using a proportion


Your Notes



Your Notes

1. What percent of 80 is 28?


Checkpoint Use a proportion to answer the question.

THE PERCENT EQUATION

You can represent “a is p percent of b” by using the equation:

a 5 p b

where a is a part of the base and p% is the .


Solution a 5 p% p b Write percent equation.

5 p% p Substitute for a and for b.

5 p% Divide each side by .

5 p% Write decimal as a percent.

98 is of 224.

CHECK

5 p% p Write original equation.

0 p Substitute for p%.

5 ✓ Multiply. Solution checks.

Example 2 Find a percent using the percent equation

The percent equation, a 5 p% p b, is derived from the proportion,

a } b 5

p }

100 .



Your Notes


35%


40%

Checkpoint Use a proportion to answer the question.

THE PERCENT EQUATION

You can represent “a is p percent of b” by using the equation:

a 5 p% p b

where a is a part of the base b and p% is the percent .



98 5 p% p 224 Substitute 98 for a and 224 for b.

0.4375 5 p% Divide each side by 224 .

43.75% 5 p% Write decimal as a percent.

98 is 43.75% of 224.

CHECK

98 5 p% p 224 Write original equation.

98 0 0.4375 p 224 Substitute 0.4375 for p%.

98 5 98 ✓ Multiply. Solution checks.

Example 2 Find a percent using the percent equation

The percent equation, a 5 p% p b, is derived from the proportion,

a } b 5

p }

100 .


Your Notes

What number is 75% of 164?

Solutiona 5 p% ? b Write percent equation.

5 p Substitute for p and for b.

5 p Write percent as a decimal.

5 Multiply.

is 75% of 164.

Example 3 Find a part of a base using the percent equation


4. What number is 35% of 80?

Checkpoint Use the percent equation to answer the question.

21 is 37.5% of what number?


5 p b Substitute for a and for p.

5 p b Write percent as a decimal.

5 b Divide each side by .

21 is 75% of .

Example 4 Find a base using the percent equation



Your Notes

What number is 75% of 164?

Solutiona 5 p% ? b Write percent equation.

5 75% p 164 Substitute 75 for p and 164 for b.

5 0.75 p 164 Write percent as a decimal.

5 123 Multiply.

123 is 75% of 164.

Example 3 Find a part of a base using the percent equation


75%

4. What number is 35% of 80?

28


21 is 37.5% of what number?


21 5 37.5% p b Substitute 21 for a and 37.5 for p.

21 5 0.375 p b Write percent as a decimal.

56 5 b Divide each side by 0.375 .

21 is 75% of 56 .

Example 4 Find a base using the percent equation




Homework

Your Notes

5. 27 is 25% of what number?



TYPES OF PERCENT PROBLEMS

Percent Problem Example Equation

Find a percent. What percent 5 p% p of 252 is 84?

Find part of What number a 5 p a base. is 30% of 90?

Find a base. 16 is 20% of 16 5 p b what number?



Homework

Your Notes


108


52


TYPES OF PERCENT PROBLEMS

Percent Problem Example Equation

Find a percent. What percent 84 5 p% p 252 of 252 is 84?

Find part of What number a 5 30% p 90 a base. is 30% of 90?

Find a base. 16 is 20% of 16 5 20% p b what number?


Find Percent of Change Goal p Solve percent of change problems.


80 3.7 Focus On Operations• Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

VOCABULARY

Percent of change

Percent of increase

Percent of decrease

PERCENT OF CHANGE

Percent of change, p% 5 }}}}

Amount of increase 5 amount 2 amount

Amount of decrease 5 amount 2 amount

Identify the percent of change as an increase or decrease. Then find the percent of change.

a. Original: 160 b. Original: 75New: 128 New: 120

Solutiona. The percent of change b. The percent of change

is . is .

160 2 }} 5 } 2 75

} 5 }

5 0. 5 % 5 0. 5 %

Percent of Percent of is %. is %.

Example 1 Find a percent of change

Your Notes


Find Percent of Change Goal p Solve percent of change problems.


80 3.7 Focus On Operations• Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

VOCABULARY

Percent of change How much a quantity increases or decreases with respect to the original amount

Percent of increase Percent of change if the new amount is greater than the original amount

Percent of decrease Percent of change if the new amount is less than the original amount

PERCENT OF CHANGE

Percent of change, p% 5 Amount of increase or decrease

}}}} Original Amount

Amount of increase 5 new amount 2 original amount

Amount of decrease 5 original amount 2 new amount

Identify the percent of change as an increase or decrease. Then find the percent of change.

a. Original: 160 b. Original: 75New: 128 New: 120

Solutiona. The percent of change b. The percent of change

is a decrease . is an increase .

160 2 128 }}

160 5

32 }

160

120 2 75 }

75 5

45 }

75

5 0. 2 5 20 % 5 0. 6 5 60 %

Percent of decrease Percent of increase is 20 %. is 60 %.

Example 1 Find a percent of change

Your Notes


1. Original: 360 New: 54 2. Original: 140 New: 189

3. Original: 10 New: 19 4. Original: 13 New: 11.83

Checkpoint Find the percent of change.

Find the sale price of a jacket if the original price is $124.00 and the discount is 30% off.

SolutionThe sale price is a decrease from the original price, so multiply the original price by ( 2 p%)

Sale price 5 124 • (100% 2 %) Substitute.

5 • and write

as a .

5 Multiply.

The sale price of the jacket is $ .

Example 2 Find a new amount

You can also findthe sale price byfirst finding thechange in price—30% of 124— andthen subtractingthis change fromthe original price.


5. Increase 34 by 18%.

6. A store is advertising televisions on sale for 45% off. If the original price was $655, what is the sale price?

Checkpoint Find the new amount.

Homework


1. Original: 360 New: 54 2. Original: 140 New: 189

85% decrease 35% increase

3. Original: 10 New: 19 4. Original: 13 New: 11.83 90% increase 9% decrease

Checkpoint Find the percent of change.

Find the sale price of a jacket if the original price is $124.00 and the discount is 30% off.

SolutionThe sale price is a decrease from the original price, so multiply the original price by ( 100% 2 p%)

Sale price 5 124 • (100% 2 30 %) Substitute.

5 124 • 0.7 Subtract and write

as a decimal .

5 86.8 Multiply.

The sale price of the jacket is $ 86.80 .

Example 2 Find a new amount

You can also findthe sale price byfirst finding thechange in price—30% of 124— andthen subtractingthis change fromthe original price.


5. Increase 34 by 18%.

40.12

6. A store is advertising televisions on sale for 45% off. If the original price was $655, what is the sale price?

$360.25

Checkpoint Find the new amount.

Homework


3.8 Rewrite Equations and Formulas Goal p Write equations in function form and rewrite

formulas.

VOCABULARY

Function form

Literal equation

Write 2x 1 2y 5 10 in function form.

SolutionSolve the equation for y.

2x 1 2y 5 10 Write original equation.

2y 5 Subtract from each side.

y 5 Divide each side by .

The equation y 5 is written in function form.

Example 1 Rewrite an equation in function form

Solve a 1 by 5 c for a.

Solutiona 1 by 5 c Write original equation.

a 5 Subtract from each side.

The solution is a 5 .

Example 2 Solve a literal equation

Your Notes



3.8 Rewrite Equations and Formulas Goal p Write equations in function form and rewrite

formulas.

VOCABULARY

Function form An equation in x and y written so the dependent variable y is isolated on one side of the equation

Literal equation An equation that contains two or more variables

Write 2x 1 2y 5 10 in function form.

SolutionSolve the equation for y.

2x 1 2y 5 10 Write original equation.

2y 5 10 2 2x Subtract 2x from each side.

y 5 5 2 x Divide each side by 2 .

The equation y 5 5 2 x is written in function form.

Example 1 Rewrite an equation in function form

Solve a 1 by 5 c for a.

Solutiona 1 by 5 c Write original equation.

a 5 c 2 by Subtract by from each side.

The solution is a 5 c 2 by .

Example 2 Solve a literal equation

Your Notes




Homework

Your Notes

The interest I on an investment of P dollars at an interest rate r for t years is given by the formula I 5 Prt.

a. Solve the formula for the time t.

b. Use the rewritten formula to find the time it takes to earn $100 interest on $1000 at a rate of 5.0%.

Solutiona. I 5 Prt Write original formula.

I 5 t Divide each side by .

b. Substitute for I, for P, and for r in the rewritten formula.

t 5 I Write rewritten formula.

5 p

Substitute.

5 Simplify.

It will take years to earn $100 in interest.

Example 3 Solve and use a formula

3. Solve a 1 by 5 c for b.

4. In Example 3, solve the equation for P. Find the investment P if I 5 $400, r 5 4%, and t 5 4 years.


1. 2x 1 y 5 5 2. 3 1 3y 5 9 2 6x

Checkpoint Write the equation in function form.



Homework

Your Notes

The interest I on an investment of P dollars at an interest rate r for t years is given by the formula I 5 Prt.

a. Solve the formula for the time t.

b. Use the rewritten formula to find the time it takes to earn $100 interest on $1000 at a rate of 5.0%.

Solutiona. I 5 Prt Write original formula.

I

Pr 5 t Divide each side by Pr .

b. Substitute 100 for I, 1000 for P, and 0.05 for r in the rewritten formula.

t 5 I

Pr Write rewritten formula.

5 1000

100

p 0.05 Substitute.

5 2 Simplify.

It will take 2 years to earn $100 in interest.

Example 3 Solve and use a formula

3. Solve a 1 by 5 c for b.

b 5 c } y 2 a }

y

4. In Example 3, solve the equation for P. Find the investment P if I 5 $400, r 5 4%, and t 5 4 years.

P 5 I } rt

; $2500


1. 2x 1 y 5 5 2. 3 1 3y 5 9 2 6x

y 5 5 2 2x y 5 2 2 2x

Checkpoint Write the equation in function form.



Inverse operations

Identity

Proportion

Scale drawing

Scale

Literal equation


Ratio

Cross product

Scale model

Function form

Percent of change





Inverse operations

addition and subtraction;

multiplication and division

Identity

2y 2 6 5 2(y 2 3)

Proportion

a } b 5 c } d

Scale drawing

A map

Scale

1 in. :10 ft

Literal equation

I 5 Prt


x 2 a 5 b

x 5 b 1 a

Ratio

a :b

Cross product

If a } b 5 c } d , then ad 5 bc.

Scale model

A three-dimensional model of an object in which the dimensions of the model are in proportion to the dimensions of the object.

Function form

y 5 9 2 2x

Percent of changePercent of increase or decrease. A price changing from $100 to $130 is a 30% increase. A price changing from $100 to $80 is a 20% decrease.




VOCABULARY

Quadrant

Goal p Identify and plot points in a coordinate plane.


4.1 Plot Points in a Coordinate Plane

Give the coordinates of the point.

a. A b. B

Solution

a. Point A is units to the

x

y

1

3

12121

23

23 3

AB

CD

E

of the origin and units .

The x-coordinate is .The y-coordinate is .The coordinates are .

b. Point B is units to the of the origin and

units .The x-coordinate is .The y-coordinate is .The coordinates are .

Example 1 Name points in a coordinate plane

Points in Quadrant l have two positive coordinates. Points in the other three quadrants have at least one negative coordinate.

1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.


Your Notes


VOCABULARY

Quadrant One of the four parts into which the axes divide a coordinate plane; labeled I, II, III, IV

Goal p Identify and plot points in a coordinate plane.


4.1 Plot Points in a Coordinate Plane

Give the coordinates of the point.

a. A b. B

Solution

a. Point A is 2 units to the

x

y

1

3

12121

23

23 3

AB

CD

E

right of the origin and 2 units up .The x-coordinate is 2 .The y-coordinate is 2 .The coordinates are (2, 2) .

b. Point B is 4 units to the left of the origin and 2 units up .The x-coordinate is 24 .The y-coordinate is 2 .The coordinates are (24, 2) .

Example 1 Name points in a coordinate plane

Points in Quadrant l have two positive coordinates. Points in the other three quadrants have at least one negative coordinate.

1. Use the coordinate plane in Example 1 to give the coordinates of points C, D, and E.

C(22, 23), D(4, 23), E(1, 24)


Your Notes


Plot the point in a coordinate plane. Describe the location of the point.

a. A(0, 3) b. B(1, 22) c. C(23, 24)

Solution

a. Begin at the .

x

y

1

3

12121

23

23 3

Move units .Point A is on the .

b. Begin at the . Move unit to the .Move units .Point B is in Quadrant .

c. Begin at the Move units to the .Move units .Point C is in Quadrant .

Example 2 Plot points in a coordinate plane

Checkpoint Plot the point in a coordinate plane. Describe the location of the point.

2. A(24, 24) 3. B(2, 0)

x

y

1

3

12121

23

23 3

x

y

1

3

12121

23

23 3


Your Notes


Plot the point in a coordinate plane. Describe the location of the point.

a. A(0, 3) b. B(1, 22) c. C(23, 24)

Solution

a. Begin at the origin .

x

y

1

3

12121

23

23 3

A(0, 3)

C(23, 24)

B(1, 22)

Move 3 units up .Point A is on the y-axis .

b. Begin at the origin . Move 1 unit to the right .Move 2 units down .Point B is in Quadrant lV .

c. Begin at the origin. Move 3 units to the left .Move 4 units down .Point C is in Quadrant lIl .

Example 2 Plot points in a coordinate plane

Checkpoint Plot the point in a coordinate plane. Describe the location of the point.

2. A(24, 24) 3. B(2, 0)

x

y

1

3

12121

23

23 3

A(24, 24)

x

y

1

3

12121

23

23 3

B(2, 0)

Quadrant III x-axis


Your Notes


Graph the function y 5 x 1 1 with domain 22, 21, 0, 1, 2. Then identify the range of the function.

Solution x y 5 x 1 1

22 y 5 22 1 1 5

21 y 5 21 1 1 5

0 y 5 0 1 1 5

1 y 5 1 1 1 5

2 y 5 2 1 1 5

Step 1 Make a table.

Step 2 List the ordered pairs:

(22, ), (21, ), (0, ), (1, ), (2, ).

Then graph the function.

x

y

1

3

12121

23

23 3

Step 3 Identify the range: .


Homework


4. Graph the function y 5 2 1 } 2 x 1 3 with domain

24, 22, 0, 2, and 4. Then identify the range.

x

y

1

3

5

12121

23

23 3


Your Notes


Graph the function y 5 x 1 1 with domain 22, 21, 0, 1, 2. Then identify the range of the function.

Solution x y 5 x 1 1

22 y 5 22 1 1 5 21

21 y 5 21 1 1 5 0

0 y 5 0 1 1 5 1

1 y 5 1 1 1 5 2

2 y 5 2 1 1 5 3


Step 2 List the ordered pairs:

(22, 21 ), (21, 0 ), (0, 1 ), (1, 2 ), (2, 3 ).

Then graph the function.

x

y

1

3

12121

23

23 3

Step 3 Identify the range: 21, 0, 1, 2, 3 .


Homework


4. Graph the function y 5 2 1 } 2 x 1 3 with domain

24, 22, 0, 2, and 4. Then identify the range.

x

y

1

3

5

12121

23

23 3

range: 1, 2, 3, 4, 5


Your Notes


Perform Transformations Goal p Perform and describe transformations in a

coordinate plane.

VOCABULARY

Transformation

Translation

Vertical stretch

Vertical shrink

Reflection

The graphs show the function y 5 x 2 1 with domain 1, 2, 3. (a) The transformation (x, y) → (x, y 1 2) moves the graph units. (b) The transformation (x, y) → (x, 23y) is a vertical a reflection in the x-axis Draw the transformed graphs.

(a). y

x1

1

(b). y

x

1

1

Original Image Original Image

( , 0) → ( ) ( , ) → ( , )

→ →

→ →

Example 1 Perform transformations

The functions represented by the original and both image graphs have the same domain. The range for the original function is 1, 2, . The ranges for image graph (a) and (b) are respectively, 2, , and , , .

88 4.1 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Focus On GraphingUse after Lesson 4.1

Your Notes


Perform Transformations Goal p Perform and describe transformations in a

coordinate plane.

VOCABULARY

Transformation Applying a rule to the coordinates of each point on a graph to produce an image

Translation Moves every point on a graph the same distance in the same direction

Vertical stretch Moves a point away from the x-axis

Vertical shrink Moves a point toward the x-axis

Reflection Flips a point in a line

The graphs show the function y 5 x 2 1 with domain 1, 2, 3. (a) The transformation (x, y) → (x, y 1 2) moves the graph up 2 units. (b) The transformation (x, y) → (x, 23y) is a vertical stretch with a reflection in the x-axis Draw the transformed graphs.

(a). y

x1

1

(b). y

x

1

1

Original Image Original Image

( 1 , 0) → ( 1 , 2 ) ( 1 , 0 ) → ( 1 , 0 )

(2, 1) → (2, 3) (2, 1) → (2, 23)

(3, 2) → (3, 4) (3, 2) → (3, 26)

Example 1 Perform transformations

The functions represented by the original and both image graphs have the same domain. The range for the original function is 1, 2, 3 . The ranges for image graph (a) and (b) are respectively, 2, 3 , 4 and 0 , 23, 26.



Your Notes


1. Use small dots to graph the function: y 5 3x 2 1 with domain 21, 0, 1. Perform the transformation (x, y) → (x, y 2 2). Graph the image using large dots. Identify the domain and range of the function represented by the image.

2. Use small dots to graph the

function: y 5 22x 1 2 with domain 22, 0, 2. Perform the transformation (x, y) → (x, 0.5y). Graph the image using large dots. Identify the domain and range of the function represented by the image.

Copyright © Holt McDougal. All rights reserved. 4.1 Focus on Graphing • Algebra 1 Notetaking Guide 89

y

x1

1

IDENTIFYING TRANSFORMATIONS

Vertical or

(x, y) → (x1h, y ) (x, y) → (x, y 1 k)

reflection(x, y) → (x, ay)

where a 0

reflection

(x, y) → (x, ay)

where a 0y

x1

1

y

x1

1

y

x1

1

y

x1

1


y

x

1

1

Homework

Your Notes


1. Use small dots to graph the function: y 5 3x 2 1 with domain 21, 0, 1. Perform the transformation (x, y) → (x, y 2 2). Graph the image using large dots. Identify the domain and range of the function represented by the image.

domain: 21, 0, 1;

range: 26, 23, 0

2. Use small dots to graph the

function: y 5 22x 1 2 with domain 22, 0, 2. Perform the transformation (x, y) → (x, 0.5y). Graph the image using large dots. Identify the domain and range of the function represented by the image.

domain: 22, 0, 2 range: 3, 1, 21


y

x1

1

IDENTIFYING TRANSFORMATIONS

Translation Vertical stretch or shrink

Horizontal (x, y) → (x1h, y )

Vertical (x, y) → (x, y 1 k)

Without reflection

(x, y) → (x, ay)

where a . 0

With reflection

(x, y) → (x, ay)

where a , 0y

x1

1

y

x1

1

y

x1

1

y

x1

1


y

x

1

1

Homework

Your Notes


4.2 Graph Linear EquationsGoal p Graph linear equations in a coordinate plane.

VOCABULARY

Solution of an equation in two variables

Graph of an equation in two variables

Linear equation

Standard form of a linear equation

Linear function

Your Notes

Graph the equation x 1 y 5 4.

SolutionStep 1 Solve the equation for y.

x 1 y 5 4

y 5


Choose a few values for x and find the values for y.

x 22 21 0 1 2

y

Example 1 Graph an equation

Use convenient values for x when making a table. These should include a combination of negative values, zero, and positive values.



4.2 Graph Linear EquationsGoal p Graph linear equations in a coordinate plane.

VOCABULARY

Solution of an equation in two variables An ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the equation

Graph of an equation in two variables The set of points in a coordinate plane that represents all solutions of the equation

Linear equation An equation whose graph is a line

Standard form of a linear equation Ax 1 By 5 C, where A, B, and C are real numbers and A and B are not both zero

Linear function The equation Ax 1 By 5 C represents a linear function provided B Þ 0 (that is, provided the graph of the equation is not a vertical line).

Your Notes

Graph the equation x 1 y 5 4.

SolutionStep 1 Solve the equation for y.

x 1 y 5 4

y 5 4 2 x


Choose a few values for x and find the values for y.

x 22 21 0 1 2

y 6 5 4 3 2

Example 1 Graph an equation

Use convenient values for x when making a table. These should include a combination of negative values, zero, and positive values.



Step 3 Plot the points.

x

y

1

3

5

12121

23 3

Step 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

Your Notes


Graph (a) y 5 23 and (b) x 5 2.

Solutiona. Regardless of the value of x, the value of y is always

. The graph of y 5 23 is a line 3 units the x-axis.

x

y

1

3

12121

23

23 3

b. Regardless of the value of y, the value of x is always . The graph of x 5 2 is a line

2 units to the of the y-axis.

x

y

1

3

12121

23

23 3

Example 2 Graph y 5 b and x 5 a

The equations y 5 23 and 0x 1 1y 5 23 are equivalent. For any value of x, the ordered pair (x, 23) is a solution of y 5 23.


Step 3 Plot the points.

x

y

1

3

5

12121

23 3

Step 4 Connect the points by drawing a line through them. Use arrows to indicate that the graph goes on without end.

Your Notes


Graph (a) y 5 23 and (b) x 5 2.

Solutiona. Regardless of the value of x, the value of y is always

23 . The graph of y 5 23 is a horizontal line 3 units below the x-axis.

x

y

1

3

12121

23

23 3

y 5 23

b. Regardless of the value of y, the value of x is always 2 . The graph of x 5 2 is a vertical line 2 units to the right of the y-axis.

x

y

1

3

12121

23

23 3

x 5 2

Example 2 Graph y 5 b and x 5 a

The equations y 5 23 and 0x 1 1y 5 23 are equivalent. For any value of x, the ordered pair (x, 23) is a solution of y 5 23.


1. y 5 2x 2 1 2. x 5 0.5

x

y

1

3

12121

23

25

23 3

x

y

1

3

12121

23

23 3

3. y 5 24x 1 1 4. y 5 21.5

x

y

2

6

12122

26

23 3

x

y

1

3

12121

23

23 3

Checkpoint Graph the equation.

EQUATIONS OF HORIZONTAL AND VERTICAL LINES

1. The graph of y 5 b is a line.

2. The line of graph y 5 b passes through the point .

3. The graph of x 5 a is a line.

4. The line of graph x 5 a passes through the point .


Your Notes


1. y 5 2x 2 1 2. x 5 0.5

x

y

1

3

12121

23

25

23 3

x

y

1

3

12121

23

23 3

x 5 0.5

3. y 5 24x 1 1 4. y 5 21.5

x

y

2

6

12122

26

23 3

x

y

1

3

12121

23

23 3

Checkpoint Graph the equation.

EQUATIONS OF HORIZONTAL AND VERTICAL LINES

1. The graph of y 5 b is a horizontal line.

2. The line of graph y 5 b passes through the point (0, b) .

3. The graph of x 5 a is a vertical line.

4. The line of graph x 5 a passes through the point (a, 0) .


Your Notes


Graph the function y 5 2x 1 2 with domain x ≥ 0. Then identify the range of the function.

Solution

Step 1 Make a .

x 0 1 2 3 4

y

Step 2 Plot the .

x

y

1

3

5

7

9

121 3 5

Step 3 Connect the points with a because the domain is .

Step 4 Identify the range. From the graph, you can see that all points have a y-coordinate of , so the range of the function is .

Example 3 Graph a linear function


Homework

5. Graph the function y 5 2x 1 4 with domain x $ 0. Then identify the range of the function.

x

y

1

3

12121

23

23 3


Your Notes


Graph the function y 5 2x 1 2 with domain x ≥ 0. Then identify the range of the function.

Solution

Step 1 Make a table .

x 0 1 2 3 4

y 2 4 6 8 10

Step 2 Plot the points .

x

y

1

3

5

7

9

121 3 5

Step 3 Connect the points with a ray because the domain is restricted .

Step 4 Identify the range. From the graph, you can see that all points have a y-coordinate of 2 or more , so the range of the function is y ≥ 2 .

Example 3 Graph a linear function


Homework

5. Graph the function y 5 2x 1 4 with domain x $ 0. Then identify the range of the function.

x

y

1

3

12121

23

23 3

range: y ≤ 4


Your Notes


Identify Discrete and Continuous Functions Goal p Graph and classify discrete and continuous

functions.

VOCABULARY

Discrete function

Continuous function

DISCRETE AND CONTINUOUS FUNCTIONS

The graph of a The graph of a

consists of isolated points. is unbroken.

y

x

y

x

Graph the function y 5 3x 2 2 with the given domain. Classify the function as discrete or continuous. Then identify the range of the function.a. Domain: x 5 21, 0, 1 b. Domain: x # 1

y

x1

1

y

x1

1

The graph consists of The graph is , points, so so the function is

the function is . The range is . Its range is .

Example 1 Graph and classify a function



Your Notes


Identify Discrete and Continuous Functions Goal p Graph and classify discrete and continuous

functions.

VOCABULARY

Discrete function A function that has a graph consisting of isolated points

Continuous function A function that has an unbroken graph

DISCRETE AND CONTINUOUS FUNCTIONS

The graph of a The graph of a discrete function continuous function consists of isolated points. is unbroken.

y

x

y

x

Graph the function y 5 3x 2 2 with the given domain. Classify the function as discrete or continuous. Then identify the range of the function.a. Domain: x 5 21, 0, 1 b. Domain: x # 1

y

x1

1

y

x1

1

The graph consists of The graph is unbroken , isolated points, so so the function isthe function is discrete . continuous The range is 25, 22, 1 . Its range is y # 1 .

Example 1 Graph and classify a function



Your Notes


1. Graph the function y 5 x 2 3 with the domain: x $ 2. Classify the function as discrete or continuous. Then identify the range.


Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, graph the function and find the value of y when x 5 0.5.

Time after planting (days), x 1 2 3Growth of plant (millimeters), y 14 28 42

Solution

It make sense to talk about the amount a plant has grown after any length of time. So the table represents a function.

Example 2 Classify and graph a real-world function

Copyright © Holt McDougal. All rights reserved. 4.2 Focus On Functions• Algebra 1 Notetaking Guide 95

y

x1

1

y

x0 1 2 3 4

10

20

30

40

Plant Growth

Time after planting(days)

Gro

wth

of

pla

nt

(mill

imet

ers)

2. Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, find the value of y when x 5 2.5. Round your answer to the nearest hundredth.

Number of songs downloaded, x 1 5 10

Cost of songs (dollars), y 1.25 6.25 12.50


Homework

Your Notes


1. Graph the function y 5 x 2 3 with the domain: x $ 2. Classify the function as discrete or continuous. Then identify the range.

Continuous; y $ −1


Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, graph the function and find the value of y when x 5 0.5.

Time after planting (days), x 1 2 3Growth of plant (millimeters), y 14 28 42

Solution

It does make sense to talk about the amount a plant has grown after any length of time. So the table represents a continuous function. When x 5 0.5 days, y 5 7 millimeters

Example 2 Classify and graph a real-world function

Copyright © Holt McDougal. All rights reserved. 4.2 Focus On Functions• Algebra 1 Notetaking Guide 95

y

x1

1

y

x0 1 2 3 4

10

20

30

40

Plant Growth

Time after planting(days)

Gro

wth

of

pla

nt

(mill

imet

ers)

2. Tell whether the function represented by the table is discrete or continuous. Explain. If continuous, find the value of y when x 5 2.5. Round your answer to the nearest hundredth.

Number of songs downloaded, x 1 5 10

Cost of songs (dollars), y 1.25 6.25 12.50

Discrete; you cannot buy part of a song.


Homework

Your Notes


4.3 Graph Using InterceptsGoal p Graph a linear equation using intercepts.

VOCABULARY

x-intercept

y-intercept

Your Notes

Find the x-intercept and the y-intercept of the graph of 8x 2 2y 5 32.

Solution

1. Substitute for y and solve for x.

8x 2 2y 5 32 Write originalequation.

8x 2 2( ) 5 32 Substitute for y.

x 5 5 Solve for .

2. Substitute for x and solve for y.


8( ) 2 2y 5 32 Substitute for x.

y 5 5 Solve for .

The x-intercept is . The y-intercept is .

Example 1 Find the intercepts of the graph of an equation



4.3 Graph Using InterceptsGoal p Graph a linear equation using intercepts.

VOCABULARY

x-intercept The x-coordinate of a point where a graph crosses the x-axis

y-intercept The y-coordinate of a point where a graph crosses the y-axis

Your Notes

Find the x-intercept and the y-intercept of the graph of 8x 2 2y 5 32.

Solution

1. Substitute 0 for y and solve for x.


8x 2 2( 0 ) 5 32 Substitute 0 for y.

x 5 8

32 5 4 Solve for x .

2. Substitute 0 for x and solve for y.


8( 0 ) 2 2y 5 32 Substitute 0 for x.

y 5 22

32 5 216 Solve for y .

The x-intercept is 4 . The y-intercept is 216 .

Example 1 Find the intercepts of the graph of an equation



Checkpoint Find the x-intercept and y-intercept of the graph of the equation.

1. 2x 1 3y 5 18 2. 212x 2 4y 5 36

Graph 3.5x 1 2y 5 14. Label the points where the line crosses the axis.

Solution

Step 1 Find the .

3.5x 1 2y 5 14 3.5x 1 2y 5 14

3.5x 1 2( ) 5 14 3.5( ) 1 2y 5 14

x 5 5

y 5 5

Step 2 Plot the points that correspond to the intercepts.

The x-intercept is , so plot the point .

The y-intercept is , so plot the point .

Step 3 the points by drawing a line through them.

x

y

1

3

5

7

12121

23 3 5

CHECK

You can check the graph of the equation by using a third point. When x 5 2, y 5 , so the ordered pair

is a third solution of the equation. You can see that lies on the graph, so the graph is correct.

Example 2 Use intercepts to graph an equation


Your Notes


Checkpoint Find the x-intercept and y-intercept of the graph of the equation.

1. 2x 1 3y 5 18 2. 212x 2 4y 5 36

x-intercept: 9 x-intercept: 23 y-intercept: 6 y-intercept: 29

Graph 3.5x 1 2y 5 14. Label the points where the line crosses the axis.

Solution

Step 1 Find the intercepts .

3.5x 1 2y 5 14 3.5x 1 2y 5 14

3.5x 1 2( 0 ) 5 14 3.5( 0 ) 1 2y 5 14

x 5

3.5

14 5 4

y 5

2

14 5 7

Step 2 Plot the points that correspond to the intercepts.

The x-intercept is 4 , so plot the point (4, 0) .

The y-intercept is 7 , so plot the point (0, 7) .

Step 3 Connect the points by drawing a line through them.

x

y

1

3

5

7

12121

23 3 5

CHECK

You can check the graph of the equation by using a third point. When x 5 2, y 5 3.5 , so the ordered pair (2, 3.5) is a third solution of the equation. You can see that (2, 3.5) lies on the graph, so the graph is correct.

Example 2 Use intercepts to graph an equation


Your Notes


3. Graph 2x 2 7y 5 14. Label the points where the line crosses the axes.

x

y

1

3

12121

23

3 5 7

4. Identify the x-intercept and y-intercept of the graph.

x

y

1

3

212321

3


Homework

Identify the x-intercept and y-intercept of the graph.

x

y

1

3

12121

23

23 3

SolutionTo find the x-intercept, look to see where the graph crosses the . The x-intercept is . To find the y-intercept, look to see where the graph crosses the . The y-intercept is .

Example 3 Use a graph to find the intercepts


Your Notes


3. Graph 2x 2 7y 5 14. Label the points where the line crosses the axes.

x

y

1

3

12121

23

3 5 7

(7, 0)

(0, 22)

4. Identify the x-intercept and y-intercept of the graph.

x

y

1

3

212321

3

x-intercept 5 1

y-intercept 5 3


Homework

Identify the x-intercept and y-intercept of the graph.

x

y

1

3

12121

23

23 3

SolutionTo find the x-intercept, look to see where the graph crosses the x-axis . The x-intercept is 22 . To find the y-intercept, look to see where the graph crosses the y-axis . The y-intercept is 2 .

Example 3 Use a graph to find the intercepts


Your Notes


4.4 Find Slope and Rate of ChangeGoal p Find the slope of a line and interpret slope as a

rate of change.

VOCABULARY

Slope

Rate of change

FINDING THE SLOPE OF A LINE

Words Symbols

The slope of the nonvertical line m 5 y2 2 y1 } x2 2 x1

passing through the two points (x1, y1) and (x2, y2) is the ratio of the (change in y) to the

(change in x).

slope 5 5 change in y

} change in x

Graph

x

y

(x1, y1)

(x2, y2)

runx22 x1

risey22 y1


Your Notes


4.4 Find Slope and Rate of ChangeGoal p Find the slope of a line and interpret slope as a

rate of change.

VOCABULARY

Slope The ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on a nonvertical line; Represented by m

Rate of change Compares a change in one quantity to a change in another quantity

FINDING THE SLOPE OF A LINE

Words Symbols

The slope of the nonvertical line m 5 y2 2 y1 } x2 2 x1

passing through the two points (x1, y1) and (x2, y2) is the ratio of the rise (change in y) to the run (change in x).

slope 5 run

rise 5

change in y } change in x

Graph

x

y

(x1, y1)

(x2, y2)

runx22 x1

risey22 y1


Your Notes


Find the slope of the line shown.

a. Let (x1, y1) 5 (21, 2) b. Let (x1, y1) 5 (1, 4) and (x2, y2) 5 (3, 5). and (x2, y2) 5 (3, 22).

x

y

1

3

5

12121

23 3

x

y

1

3

5

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1

Write formula for slope.

5 2 2

2 (21) Substitute.

5 Simplify. The line from left to right. The slope is

.

b. m 5 y2 2 y1 } x2 2 x1


5 2 4

2 1 Substitute.

5 5 Simplify. The line from left to right. The slope is

.

Example 1 Find a slope

Checkpoint Find the slope of the line passing through the points.

1. (23, 21) and (22, 1) 2. (26, 3) and (5, 22)

Keep the x- and y-coordinates in the same order in the numerator and denominator when calculating slope. This will help avoid error.


Your Notes




x

y

1

3

5

12121

23 3

3

4

m 5 34

x

y

1

3

5

12121

23

23 3 5

2

26 m 5 23

Solution

a. m 5 y2 2 y1 } x2 2 x1


5

5 2 2

3 2 (21) Substitute.

5 3 } 4 Simplify. The line rises from left to right. The slope is

positive .

b. m 5 y2 2 y1 } x2 2 x1


5

22 2 4

3 2 1 Substitute.

5 26 } 2 5 23 Simplify. The line falls from left to right. The slope is

negative .

Example 1 Find a slope

Checkpoint Find the slope of the line passing through the points.

1. (23, 21) and (22, 1) 2. (26, 3) and (5, 22)

2 2 5 } 11

Keep the x- and y-coordinates in the same order in the numerator and denominator when calculating slope. This will help avoid error.


Your Notes




x

y

1

3

12121

23 3

x

y

1

3

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1


5 5 2

24 2 Substitute.

5 Simplify. The line is . The slope is .

b. m 5 y2 2 y1 } x2 2 x1


5 3 2

4 2 Substitute.

5 Simplify. The line is . The slope is .

Example 2 Find the slope of a line

Checkpoint Find the slope of the line passing through the points. Then classify the line by its slope.

3. (1, 22) and (1, 3) 4. (23, 7) and (4, 7)


Your Notes




x

y

1

3

12121

23 3

x

y

1

3

12121

23

23 3 5

Solution

a. m 5 y2 2 y1 } x2 2 x1


5

55 2

224 2 Substitute.

5 0 } 26 Simplify.

The line is horizontal . The slope is zero .

b. m 5 y2 2 y1 } x2 2 x1


5

(22)3 2

4 4 2 Substitute.

5 5 } 0 Simplify. The line is vertical . The slope is undefined .

Example 2 Find the slope of a line

Checkpoint Find the slope of the line passing through the points. Then classify the line by its slope.

3. (1, 22) and (1, 3) 4. (23, 7) and (4, 7)

undefined; vertical 0; horizontal


Your Notes


Gas Prices The table shows the cost of a gallon of gas for a number of days. Find the rate of change with respect to time.

Time (days) Day 1 Day 3 Day 5

Price/gal ($) 1.99 2.09 2.19

Rate of change 5 change in cost

}} change in time Write formula.

5 2.09 2

3 2 Substitute.

5 5 Simplify.

The rate of change in price is per day.

Example 3 Find a rate of change

5. The table shows the change in temperature over time. Find the rate of change in degrees Fahrenheit with respect to time.

Temperature (8F) Time (hours)

38 0

43 2

48 4

53 6

Checkpoint

Homework


Your Notes


Gas Prices The table shows the cost of a gallon of gas for a number of days. Find the rate of change with respect to time.

Time (days) Day 1 Day 3 Day 5

Price/gal ($) 1.99 2.09 2.19

Rate of change 5 change in cost

}} change in time Write formula.

5 1.992.09 2

13 2 Substitute.

5 2

0.1 5 0.05 } 1 Simplify.

The rate of change in price is 5 cents per day.

Example 3 Find a rate of change

5. The table shows the change in temperature over time. Find the rate of change in degrees Fahrenheit with respect to time.

Temperature (8F) Time (hours)

38 0

43 2

48 4

53 6

2.58F per hour

Checkpoint

Homework


Your Notes


4.5 Graph Using Slope-Intercept FormGoal p Graph linear equations using slope-intercept form.

VOCABULARY

Slope-intercept form

Parallel


FINDING THE SLOPE AND Y-INTERCEPT OF A LINE

Words Symbols

A linear equation of y 5 mx 1 bthe form y 5 mx 1 b is written in

where is the slope and is the y-intercept of the equation's graph. y 5 2x 1 1

Graph

x

y

1

3

12121

23

23 3 5

Your Notes


4.5 Graph Using Slope-Intercept FormGoal p Graph linear equations using slope-intercept form.

VOCABULARY

Slope-intercept form A linear equation written in the form y 5 mx 1 b

Parallel Two lines in the same plane that do not intersect


FINDING THE SLOPE AND Y-INTERCEPT OF A LINE

Words Symbols

A linear equation of y 5 mx 1 bthe form y 5 mx 1 b is written in slope-intercept form slope y-intercept where m is the slope and b is the y-intercept of the equation's graph. y 5 2x 1 1

Graph

x

y

1

3

12121

23

23 3 5

2

1 y 5 2x 1 1

Your Notes


Identify the slope and y-intercept of the line with the given equation.

a. y 5 x 1 3 b. 22x 1 y 5 5

Solution

a. The equation is in the form . So, the slope of the line is , and the y-intercept is .

b. Rewrite the equation in slope-intercept form by solving for .

22x 1 y 5 5 Write original equation.

y 5 Subtract from each side.

The line has a slope of and a y-intercept of .

Example 1 Identify slope and y-intercept

Checkpoint Identify the slope and y-intercept of the line with the given equation.

1. y 5 4x 2 1 2. 4x 2 2y 5 8

3. 4y 5 3x 1 16 4. 6x 1 3y 5 221


Your Notes


Identify the slope and y-intercept of the line with the given equation.

a. y 5 x 1 3 b. 22x 1 y 5 5

Solution

a. The equation is in the form y 5 mx 1 b . So, the slope of the line is 1 , and the y-intercept is 3 .

b. Rewrite the equation in slope-intercept form by solving for y .

22x 1 y 5 5 Write original equation.

y 5 2x 1 5 Subtract 22x from each side.

The line has a slope of 2 and a y-intercept of 5 .

Example 1 Identify slope and y-intercept

Checkpoint Identify the slope and y-intercept of the line with the given equation.

1. y 5 4x 2 1 2. 4x 2 2y 5 8

slope: 4 slope: 2

y-intercept: 21 y-intercept: 24

3. 4y 5 3x 1 16 4. 6x 1 3y 5 221

slope: 3 } 4 slope: 22

y-intercept: 4 y-intercept: 27


Your Notes



5. Graph the equation 2 1 } 2 x 1 y 5 1.

x

y

1

3

12121

23

23 3


Graph the equation 4x 1 y 5 2.

SolutionStep 1 Rewrite the equation in slope-intercept form.

Step 2 the slope and the y-intercept.

m 5 b 5

Step 3 the point that corresponds to the y-intercept, ( ).

Step 4 Use the slope to locate a second point on the line. Draw a line through the two points.

x

y

1

3

12121

23

23 3

Example 2 Graph an equation using slope-intercept formYour Notes



5. Graph the equation 2 1 } 2 x 1 y 5 1.

m 5 1 } 2

x

y

1

3

12121

23

23 3

21

(0, 1)(2, 2) b 5 1


Graph the equation 4x 1 y 5 2.

SolutionStep 1 Rewrite the equation in slope-intercept form.

y 5 24x 1 2

Step 2 Identify the slope and the y-intercept.

m 5 –4 b 5 2

Step 3 Plot the point that corresponds to the y-intercept, ( 0, 2 ).

Step 4 Use the slope to locate a second point on the line. Draw a line through the two points.

x

y

1

3

12121

23

23 3

1

24

Example 2 Graph an equation using slope-intercept formYour Notes


Homework

Determine which of the lines are parallel.

x

y

1

3

12121

25

23 5

Line a

Line c

Line b

Solution

Find the slope of each line.

Line a: m 5 2 3

2 4 5 5

Line b: m 5 2 4

2 2 5 5

Line c: m 5 2 2

2 6 5 5

Lines and have the same slope. They are parallel.

Example 3 Identify parallel lines

6. Determine which lines are parallel.

Line a: through (2, 5) and (22, 2)

Line b: through (4, 1) and (23, 24)

Line c: through (2, 3) and (22, 0)



Your Notes


Homework

Determine which of the lines are parallel.

x

y

1

3

12121

25

23 5

Line a

Line c

Line b

Solution

Find the slope of each line.

Line a: m 5

23 2 3

22 2 4 5

26

26 5 1

Line b: m 5

22 2 4

24 2 2 5

26

26 5 1

Line c: m 5

25 2 2

23 2 6 5

29

27 5

7 } 9

Lines a and b have the same slope. They are parallel.

Example 3 Identify parallel lines

6. Determine which lines are parallel.

Line a: through (2, 5) and (22, 2)

Line b: through (4, 1) and (23, 24)

Line c: through (2, 3) and (22, 0)

Lines a and c are parallel with slope 3 } 4 .



Your Notes



Solve Linear Equations by Graphing Goal p Use graphs to solve linear equations.

SOLVING LINEAR EQUATIONS GRAPHICALLY

Step 1 Write the equation in the form ax 1 b 5 0.

Step 2 Write the related function .

Step 3 Graph the equation .

The solution of ax 1 b 5 0 is the of the graph of .

Solve 4 } 3 x 1 1 5 2x graphically. Check your solution

algebraically.

Solution

Step 1 Write the equation in the form 1 5 0.

4 } 3 x 1 1 5 2x Write original equation.

1 5 0 Subtract from each side.

Step 2 Write the y 5 .

Step 3 Graph y 5 .

The is .

The solution of 4 } 3 x 1 1 5 2x is .

CHECK

4 } 3 ( ) 1 1 5 2( ) Substitute.

1 1 5 Simplify.

5 Solution checks.

Example 1 Solve an equation graphically


y

x1

1

Your Notes



Solve Linear Equations by Graphing Goal p Use graphs to solve linear equations.

SOLVING LINEAR EQUATIONS GRAPHICALLY


Step 2 Write the related function y 5 ax 1 b .

Step 3 Graph the equation y 5 ax 1 b .

The solution of ax 1 b 5 0 is the x-intercept of the graph of y 5 ax 1 b .

Solve 4 } 3 x 1 1 5 2x graphically. Check your solution

algebraically.

Solution


4 } 3 x 1 1 5 2x Write original equation.

2 2 } 3 x 1 1 5 0 Subtract 2x from each side.

Step 2 Write the related function y 5 2 2 } 3 x 1 1.

Step 3 Graph y 5 2 2 } 3 x 1 1.

The x-intercept is 3 } 2 .

The solution of 4 } 3 x 1 1 5 2x is 3 } 2 .

CHECK

4 } 3 ( 3 } 2 ) 1 1 5 2( 3 } 2 ) Substitute.

2 1 1 5 3 Simplify.

3 5 3 Solution checks.

Example 1 Solve an equation graphically


y

x1

1

Your Notes


1. Solve the equation graphically. Check your solution. 7x − 3 5 4x

2. Internet The number s (in millions) of subscribers to an Internet service can be modeled by the function s 5 1.85t 1 63.4 where t is the number of years since 1997. Use a graphing calculator to approximate the year when the number of subscribers will be 100 million.

The number c (in millions) of cell phone subscribers can be modeled by the function c 5 2.79t 1 45.1 where t is the number of years since 1999. In approximately what year was the total number of users 50 million?

Solution

Substitute for c in the linear model. The resulting linear equation is 5 1 .

Step 1 Write the equation in the form 1 5 0.

0 5 Subtract from each side.

0 5 Substitute for .

Step 2 Write the related function: y 5 .

Step 3 Graph the related function on a graphing calculator. Draw the line you obtain. The x-intercept is closest to .

Because x is the number of years since , you can estimate that the number

of cell phone users reached 50 million in about years, or in .

Example 2 Approximate a real-world solution



Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1

y

x1

1

Homework

Your Notes


1. Solve the equation graphically. Check your solution. 7x − 3 5 4x

The solution is 1.

7(1) 2 3 5 4(1)

4 5 4

2. Internet The number s (in millions) of subscribers to an Internet service can be modeled by the function s 5 1.85t 1 63.4 where t is the number of years since 1997. Use a graphing calculator to approximate the year when the number of subscribers will be 100 million.

2017

The number c (in millions) of cell phone subscribers can be modeled by the function c 5 2.79t 1 45.1 where t is the number of years since 1999. In approximately what year was the total number of users 50 million?

Solution

Substitute 50 for c in the linear model. The resulting linear equation is 50 5 2.79t 1 45.1 .


0 5 2.79t 2 4.9 Subtract 50 from each side.

0 5 2.79x 2 4.9 Substitute x for t .

Step 2 Write the related function: y 5 2.79x 2 4.9 .

Step 3 Graph the related function on a graphing calculator. Draw the line you obtain. The x-intercept is closest to 2 .

Because x is the number of years since 1999 , you can estimate that the number of cell phone users reached 50 million in about 2 years, or in 2001 .



X�1.76 Y�0.01


Use this viewing window:Xmin 5 –1Xmax 5 3Xsci 5 1Ymin 5 –7Ymax 5 3Ysci 5 1

y

x1

1

Homework

Your Notes


4.6 Model Direct VariationGoal p Write and graph direct variation equations.

VOCABULARY

Direct variation

Constant of variation

Tell whether the equation represents direct variation. If so, identify the constant of variation.

a. 4x 1 2y 5 0 b. 22x 1 y 5 3

SolutionTo tell whether an equation represents direct variation, try to rewrite the equation in the form y 5 ax.

a. 4x 1 2y 5 0 Write original equation.

2y 5 Subtract from each side.

y 5 Simplify.

Because the equation 4x 1 2y 5 0 be rewritten in the form y 5 ax, it direct variation. The constant of variation is .

b. 22x 1 y 5 3 Write original equation.

y 5 1 3 Add to each side.

Because the equation 22x 1 y 5 3 be rewritten in the form y 5 ax, it direct variation.

Example 1 Identify direct variation equations


Your Notes


4.6 Model Direct VariationGoal p Write and graph direct variation equations.

VOCABULARY

Direct variation The relationship of two variables, x and y, provided y 5 ax and a Þ 0

Constant of variation In y 5 ax, a is called the constant of variation.

Tell whether the equation represents direct variation. If so, identify the constant of variation.

a. 4x 1 2y 5 0 b. 22x 1 y 5 3

SolutionTo tell whether an equation represents direct variation, try to rewrite the equation in the form y 5 ax.

a. 4x 1 2y 5 0 Write original equation.

2y 5 24x Subtract 4x from each side.

y 5 22x Simplify.

Because the equation 4x 1 2y 5 0 can be rewritten in the form y 5 ax, it represents direct variation. The constant of variation is 22 .

b. 22x 1 y 5 3 Write original equation.

y 5 2x 1 3 Add 2x to each side.

Because the equation 22x 1 y 5 3 cannot be rewritten in the form y 5 ax, it does not represent direct variation.

Example 1 Identify direct variation equations


Your Notes


Checkpoint Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. 3x 1 4y 5 0 2. 5x 1 y 5 1

Graph the direct variation equation.

a. y 5 25x b. y 5 3 } 5 x

Solutiona. Plot a point at the origin. The slope is equal to the

constant of variation, or . Find and plot a second point, then draw a line through the points.

x

y

1

12121

23

25

23 3

b. Plot a point at the origin. The slope is equal to the

constant of variation, or . Find and plot a second point, then draw a line through the points.

x

y

1

3

5

12121

23 3 5

Example 2 Graph direct variation equations

The graph of a direct variation equation is a line with a slope of a and a y-intercept of 0. This line passes through the origin.


Your Notes


Checkpoint Tell whether the equation represents direct variation. If so, identify the constant of variation.

1. 3x 1 4y 5 0 2. 5x 1 y 5 1

yes; 2 3 } 4 no

Graph the direct variation equation.

a. y 5 25x b. y 5 3 } 5 x

Solutiona. Plot a point at the origin. The slope is equal to the

constant of variation, or 25 . Find and plot a second point, then draw a line through the points.

x

y

1

12121

23

25

23 3

1

25

(0, 0)

(1, 25)

b. Plot a point at the origin. The slope is equal to the

constant of variation, or 3 } 5 . Find and plot a second

point, then draw a line through the points.

x

y

1

3

5

12121

23 3 5

5

3

(0, 0)

(5, 3)

Example 2 Graph direct variation equations

The graph of a direct variation equation is a line with a slope of a and a y-intercept of 0. This line passes through the origin.


Your Notes


Homework

Your Notes

The graph of a direct variation

x

y3

121

23

25

23 3

(0, 0)

(24, 23)

equation is shown.

a. Write the direct variation equation.

b. Find the value of y when x 5 80.

Solutiona. Because y varies directly with x, the equation has the

form y 5 ax. Use the fact that y 5 23 when x 5 24 to find a.

y 5 ax Write direct variation equation.

5 a( ) Substitute.

5 a Solve for a. A direct variation equation that relates x and y is

y 5 .

b. When x 5 80, y 5 5 .

Example 3 Write and use a direct variation equation

3. Graph the direct variation equation y 5 1 } 2 x.

x

y

1

3

12121

23 3

4. The graph of a direct variation equation passes through the point (3, 24). Write the direct variation equation and find the value of y when x 5 15.




Homework

Your Notes

The graph of a direct variation

x

y3

121

23

25

23 3

(0, 0)

(24, 23)

equation is shown.

a. Write the direct variation equation.

b. Find the value of y when x 5 80.

Solutiona. Because y varies directly with x, the equation has the

form y 5 ax. Use the fact that y 5 23 when x 5 24 to find a.

y 5 ax Write direct variation equation.

23 5 a( 24 ) Substitute.

3 } 4 5 a Solve for a.

A direct variation equation that relates x and y is

y 5 3 } 4 x .

b. When x 5 80, y 5 3 } 4 (80) 5 60 .

Example 3 Write and use a direct variation equation

3. Graph the direct variation equation y 5 1 } 2 x.

x

y

1

3

12121

23 3

21

(0, 0)

(2, 1)

4. The graph of a direct variation equation passes through the point (3, 24). Write the direct variation equation and find the value of y when x 5 15.

y 5 2 4 } 3 x; y 5 220




4.7 Graph Linear FunctionsGoal p Use function notation.

VOCABULARY

Function notation

Family of functions

Parent linear function

For the function f(x) 5 3x 1 1, find the value of x so that f(x) 5 10.

Solution f(x) 5 3x 1 1 Write original equation.

5 3x 1 1 Substitute for f(x).

5 x Solve for x.

When x 5 , f(x) 5 10.

Example 1 Find an x-value

1. For f(x) 5 6x 2 6, find the value of x so that f(x) 5 24.



Your Notes



4.7 Graph Linear FunctionsGoal p Use function notation.

VOCABULARY

Function notation A way to name a function that is defined by an equation; f(x) 5 mx 1 b

Family of functions A group of functions with similar characteristics

Parent linear function The most basic linear function in a family of linear equations: f (x) 5 x

For the function f(x) 5 3x 1 1, find the value of x so that f(x) 5 10.

Solution f(x) 5 3x 1 1 Write original equation.

10 5 3x 1 1 Substitute 10 for f(x).

3 5 x Solve for x.

When x 5 3 , f(x) 5 10.

Example 1 Find an x-value


x 5 5


x 5 2


Your Notes



Text Messages A wireless communication provider estimates that the number of text messages m (in millions) sent over several years can be modeled by the function m 5 120t 1 95 where t represents the number of years since 2002. Graph the function and identify its domain and range.

t m

0

1

2

3

t

m

10 2 3 4 5

1000

200

300

400

500

600

Years since 2002

Text

messag

es (

millio

ns)

The domain of the function is t ≥ . From the graph or table, you can see that the range of the function is m ≥ .


3. Use the model from Example 2 to find the value of t so that m 5 1055. Explain what the solution means in this situation.


PARENT FUNCTION FOR LINEAR FUNCTIONS

1. The is the most basic linear function.

2. is the form of the parent linear function.


Your Notes


Text Messages A wireless communication provider estimates that the number of text messages m (in millions) sent over several years can be modeled by the function m 5 120t 1 95 where t represents the number of years since 2002. Graph the function and identify its domain and range.

t m

0 95

1 215

2 335

3 455

t

m

10 2 3 4 5

1000

200

300

400

500

600

Years since 2002

Text

messag

es (

millio

ns)

The domain of the function is t ≥ 0 . From the graph or table, you can see that the range of the function is m ≥ 95 .


3. Use the model from Example 2 to find the value of t so that m 5 1055. Explain what the solution means in this situation.

8; There will be over one billion text messages in 2010.


PARENT FUNCTION FOR LINEAR FUNCTIONS

1. The parent linear function is the most basic linear function.

2. f (x) 5 x is the form of the parent linear function.


Your Notes


Graph the function. Compare the graph with the graph of f(x) 5 x.

a. p(x) 5 x 2 4 b. q(x) 5 22x

Solutiona.

x

y

1

121

23

23 3 5

p(x) 5 x 2 4

b.

x

y

3

121

23

23 3

q(x) 5 22x

Because the graphs of Because the slope of p and f have the same the graph of q slope, m 5 1, the lines from left to right and are . Also, the slope of the graphthe y-intercept of the of f from left graph of p is less to right, the slope of than the y-intercept of q is . Thethe graph of f. y-intercept of both graphs is .

Example 3 Compare graphs with the graph f(x) 5 x

4. Graph r(x) 5 1 } 2 x. Compare the graph with the graph

of f (x) 5 x.

x

y

3

1

1

23

23 3



Your Notes


Graph the function. Compare the graph with the graph of f(x) 5 x.

a. p(x) 5 x 2 4 b. q(x) 5 22x

Solutiona.

x

y

1

121

23

23 3 5

p(x) 5 x 2 4

b.

x

y

3

121

23

23 3

q(x) 5 22x

Because the graphs of Because the slope of p and f have the same the graph of q falls slope, m 5 1, the lines from left to right and are parallel . Also, the slope of the graphthe y-intercept of the of f rises from left graph of p is 4 less to right, the slope of than the y-intercept of q is negative . Thethe graph of f. y-intercept of both graphs is 0 .

Example 3 Compare graphs with the graph f(x) 5 x

4. Graph r(x) 5 1 } 2 x. Compare the graph with the graph

of f (x) 5 x.

x

y

3

1

1

23

23 3

r (x) 5 x12

The slope of the graph of r is less than the slope of f. The y-intercept for both graphs is 0.



Your Notes


Homework


COMPARING GRAPHS OF LINEAR FUNCTIONS WITH THE GRAPH OF f(x) 5 x

g(x) 5 x 1 b The graphs have the same

x

y

3

1

121

23

23 3

.

The graphs have different .

Graphs of this family are of

the graph of f(x) 5 x.

g(x) 5 mx where m > 0 The graphs have different

x

y

3

1

1

23

23 3

(positive) .

The graphs have the same .

Graphs of this family are vertical or of the graph of f(x) 5 x.

g(x) 5 mx where m < 0 The graphs have different

x

y

3

121

23

23 3

(negative) .

The graphs have the same .

Graphs of this family are vertical or or of the graph of f(x) 5 x.

Your Notes


Homework


COMPARING GRAPHS OF LINEAR FUNCTIONS WITH THE GRAPH OF f(x) 5 x

g(x) 5 x 1 b The graphs have the same

x

y

3

1

121

23

23 3

slope .

The graphs have different y-intercepts .

Graphs of this family are vertical translations of

the graph of f(x) 5 x.

g(x) 5 mx where m > 0 The graphs have different

x

y

3

1

1

23

23 3

(positive) slopes .

The graphs have the same y-intercept .

Graphs of this family are vertical stretches or shrinks of the graph of f(x) 5 x.

g(x) 5 mx where m < 0 The graphs have different

x

y

3

121

23

23 3

(negative) slopes .

The graphs have the same y-intercept .

Graphs of this family are vertical stretches or shrinks or reflections of the graph of f(x) 5 x.

Your Notes


Words to Review


Quadrant Transformation

Translation Vertical stretch

Vertical shrink Reflection

Give an example of the vocabulary word.


Words to Review


Quadrant

x

y

1

12123 321

Quadrant I

Quadrant IV

Quadrant II

Quadrant III

Transformation

(x, y) → (image x, image y)

Translation

(x, y) → (x 2 2, y)

y

x1

1

Vertical stretch

(x, y) → (x, 2y)

y

x

1

1

Vertical shrink

(x, y) → (x, 1 } 3 y)

y

x

11

Reflection

(x, y) → (x, –y)y

x1

1

Give an example of the vocabulary word.



Solution of an equation in two variables.


Linear equation Standard form of a linear equation

Linear function Discrete function

Continuous function x-intercept

y-intercept Slope



Solution of an equation in two variables.

x 1 y 5 2

(2, 0)


x

y

1

3

12123 321

23

(2, 1)

(1, 21)

(21, 25)

Linear equation

2x 1 4y 5 8

Standard form of a linear equation

Ax 1 By 5 C

Linear function

y 5 x 1 1

Discrete functiony

x1

1

Continuous functiony

x1

1

x-intercept

The x-intercept of 2x 1 3y 5 18 is 9.

y-intercept

The y-intercept of 2x 1 3y 5 18 is 6.

Slope

slope 5 rise } run

5 change in y

} change in x



Rate of change Slope-intercept form

Parallel

Direct variation

Constant of variation Function notation

Family of functions Parent linear function




Rate of change

change in cost

}} change in time

Slope-intercept form

y 5 mx 1 b

Parallel

Lines in the same plane that do not intersect

Direct variation

22x 1 y 5 0

y 5 2x

Constant of variation

y 5 2x

constant of variation 5 2

Function notation

f (x) 5 mx 1 b

Family of functions

Of the form:

f (x) 5 mx 1 b

family of linear functions

Parent linear function

f (x) 5 x




Write Linear Equations inSlope-Intercept FormGoal p Write equations of lines.

5.1

Your Notes

Write an equation of the line with a slope of 24 and a y-intercept of 6.

Solutiony 5 mx 1 b Write slope-intercept form.

y 5 x 1 Substitute for m and for b.

Example 1 Use slope and y-intercept to write an equation

Checkpoint Write an equation of the line with the given slope and y-intercept.

1. Slope is 8; 2. Slope is 2 } 3 ;

y-intercept is 25. y-intercept is 22.

3. Slope is 23; 4. Slope is 2 5 } 2 ;


Use the slope-intercept form (y 5 mx 1 b) to write an equation of a line if slope and y-intercept are given.



Write Linear Equations inSlope-Intercept FormGoal p Write equations of lines.

5.1

Your Notes

Write an equation of the line with a slope of 24 and a y-intercept of 6.

Solutiony 5 mx 1 b Write slope-intercept form.

y 5 24 x 1 6 Substitute 24 for m and 6 for b.

Example 1 Use slope and y-intercept to write an equation

Checkpoint Write an equation of the line with the given slope and y-intercept.

1. Slope is 8; 2. Slope is 2 } 3 ;


y 5 8x 1 (25) y 5 2 } 3 x 1 (22)

3. Slope is 23; 4. Slope is 2 5 } 2 ;


y 5 23x 1 7 y 5 2 5 } 2 x 1 9

Use the slope-intercept form (y 5 mx 1 b) to write an equation of a line if slope and y-intercept are given.



Write an equation of the line shown.

SolutionStep 1 Calculate the slope.

x

y

1

3

12121

23

23 3

(3, 22)

(0, 4)

m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

Step 2 Write an equation of the line. The line crosses the y-axis at . So, the y-intercept is .

y 5 mx 1 b Write slope-intercept form.

y 5 x 1 Substitute for m and for b.

Example 2 Write an equation of a line given two points

You can write an equation of a line if you know the y-intercept and any other point on the line.

5. Write an equation of the line shown.

x

y1

12121

23

25

3 5

(5, 23)

(0, 26)


Your Notes



Write an equation of the line shown.


x

y

1

3

12121

23

23 3

(3, 22)

(0, 4)

m 5 y2 2 y1 } x2 2 x1

5 3

22

0

4

2

2

5 3

26 5 22

Step 2 Write an equation of the line. The line crosses the y-axis at (0, 4) . So, the y-intercept is 4 .


y 5 22 x 1 4 Substitute 22 for m and 4 for b.

Example 2 Write an equation of a line given two points

You can write an equation of a line if you know the y-intercept and any other point on the line.

5. Write an equation of the line shown.

y 5 3 } 5 x 2 6

x

y1

12121

23

25

3 5

(5, 23)

(0, 26)


Your Notes


Write an equation for the linear function f with the values f(0) 5 4 and f(2) 5 12.

Solution

Step 1 Write f(0) 5 4 as and f(2) 5 12 as .

Step 2 Calculate the slope of the line that passes through and .

m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Step 3 Write an equation of the line. The line crosses the y-axis at (0, ). So, the y-intercept is .


y 5 Substitute for m and for b.

The function is .

Example 3 Write a linear function

Homework

6. Write an equation for the linear function with the values f(0) 5 3 and f(3) 5 15.



Your Notes


Write an equation for the linear function f with the values f(0) 5 4 and f(2) 5 12.

Solution

Step 1 Write f(0) 5 4 as (0, 4) and f(2) 5 12 as (2, 12).

Step 2 Calculate the slope of the line that passes through (0, 4) and (2, 12).

m 5 y2 2 y1 } x2 2 x1

5 2

12

0

4

2

2

5 2

8

5 4

Step 3 Write an equation of the line. The line crosses the y-axis at (0, 4 ). So, the y-intercept is 4 .


y 5 4x 1 4 Substitute 4 for m and 4 for b.

The function is f (x) 5 4x 1 4.

Example 3 Write a linear function

Homework

6. Write an equation for the linear function with the values f(0) 5 3 and f(3) 5 15.

y 5 4x 1 3



Your Notes


5.2 Use Linear Equations inSlope-Intercept FormGoal p Write an equation of a line using points

on the line.

WRITING AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM

Step 1 Identify the slope . You can use the to calculate the slope if you know

two points on the line.

Step 2 Find the . You can substitute the and the of a point (x, y) on the line into y 5 mx 1 b. Then solve for .

Step 3 Write an equation using .

Your Notes

Write an equation of the line that passes through the point (1, 2) and has a slope of 3.

Solution

Step 1 Identify the slope. The slope is .

Step 2 Find the y-intercept. Substitute the slope and the coordinates of the given point into y 5 mx 1 b. Solve for b.


5 ( ) 1 b Substitute for m, for x, and for y.

5 b Solve for .

Step 3 Write an equation of the line.



Example 1 Write an equation given the slope and a point

Be careful not to mix up the x- and y-values when you substitute.

Your NotesYour Notes



5.2 Use Linear Equations inSlope-Intercept FormGoal p Write an equation of a line using points

on the line.

WRITING AN EQUATION OF A LINE IN SLOPE-INTERCEPT FORM

Step 1 Identify the slope m . You can use the slope formula to calculate the slope if you know two points on the line.

Step 2 Find the y-intercept . You can substitute the slope and the coordinates of a point (x, y) on the line into y 5 mx 1 b. Then solve for b .

Step 3 Write an equation using y 5 mx 1 b .

Your Notes

Write an equation of the line that passes through the point (1, 2) and has a slope of 3.

Solution

Step 1 Identify the slope. The slope is 3 .

Step 2 Find the y-intercept. Substitute the slope and the coordinates of the given point into y 5 mx 1 b. Solve for b.


2 5 3 ( 1 ) 1 b Substitute 3 for m, 1 for x, and 2 for y.

21 5 b Solve for b .




Example 1 Write an equation given the slope and a point

Be careful not to mix up the x- and y-values when you substitute.

Your NotesYour Notes



1. Write an equation of the line that passes through the point (2, 2) and has a slope of 4.


Write an equation of the line that passes through (2, 23) and (22, 1).


m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

Step 2 Find the y-intercept. Use the slope and the point (2, 23).


23 5 ( ) 1 b Substitute for m, for x, and

for y.

5 b Solve for b.




Example 2 Write an equation given two points

You can also find the y-intercept using the coordinates of the other given point.


Your Notes


1. Write an equation of the line that passes through the point (2, 2) and has a slope of 4.

y 5 4x 2 6


Write an equation of the line that passes through (2, 23) and (22, 1).


m 5 y2 2 y1 } x2 2 x1

5 22

1

2

(23)

2

2

5 24

4 5 21

Step 2 Find the y-intercept. Use the slope and the point (2, 23).



21 5 b Solve for b.




Example 2 Write an equation given two points

You can also find the y-intercept using the coordinates of the other given point.


Your Notes


2. Write an equation for the line that passes through (28, 213) and (4, 2).



HOW TO WRITE EQUATIONS IN SLOPE-INTERCEPT FORM

1. Given slope m and y-intercept b. Substitute and in the equation

.

2. Given slope m and one point. Substitute and the of the

point in . Solve for . Write the .

3. Given two points. Use the points to find the slope . Then

substitute and the of in . Solve for . Write

the .

Homework


Your Notes



y 5 5 } 4 x 2 3


y 5 2 3 } 2 x 2 1 } 2


HOW TO WRITE EQUATIONS IN SLOPE-INTERCEPT FORM

1. Given slope m and y-intercept b. Substitute m and b in the equation

y 5 mx 1 b .

2. Given slope m and one point. Substitute m and the coordinates of the

point in y 5 mx 1 b . Solve for b . Write the equation .

3. Given two points. Use the points to find the slope m . Then

substitute m and the coordinates of one point in y 5 mx 1 b . Solve for b . Write the equation .

Homework


Your Notes


5.3 Write Linear Equations inPoint-Slope FormGoal pWrite linear equations in point-slope form.

VOCABULARY

Point-slope form

POINT-SLOPE FORM

The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is

.

Write an equation in point-slope form on the line that passes through the point (3, 2) and has a slope of 2.

Solution y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 5 (x 2 ) Substitute for m, for x1, and for y1.

Example 1 Write an equation in point-slope form


Your Notes


5.3 Write Linear Equations inPoint-Slope FormGoal pWrite linear equations in point-slope form.

VOCABULARY

Point-slope form The equation of a nonvertical line y 2 y1 5 m(x 2 x1) that passes through a given point (x1, y1) with a slope m

POINT-SLOPE FORM

The point-slope form of the equation of the nonvertical line through a given point (x1, y1) with a slope of m is y 2 y1 5 m(x 2 x1) .

Write an equation in point-slope form on the line that passes through the point (3, 2) and has a slope of 2.

Solution y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 2 5 2 (x 2 3 ) Substitute 2 for m, 3 for x1, and 2 for y1.

Example 1 Write an equation in point-slope form


Your Notes


Graph the equation y 2 2 5 1 } 2 (x 2 2).

SolutionBecause the equation is in point-slope form, you know

that the line has a slope of and passes through the point .

Plot the point Find a second point on the line using the . Draw a line through the points.

x

y

1

3

5

12121

23 3 5

Example 2 Graph an equation in point-slope form

1. Write an equation in point-slope form of the line that passes through the point (23, 5) and has a slope of 4.

2. Graph the equation y 1 1 5 2(x 2 1).

x

y

1

3

12121

23

23 3



Your Notes


Graph the equation y 2 2 5 1 } 2 (x 2 2).

SolutionBecause the equation is in point-slope form, you know

that the line has a slope of 1 } 2 and passes through the point (2, 2) .

Plot the point (2, 2) Find a second point on the line using the slope . Draw a line through the points.

x

y

1

3

5

12121

23 3 5

12

(2, 2)(4, 3)

Example 2 Graph an equation in point-slope form

1. Write an equation in point-slope form of the line that passes through the point (23, 5) and has a slope of 4.

y 2 5 5 4(x 1 3)

2. Graph the equation y 1 1 5 2(x 2 1).

x

y

1

3

12121

23

23 3

12 (2, 1)

(1, 21)



Your Notes


Homework


Write an equation in point-slope form of the line shown.

x

y

1

3

2121

23

23 3

(4, 2)

(22, 23)

SolutionStep 1 Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 22

23

4

2

2

2

5 26

25 5

5 } 6

Step 2 Write the equation in point-slope form. You can use either given point.

Method 1 Use (22, 23). Method 2 Use (4, 2).

y 2 y1 5 m(x 2 x1) y 2 y1 5 m(x 2 x1)

y 1 3 5 5 } 6 (x 1 2) y 2 2 5 5 }

6 (x 2 4)

CHECK Check that the equations are equivalent by writing them in slope-intercept form.

y 1 3 5 5 } 6 x 1 5 } 3 y 2 2 5 5 } 6 x 2 10 } 3

y 5 5 } 6 x 2 4 } 3 y 5 5 } 6 x 2 4 } 3

Example 3 Use point-slope form to write an equationYour Notes


Homework


Write an equation in point-slope form of the line shown.

x

y

1

3

2121

23

23 3

(4, 2)

(22, 23)

SolutionStep 1 Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 2

2

5 5

Step 2 Write the equation in point-slope form. You can use either given point.

Method 1 Use (22, 23). Method 2 Use (4, 2).

y 2 y1 5 m(x 2 x1) y 2 y1 5 m(x 2 x1)

CHECK Check that the equations are equivalent by writing them in slope-intercept form.

y 5 x y 5 x

y 5 y 5

Example 3 Use point-slope form to write an equationYour Notes



Relate Arithmetic Sequences to Linear Functions Goal p Identify, graph, and write the general form

of arithmetic sequences.

VOCABULARY

Sequence

Arithmetic sequence

Common difference

Tell whether the sequence is arithmetic. If it is, find the next two terms.

26, 1, 8, 15, 22, …Solution

The first term is a1 5 . Find the of terms.

a2 2 a1 5 1 2 (26) 5 a32 a2 5 2 5

a4 2 a 5 2 5 a 2 a 5 2 5

The terms have a , d 5 . The sequence arithmetic. The next two terms are a6 5 and a7 5 .

Example 1 Identify an arithmetic sequence

Graph the sequence 26, 1, 8, 15, 22, …

Make a table pairing each term with its position number.

Solution

Position, x 1 2 3Term, y −6

Plot the pairs of numbers as points.

Example 2 Graph a sequence

The sequence in this example is a function. The domain is the set of positive numbers. The range is the set of terms.


x

y

1

5

Your Notes



Relate Arithmetic Sequences to Linear Functions Goal p Identify, graph, and write the general form

of arithmetic sequences.

VOCABULARY

Sequence an ordered list of numbers

Arithmetic sequence sequence with a constant difference between consecutive terms

Common difference the constant difference in an arithmetic sequence

Tell whether the sequence is arithmetic. If it is, find the next two terms.

26, 1, 8, 15, 22, …Solution

The first term is a1 5 26. Find the difference of consecutive terms.

a2 2 a1 5 1 2 (26) 5 7 a32 a2 5 8 2 1 5 7

a4 2 a 3 5 15 2 8 5 7 a 5 2 a4 5 22 2 15 5 7

The terms have a common difference , d 5 7 . The sequence is arithmetic. The next two terms are a6 5 29 and a7 5 36 .

Example 1 Identify an arithmetic sequence

Graph the sequence 26, 1, 8, 15, 22, …

Make a table pairing each term with its position number.

Solution

Position, x 1 2 3 4 5 Term, y −6 1 8 15 22

Plot the pairs of numbers as points.

Example 2 Graph a sequence

The sequence in this example is a function. The domain is the set of positive numbers. The range is the set of terms.


x

y

1

5

Your Notes



1. Tell whether the sequence is arithmetic. If it is, find the next two terms. 3, 22, 27, 212, 217, …

2. Graph the sequence 24, 21, 2, 5, 8,…. Identify the domain and range.

Domain:

Range:


KEY CONCEPT

Rule for an Arithmetic Sequence

The nth of an arithmetic with first term a1 and d is given by an 5 1 ( 2 ) .

Write a rule for the nth term of the sequence 25, 0, 5, 10, 15, …. Find a50.

Solution

The first term a1 5 . The common difference is d 5 .Substituting into the general rule an 5 1 ( 2 )

gives the rule an 5 1 ( 2 ) .

a50 5 1 ( 2 ) a50 5

Example 3 Write a rule for the nth term of a sequence

x

y

1

1

3. Write a rule for the nth term of the sequence and find a30 for: 18, 7, 24, 215, −26, …

a30 5

Checkpoint Complete the following exercises.Homework

Your Notes



1. Tell whether the sequence is arithmetic. If it is, find the next two terms. 3, 22, 27, 212, 217, …

Yes; next terms: 222, 227

2. Graph the sequence 24, 21, 2, 5, 8,…. Identify the domain and range.

Domain: 1, 2, 3, 4, 5,…

Range: 24, 21, 2, 5, 8,…


KEY CONCEPT

Rule for an Arithmetic Sequence

The nth term of an arithmetic sequence with first term a1 and common difference d is given by an 5 a1 1 ( n 2 1 ) d .

Write a rule for the nth term of the sequence 25, 0, 5, 10, 15, …. Find a50.

Solution

The first term a1 5 25 . The common difference is d 5 5 .Substituting into the general rule an 5 a1 1 ( n 2 1 ) d

gives the rule an 5 25 1 ( n 2 1)5.

a50 5 25 1 ( 50 2 1 )5 a50 5 240

Example 3 Write a rule for the nth term of a sequence

x

y

1

1

3. Write a rule for the nth term of the sequence and find a30 for: 18, 7, 24, 215, −26, …

an 5 18 1 (n 2 1)(211) a30 5 2301

Checkpoint Complete the following exercises.Homework

Your Notes


5.4 Write Linear Equations inStandard FormGoal p Write equations in standard form.

Write two equations in standard form that are equivalent to 4x 1 2y 5 12.

SolutionTo write one equivalent equation, multiply each side by .

To write one equivalent equation, multiply each side by .

Example 1 Write equivalent equations in standard form

1. Write two equations in standard form that are equivalent to 6x 2 4y 5 6.



Your Notes



5.4 Write Linear Equations inStandard FormGoal p Write equations in standard form.

Write two equations in standard form that are equivalent to 4x 1 2y 5 12.

SolutionTo write one equivalent equation, multiply each side by 0.5 .

2x 1 y 5 6

To write one equivalent equation, multiply each side by 2 .

8x 1 4y 5 24

Example 1 Write equivalent equations in standard form


3x 2 2y 5 3; 12x 2 8y 5 12


24x 1 2y 5 23; 224x 1 12y 5 218


Your Notes




Write an equation in standard form of the line shown.

x

y

1

3

2121

23

3 5 71

(2, 4)

(6, 24)


m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Step 2 Write an equation in point-slope form. Use (2, 4).

y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 5 (x 2 ) Substitute for y1, for m, and

for x1.

Step 3 Rewrite the equation in standard form.

y 2 5 x 1 Distributive property

y 1 x 5 Collect variable terms on one side, constants on the other.

Example 2 Write an equation from a graph

All linear equations can be written in standard form, Ax 1 By 5 C.

Your Notes



Write an equation in standard form of the line shown.

x

y

1

3

2121

23

3 5 71

(2, 4)

(6, 24)


m 5 y2 2 y1 } x2 2 x1

5 6

24

2

4

2

2

5 4

28

5 22

Step 2 Write an equation in point-slope form. Use (2, 4).

y 2 y1 5 m(x 2 x1) Write point-slope form.

y 2 4 5 22 (x 2 2 ) Substitute 4 for y1, 22 for m, and 2 for x1.

Step 3 Rewrite the equation in standard form.

y 2 4 5 22 x 1 4 Distributive property

y 1 2 x 5 8 Collect variable terms on one side, constants on the other.

Example 2 Write an equation from a graph

All linear equations can be written in standard form, Ax 1 By 5 C.

Your Notes


3. Write an equation in standard form of the line through (3, 21) and (2, 24).


Write an equation of the specified line.

a. Line A

x

y

1

212321

23

25

1

(3, 2)

(24, 26)Line B

Line Ab. Line B

Solutiona. The x-coordinate of the

given point on Line A is. This means that all

points on the line have an x-coordinate of . An equation of the line is .

b. The y-coordinate of the given point on Line B is . This means that all points on the line have a y-coordinate of . An equation of the line is

.

Example 3 Write an equation of a line


Your Notes


3. Write an equation in standard form of the line through (3, 21) and (2, 24).

y 2 3x 5 210


Write an equation of the specified line.

a. Line A

x

y

1

212321

23

25

1

(3, 2)

(24, 26)Line B

Line Ab. Line B

Solutiona. The x-coordinate of the

given point on Line A is 3 . This means that all points on the line have an x-coordinate of 3 . An equation of the line is x 5 3 .

b. The y-coordinate of the given point on Line B is 26 . This means that all points on the line have a y-coordinate of 26 . An equation of the line is y 5 26 .

Example 3 Write an equation of a line


Your Notes


Find the missing coefficient in the equation of the line shown. Write the completed equation.

x

y

1

3

5

21

23

23 3 51

(2, 1)

Ax 1 5y 5 23

SolutionStep 1 Find the value of A. Substitute the coordinates of

the given point for x and y in the equation.

Ax 1 5y 5 23 Write equation.

A( ) 1 5( ) 5 23 Substitute for x and for y.

A 1 5 23 Simplify.

A 5 Subtract from each side.

A 5 Divide by .

Step 2 Complete the equation.

x 1 5y 5 23 Substitute for A.

Example 4 Complete an equation in standard form

4. Write equations of the horizontal and vertical lines that pass through (210, 5).

5. Find the missing coefficient in the equation of the line that passes through (22, 2). Write the completed equation.

6x 1 By 5 4


Homework


Your Notes


Find the missing coefficient in the equation of the line shown. Write the completed equation.

x

y

1

3

5

21

23

23 3 51

(2, 1)

Ax 1 5y 5 23

SolutionStep 1 Find the value of A. Substitute the coordinates of

the given point for x and y in the equation.

Ax 1 5y 5 23 Write equation.

A( 2 ) 1 5( 1 ) 5 23 Substitute 2 for x and 1 for y.

2 A 1 5 5 23 Simplify.

2 A 5 28 Subtract 5 from each side.

A 5 24 Divide by 2 .

Step 2 Complete the equation.

24 x 1 5y 5 23 Substitute 24 for A.

Example 4 Complete an equation in standard form

4. Write equations of the horizontal and vertical lines that pass through (210, 5).

Horizontal: y 5 5; Vertical: x 5 210

5. Find the missing coefficient in the equation of the line that passes through (22, 2). Write the completed equation.

6x 1 By 5 4

B 5 8; 6x 1 8y 5 4


Homework


Your Notes


5.5 Write Equations of Parallel and Perpendicular LinesGoal p Write equations of parallel and perpendicular lines.

VOCABULARY

Converse

Perpendicular lines

PARALLEL LINES

If two nonvertical lines have the same , then they are .

If two nonvertical lines are , then they have the same .

Write an equation of the line that passes through (2, 4) and is parallel to the line y 5 4x 1 1.

SolutionStep 1 Identify the slope. The graph of the given equation

has a slope of . So, the parallel line through (2, 4) has a slope of .

Step 2 Find the y-intercept. Use the slope and the given point.



5 b Solve for b.

Step 3 Write an equation. Use y 5 mx 1 b.


Example 1 Write an equation of a parallel line


Your Notes


5.5 Write Equations of Parallel and Perpendicular LinesGoal p Write equations of parallel and perpendicular lines.

VOCABULARY

Converse A statement in which the hypothesis and conclusion of a conditional statement are interchanged

Perpendicular lines Two lines in a plane that intersect each other and form a right angle

PARALLEL LINES

If two nonvertical lines have the same slope , then they are parallel .

If two nonvertical lines are parallel , then they have the same slope .

Write an equation of the line that passes through (2, 4) and is parallel to the line y 5 4x 1 1.


has a slope of 4 . So, the parallel line through (2, 4) has a slope of 4 .




24 5 b Solve for b.

Step 3 Write an equation. Use y 5 mx 1 b.


Example 1 Write an equation of a parallel line


Your Notes


PERPENDICULAR LINES

If two nonvertical lines have slopes that are , then the lines are .

If two nonvertical lines are , then their slopes are .


Determine which of the following lines, if any, are parallel or perpendicular:

Line a: 12x 2 3y 5 3

Line b: y 5 4x 1 2

Line c: 4y 1 x 5 8

Solution

Find the slopes of the lines.

Line b: The equation is in slope-intercept form. The slope is .

Write the equations for lines a and c in slope-intercept form.

Line a: 12x 2 3y 5 3

23y 5 1 3

y 5

Line c: 4y 1 x 5 8

4y 5 1 8

y 5

Lines a and b have a slope of , so they are .

Line c has a slope of , the negative reciprocal of , so it is to lines a and b.

Example 2 Determine parallel or perpendicular lines

Your Notes


PERPENDICULAR LINES

If two nonvertical lines have slopes that are negative reciprocals , then the lines are perpendicular .

If two nonvertical lines are perpendicular , then their slopes are negative reciprocals .


Determine which of the following lines, if any, are parallel or perpendicular:

Line a: 12x 2 3y 5 3

Line b: y 5 4x 1 2

Line c: 4y 1 x 5 8

Solution

Find the slopes of the lines.

Line b: The equation is in slope-intercept form. The slope is 4 .

Write the equations for lines a and c in slope-intercept form.

Line a: 12x 2 3y 5 3

23y 5 212x 1 3

y 5 4x 2 1

Line c: 4y 1 x 5 8

4y 5 2x 1 8

y 5 2 1 } 4 x 1 2

Lines a and b have a slope of 4 , so they are parallel .

Line c has a slope of 2 1 } 4 , the negative reciprocal of 4 , so it is perpendicular to lines a and b.

Example 2 Determine parallel or perpendicular lines

Your Notes


1. Write an equation of the line that passes through (24, 6) and is parallel to the line y 5 23x 1 2.

2. Determine which of the following lines, if any, are parallel or perpendicular.

Line a: 4x 1 y 5 2

Line b: 5y 1 20x 5 10

Line c: 8y 5 2x 1 8


Determine if the following lines are perpendicular.

Line a: 6y 5 5x 1 8

Line b: 210y 5 12x 1 10

SolutionFind the slopes of the lines. Write the equations in slope-intercept form.

Line a: 6y 5 5x 1 8

y 5

Line b: 210y 5 12x 1 10

y 5

The slope of line a is . The slope of line b is .

The two slopes negative reciprocals, so lines a and b perpendicular.

Example 3 Determine whether lines are perpendicular


Your Notes


1. Write an equation of the line that passes through (24, 6) and is parallel to the line y 5 23x 1 2.

y 5 23x 2 6

2. Determine which of the following lines, if any, are parallel or perpendicular.

Line a: 4x 1 y 5 2

Line b: 5y 1 20x 5 10

Line c: 8y 5 2x 1 8

Lines a and b are parallel with a slope of 24. Line c is perpendicular to lines a and b with

a slope of 1 } 4 .


Determine if the following lines are perpendicular.

Line a: 6y 5 5x 1 8

Line b: 210y 5 12x 1 10

SolutionFind the slopes of the lines. Write the equations in slope-intercept form.

Line a: 6y 5 5x 1 8

y 5 5 } 6 x 1 4 }

3

Line b: 210y 5 12x 1 10

y 5 2 6 } 5 x 2 1

The slope of line a is 5 } 6 . The slope of line b is 2 6 }

5 .

The two slopes are negative reciprocals, so lines a and b are perpendicular.

Example 3 Determine whether lines are perpendicular


Your Notes


Write an equation of the line that passes through (23, 4)

and is perpendicular to the line y 5 1 } 3 x 1 2.


has a slope of . Because the slopes of

perpendicular lines are negative reciprocals, the slope of the perpendicular line through (23, 4) is .




5 b Solve for b.

Step 3 Write an equation.



Example 4 Write an equation of a perpendicular line

3. Determine whether line a through (1, 3) and (3, 4) is perpendicular to line b through (1, 23) and (2, 25). Justify your answer using slopes.

4. Write an equation of the line that passes through (4, 22) and is perpendicular to the line y 5 5x 1 2.



Homework

Your Notes


Write an equation of the line that passes through (23, 4)

and is perpendicular to the line y 5 1 } 3 x 1 2.


has a slope of 1 } 3 . Because the slopes of

perpendicular lines are negative reciprocals, the

slope of the perpendicular line through (23, 4) is 23 .




25 5 b Solve for b.

Step 3 Write an equation.



Example 4 Write an equation of a perpendicular line

3. Determine whether line a through (1, 3) and (3, 4) is perpendicular to line b through (1, 23) and (2, 25). Justify your answer using slopes.

Line a: m 5 1 } 2 ; Line b: m 5 22; perpendicular

4. Write an equation of the line that passes through (4, 22) and is perpendicular to the line y 5 5x 1 2.

y 5 2 1 } 5 x 2 6 }

5



Homework

Your Notes


5.6 Fit a Line to DataGoal p Make scatter plots and write equations to

model data.

VOCABULARY

Scatter plot

Correlation

Line of fit

CORRELATION

• If y tends to increase as x increases, the paired data are said to have a correlation.

• If y tends to decrease as x increases, the paired data are said to have a correlation.

• If x and y have no apparent relationship, the paired data are said to have correlation.

Describe the correlation of data graphed in the scatter plot.

a.

x

y

2

6

2 6

b.

x

y

2

6

2 6

Solution

a. b. correlation correlation

Example 1 Describe the correlation of data

Your Notes



5.6 Fit a Line to DataGoal p Make scatter plots and write equations to

model data.

VOCABULARY

Scatter plot A graph used to determine whether there is a relationship between paired data

Correlation The relationship between two data sets

Line of fit A model used to represent the trend in data showing a positive or negative correlation

CORRELATION

• If y tends to increase as x increases, the paired data are said to have a positive correlation.

• If y tends to decrease as x increases, the paired data are said to have a negative correlation.

• If x and y have no apparent relationship, the paired data are said to have relatively no correlation.

Describe the correlation of data graphed in the scatter plot.

a.

x

y

2

6

2 6

b.

x

y

2

6

2 6

Solution

a. negative b. relatively no correlation correlation

Example 1 Describe the correlation of data

Your Notes



a. Make a scatter plot of the data in the table.

x 1 1.5 2 2 3 3.5 4

y 3 1 1 20.5 21 20.5 22

b. Describe the correlation of the data.

Solutiona. Treat the data as ordered

x

y

1

21

3

121 3

pairs. Plot the ordered pairs as in a coordinate plane.

b. The scatter plot shows a correlation.

Example 2 Make a scatter plot

USING A LINE OF FIT TO MODEL DATA

Step 1 Make a of the data.

Step 2 Decide whether the data can be modeled by a .

Step 3 Draw a line that appears to the data closely. There should be approximately as many points the line as it.

Step 4 Write an equation using points on the line. The points do not have to represent actual data pairs, but they must lie on the line of fit.


Your Notes


a. Make a scatter plot of the data in the table.

x 1 1.5 2 2 3 3.5 4

y 3 1 1 20.5 21 20.5 22

b. Describe the correlation of the data.

Solutiona. Treat the data as ordered

x

y

1

21

3

121 3

pairs. Plot the ordered pairs as points in a coordinate plane.

b. The scatter plot shows a negative correlation.

Example 2 Make a scatter plot

USING A LINE OF FIT TO MODEL DATA

Step 1 Make a scatter plot of the data.

Step 2 Decide whether the data can be modeled by a line .

Step 3 Draw a line that appears to fit the data closely. There should be approximately as many points above the line as below it.

Step 4 Write an equation using two points on the line. The points do not have to represent actual data pairs, but they must lie on the line of fit.


Your Notes


Your Notes

Game Attendance The table shows the average attendance at a school's varsity basketball games for various years. Write an equation that models the average attendance at varsity basketball games as a function of the number of years since 2000.

Year 2000 2001 2002 2003 2004 2005 2006

Avg. Game Attendance 488 497 525 567 583 621 688

Solution

Step 1 Make a

x

y

0 1 2 3 4 5 6

4500

500

550

600

650

700

Avera

ge a

tten

dan

ce

Years since 2000

Game Attendance

of the data. Let x represent the number of years since 2000. Let y represent average game attendance.

Step 2 Decide whether the data can be modeled by a line. Because the scatter plot shows a correlation, you can fit a line to the data.

Step 3 Draw a line that appears to fit the points in the scatter plot .

Step 4 Write an equation using two points on the line. Use (1, 493) and (5, 621).

Find the of the line.

m 5 y2 2 y1 } x2 2 x1

5 2

2

5

5

Example 3 Write an equation to model data



Your Notes

Game Attendance The table shows the average attendance at a school's varsity basketball games for various years. Write an equation that models the average attendance at varsity basketball games as a function of the number of years since 2000.

Year 2000 2001 2002 2003 2004 2005 2006

Avg. Game Attendance 488 497 525 567 583 621 688

Solution

Step 1 Make a scatter plot

x

y

0 1 2 3 4 5 6

4500

500

550

600

650

700

Avera

ge a

tten

dan

ce

Years since 2000

Game Attendance

of the data. Let x represent the number of years since 2000. Let y represent average game attendance.

Step 2 Decide whether the data can be modeled by a line. Because the scatter plot shows a positive correlation, you can fit a line to the data.

Step 3 Draw a line that appears to fit the points in the scatter plot closely .

Step 4 Write an equation using two points on the line. Use (1, 493) and (5, 621).

Find the slope of the line.

m 5 y2 2 y1 } x2 2 x1

5 5

621

1

493

2

2

5 4

128

5 32

Example 3 Write an equation to model data



Homework

1. Make a scatter plot of the data in the table. Describe the correlation of the data.

x 1 2 2 3 4 5

y 5 5 6 7 8 8

x

y

1

3

5

7

121 3 5 7

2. Use the data in the table to write an equation that models y as a function of x.

x 1 2 3 4 5 6

y 65 76 82 86 92 97


Find the y-intercept of the line. Use the point (5, 621).



5 b Solve for b.

An equation of the line of fit is .

The average attendance y of varsity basketball games can be modeled by the function where x is the number of years since 2000.


Your Notes


Homework

1. Make a scatter plot of the data in the table. Describe the correlation of the data.

x 1 2 2 3 4 5

y 5 5 6 7 8 8

x

y

1

3

5

7

121 3 5 7

positive correlation

2. Use the data in the table to write an equation that models y as a function of x.

x 1 2 3 4 5 6

y 65 76 82 86 92 97

y 5 6x 1 62


Find the y-intercept of the line. Use the point (5, 621).



461 5 b Solve for b.

An equation of the line of fit is y 5 32x 1 461.

The average attendance y of varsity basketball games can be modeled by the function y 5 32x 1 461 where x is the number of years since 2000.


Your Notes


5.7 Predict with Linear ModelsGoal p Make predictions using best-fitting lines.

VOCABULARY

Best-fitting line

Interpolation

Extrapolation

Zero of a function

Your Notes



5.7 Predict with Linear ModelsGoal p Make predictions using best-fitting lines.

VOCABULARY

Best-fitting line The line that most closely follows a trend in data

Interpolation Use of a line or its equation to approximate a value between two known values

Extrapolation Use of a line or its equation to approximate a value outside the range of known values

Zero of a function A zero of a function y 5 f (x) is an x-value for which f(x) 5 0 (or y 5 0).

Your Notes




NFL Salaries The table shows the average National Football League (NFL) player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001

Average Player's Salary (in thousands of dollars) 585 708 787 986

a. Make a scatter plot of the data.

b. Find an equation that models the average NFL player's salary (in thousands of dollars) as a function of the number of years since 1997.

c. Approximate the average NFL player's salary in 1998.

Solutiona. Enter the data into lists on

a graphing calculator. Make a scatter plot, letting the number of years since 1997 be the (0, 2, 3, 4) and the average player's salary be the .

b. Perform

X=1 Y=649

Y1=94X+555

using the paired data. The equation of the best-fitting line is y 5 .

c. Graph the best-fitting line. Use the trace feature and the arrow keys to find the value of the equation when x 5 .

The average NFL player's salary in 1998 was thousand dollars.

Example 1 Interpolate using an equation

The slope, of the best-fitting line indicates that an NFL player’s salary increased by about thousand dollars per year.

Your Notes



NFL Salaries The table shows the average National Football League (NFL) player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


a. Make a scatter plot of the data.

b. Find an equation that models the average NFL player's salary (in thousands of dollars) as a function of the number of years since 1997.

c. Approximate the average NFL player's salary in 1998.

Solutiona. Enter the data into lists on

a graphing calculator. Make a scatter plot, letting the number of years since 1997 be the x-values (0, 2, 3, 4) and the average player's salary be the y-values .

b. Perform linear regression

X=1 Y=649

Y1=94X+555

using the paired data. The equation of the best-fitting line is y 5 94x 1 555 .

c. Graph the best-fitting line. Use the trace feature and the arrow keys to find the value of the equation when x 5 1 .

The average NFL player's salary in 1998 was 649 thousand dollars.

Example 1 Interpolate using an equation

The slope, 94 of the best-fitting line indicates that an NFL player’s salary increased by about 94 thousand dollars per year.

Your Notes


NFL Salaries Look back at Example 1.

a. Use the equation from Example 1 to approximate the average NFL player's salary in 2002 and 2003.

b. In 2002, the average NFL player's salary was actually 1180 thousand dollars. In 2003, the average NFL player's salary was actually 1259 thousand dollars. Describe the accuracy of the extrapolations made in part (a).

Solution Y1(5) 1025Y1(6) 1119 a. Evaluate the equation of the

best-fitting line from Example 1 for x 5 and x 5 .

The model predicts the average NFL player's salary as thousand dollars in 2002 and thousand dollars in 2003.

b. The differences between the predicted average NFL player's salary and the actual average NFL player's salary in 2002 and 2003 are thousand dollars and thousand dollars, respectively. The equation of the best-fitting line gives a less accurate prediction for the years outside of the given years.

Example 2 Extrapolate using an equation

RELATING SOLUTIONS OF EQUATIONS, ZEROS OF FUNCTIONS, AND x-INTERCEPTS OF GRAPHS

In Chapter 3,you learnedto solve an equation like

4x 2 4 5 0:

4x 2 4 5 0

4x 5

x 5

The solution of 4x 2 4 5 0 is .

In Chapter 4,you found the

of the graph ofa function like y 5 4x 2 4:

x

y1

2121

23

31

Now you arefinding the zero of a function like

f(x) 5 4x 2 4:

f(x) 5 0

5 0

x 5

The zero of f(x) 5 4x 2 4 is .


Your Notes


NFL Salaries Look back at Example 1.

a. Use the equation from Example 1 to approximate the average NFL player's salary in 2002 and 2003.

b. In 2002, the average NFL player's salary was actually 1180 thousand dollars. In 2003, the average NFL player's salary was actually 1259 thousand dollars. Describe the accuracy of the extrapolations made in part (a).

Solution Y1(5) 1025Y1(6) 1119 a. Evaluate the equation of the

best-fitting line from Example 1 for x 5 5 and x 5 6 .

The model predicts the average NFL player's salary as 1025 thousand dollars in 2002 and 1119 thousand dollars in 2003.

b. The differences between the predicted average NFL player's salary and the actual average NFL player's salary in 2002 and 2003 are 155 thousand dollars and 140 thousand dollars, respectively. The equation of the best-fitting line gives a less accurate prediction for the years outside of the given years.

Example 2 Extrapolate using an equation

RELATING SOLUTIONS OF EQUATIONS, ZEROS OF FUNCTIONS, AND x-INTERCEPTS OF GRAPHS

In Chapter 3,you learnedto solve an equation like

4x 2 4 5 0:

4x 2 4 5 0

4x 5 4

x 5 1

The solution of 4x 2 4 5 0 is 1 .

In Chapter 4,you found the x-intercept of the graph ofa function like y 5 4x 2 4:

x

y1

2121

23

31

Now you arefinding the zero of a function like

f(x) 5 4x 2 4:

f(x) 5 0

4x 2 4 5 0

x 5 1

The zero of f(x) 5 4x 2 4 is 1 .


Your Notes


Homework

Public Transit The percentage y of people in the U.S. that use public transit to commute to work can be modeled by the function y 5 20.045x 1 5.7 where x is the number of years since 1983. Find the zero of the function. Explain what the zero means in this situation.

Solution

Substitute for y in the equation of the and solve for x.

y 5 20.045x 1 5.7 Write the equation.

5 20.045x 1 5.7 Substitute for y.

Solve for x.

The zero of the function is about . The function has a slope, which means that the percentage of people using public transit to commute to work is . According to the model there will be no people who use public transit to commute to work years after , or in .

Example 3 Find the zero of a function

1. Baseball Salaries The table shows the average major league baseball player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


Find an equation that models the average major league baseball player's salary (in thousands of dollars) as a function of the number of years since 1997. Approximate the average major league baseball player's salary in 1998, 2002, and 2003.



Your Notes


Homework

Public Transit The percentage y of people in the U.S. that use public transit to commute to work can be modeled by the function y 5 20.045x 1 5.7 where x is the number of years since 1983. Find the zero of the function. Explain what the zero means in this situation.

Solution

Substitute 0 for y in the equation of the best-fitting line and solve for x.

y 5 20.045x 1 5.7 Write the equation.

0 5 20.045x 1 5.7 Substitute 0 for y.

x ø 127 Solve for x.

The zero of the function is about 127 . The function has a negative slope, which means that the percentage of people using public transit to commute to work is decreasing . According to the model there will be no people who use public transit to commute to work 127 years after 1983 , or in 2110 .

Example 3 Find the zero of a function

1. Baseball Salaries The table shows the average major league baseball player's salary (in thousands of dollars) from 1997 to 2001.

Year 1997 1999 2000 2001


Find an equation that models the average major league baseball player's salary (in thousands of dollars) as a function of the number of years since 1997. Approximate the average major league baseball player's salary in 1998, 2002, and 2003.

Possible equation of the line of fit: y 5201x 1 1293

1998: 1494; 2002: 2298; 2003: 2499



Your Notes


Point-slope form

Arithmetic sequence

Converse

Scatter plot

Line of fit

Common difference

Sequence

Perpendicular lines

Correlation

Best-fitting line




Point-slope form

y 2 y1 5 m(x 2 x1)

Arithmetic sequence

{1, 31, 61, 91, 121,... }

Converse

If two nonvertical lines have the same slope, then they are parallel.

If two nonvertical lines are parallel, then they have the same slope.

Scatter plot

x

y

2

6

2 6

Line of fit

A model used to represent the trend in data showing a positive or negative correlation.

Common difference

23 for { 7, 4, 1, –2, –5,... }

Sequence

{31, 28, 31, 30, 31, 30, 31, 31, 30, 31, 30, 31}

Perpendicular lines

y 5 5x 2 3

y 5 2 1 } 5 x 1 2

Correlation

The scatter plot shows a negative correlation.

x

y

2

6

2 6

Best-fitting line

A line that best fits the data points on a scatter plot.






Interpolation

Zero of a function

Extrapolation




InterpolationEstimating a value for 1997 when you are given values for 1996 and 1998.

Zero of a function

Extrapolation

Estimating a value for 2002 when you are given values for 1999 to 2001.

The zero of f(x) 5 4x 2 4 is 1.


Solve Inequalities UsingAddition and Subtraction

VOCABULARY

Graph of a linear inequality in one variable

Equivalent inequalities

Goal p Solve inequalities using addition and subtraction.

Food Drive Your school wants to collect at least 5000 pounds of food for a food drive. Write and graph an inequality to describe the amount of food that your school hopes to collect.

Solution

Let p represent the . The value of p must

be 5000 pounds. So, an inequality is .

0 600050004000300020001000 7000 8000

Example 1 Write and graph an inequality

1. You must be 16 years old or older to get your driver's license. Write and graph an inequality to describe the ages of people who may get their driver's license.

13 191817161514 20 21


Remember to use an open circle for < or > and a closed circle for ≤ or ≥.


6.1

Your Notes


Solve Inequalities UsingAddition and Subtraction

VOCABULARY

Graph of a linear inequality in one variable The set of points on a number line that represents all solutions of the inequality

Equivalent inequalities Inequalities that have the same solutions

Goal p Solve inequalities using addition and subtraction.

Food Drive Your school wants to collect at least 5000 pounds of food for a food drive. Write and graph an inequality to describe the amount of food that your school hopes to collect.

Solution

Let p represent the number of pounds of food that the school hopes to collect . The value of p must be greater than or equal to 5000 pounds. So, an inequality is p ≥ 5000 .

0 600050004000300020001000 7000 8000

Example 1 Write and graph an inequality

1. You must be 16 years old or older to get your driver's license. Write and graph an inequality to describe the ages of people who may get their driver's license.

a ≥ 16

13 191817161514 20 21


Remember to use an open circle for < or > and a closed circle for ≤ or ≥.


6.1

Your Notes


ADDITION PROPERTY OF INEQUALITY

Words Adding the same number to each side of an inequality produces an

.

Algebra If a > b, then a 1 c > .

If a < b, then a 1 c < .

If a ≥ b, then a 1 c ≥ .

If a ≤ b, then a 1 c ≤ .

Solve n 2 3.5 < 2.5. Graph your solution.

Solution n 2 3.5 < 2.5 Write original inequality.

n 2 3.5 1 < 2.5 1 Use addition property of inequality: Add to each side.

Simplify.

The solutions are all real numbers . Check by substituting a number for n in the original inequality.

0 654321 7 8

Example 2 Solve an inequality using addition

Checkpoint Solve the inequality. Graph your solution.

2. 6 > y 2 3.3 3. z 2 7 ≥ 4

5 86 1097 11

7 108 12119 13


Your Notes


ADDITION PROPERTY OF INEQUALITY

Words Adding the same number to each side of an inequality produces an equivalent inequality .

Algebra If a > b, then a 1 c > b 1 c .

If a < b, then a 1 c < b 1 c .

If a ≥ b, then a 1 c ≥ b 1 c .

If a ≤ b, then a 1 c ≤ b 1 c .

Solve n 2 3.5 < 2.5. Graph your solution.

Solution n 2 3.5 < 2.5 Write original inequality.

n 2 3.5 1 3.5 < 2.5 1 3.5 Use addition property of inequality: Add 3.5 to each side.

n < 6 Simplify.

The solutions are all real numbers less than 6 . Check by substituting a number less than 6 for n in the original inequality.

0 654321 7 8

Example 2 Solve an inequality using addition


2. 6 > y 2 3.3 3. z 2 7 ≥ 4

y < 9.3 z ≥ 11

5 86 1097 11

9.3 7 108 12119 13


Your Notes


SUBTRACTION PROPERTY OF INEQUALITY

Words Subtracting the same number from each side of an inequality produces an

.

Algebra If a > b, then a 2 c > .

If a < b, then a 2 c < .

If a ≥ b, then a 2 c ≥ .

If a ≤ b, then a 2 c ≤ .

Solve 3 ≤ y 1 8. Graph your solution.

Solution 3 ≤ y 1 8 Write original inequality.

3 2 ≤ y 1 8 2 Use subtraction property of inequality: Subtract from each side.

Simplify.

You can rewrite as .The solutions are all real numbers .

28 27 26 25 24 23 22 21 0

Example 3 Solve an inequality using subtraction


4. r 1 3 1 } 4 < 5 5. 3 1 m ≥ 7.2

22 121 320

5 76432

Homework


Your Notes


SUBTRACTION PROPERTY OF INEQUALITY

Words Subtracting the same number from each side of an inequality produces an equivalent inequality .

Algebra If a > b, then a 2 c > b 2 c .

If a < b, then a 2 c < b 2 c .

If a ≥ b, then a 2 c ≥ b 2 c .

If a ≤ b, then a 2 c ≤ b 2 c .

Solve 3 ≤ y 1 8. Graph your solution.

Solution 3 ≤ y 1 8 Write original inequality.

3 2 8 ≤ y 1 8 2 8 Use subtraction property of inequality: Subtract 8 from each side.

25 ≤ y Simplify.

You can rewrite 25 ≤ y as y ≥ 25 .The solutions are all real numbers greater than or equal to 25 .

28 27 26 25 24 23 22 21 0

Example 3 Solve an inequality using subtraction


4. r 1 3 1 } 4 < 5 5. 3 1 m ≥ 7.2

r < 1 3 } 4 m ≥ 4.2

22 121 320

5 76432

Homework


Your Notes


6.2 Solve Inequalities UsingMultiplication and DivisionGoal p Solve inequalities using multiplication

and division.

MULTIPLICATION PROPERTY OF INEQUALITY

Words Multiplying each side of an inequality by a number produces an

.

Multiplying each side of an inequality by a number and

produces an equivalent inequality.

Algebra If a 0, then .

If a b and c > 0, then .

If a > b and c < 0, then .

This property is also true for inequalities involving ≤ and ≥.

Solve y }

9 > 3. Graph your solution.

Solution

y } 9 > 3 Write original inequality.

p y } 9 > p 3 Use multiplication property

of inequality: Multiply each side by .

Simplify.

The solutions are all real numbers .

24 302928272625 31 32

Example 1 Solve an inequality using multiplication

Your Notes



6.2 Solve Inequalities UsingMultiplication and DivisionGoal p Solve inequalities using multiplication

and division.

MULTIPLICATION PROPERTY OF INEQUALITY

Words Multiplying each side of an inequality by a positive number produces an equivalent inequality .

Multiplying each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality.

Algebra If a 0, then ac < bc .

If a bc .

If a > b and c > 0, then ac > bc .

If a > b and c < 0, then ac < bc .


Solve y }

9 > 3. Graph your solution.

Solution

y } 9 > 3 Write original inequality.

9 p y } 9 > 9 p 3 Use multiplication property

of inequality: Multiply each side by 9 .

y > 27 Simplify.

The solutions are all real numbers greater than 27 .

24 302928272625 31 32


Your Notes



Solve m } 22

< 5. Graph your solution.

Solution

m } 22 < 5 Write original inequality.

p m } 22 > p 5 Multiply each side by

and the inequality symbol.

Simplify.


212 211 210 29 28 27 26 25 24


1. r } 7 ≤ 6 2. s } 24 > 0.4

36 4238 464440

22.2 22.0 21.8 21.6 21.4 21.2

3. n } 25 ≥ 22 4. w } 6 < 20.8

7 108 12119

25.0 24.9 24.8 24.7 24.6 24.5



Your Notes


Solve m } 22

< 5. Graph your solution.

Solution

m } 22 < 5 Write original inequality.

22 p m } 22 > 22 p 5 Multiply each side by 22

and reverse the inequality symbol.

m > 210 Simplify.


212 211 210 29 28 27 26 25 24


1. r } 7 ≤ 6 2. s } 24 > 0.4

r ≤ 42 s < 21.6

36 4238 464440

22.2 22.0 21.8 21.6 21.4 21.2

3. n } 25 ≥ 22 4. w } 6 < 20.8

n ≤ 10 w < 24.8

7 108 12119

25.0 24.9 24.8 24.7 24.6 24.5



Your Notes


DIVISION PROPERTY OF INEQUALITY

Words Dividing each side of an inequality by a number produces an

.

Dividing each side of an inequality by a number and

produces an equivalent inequality.

Algebra If a 0, then .

If a b and c > 0, then .

If a > b and c < 0, then .


Solve 24x < 36. Graph your solution.

Solution 24x < 36 Write original inequality.

> Use division property of inequality:Divide each side by and

the inequality symbol.

Simplify.


213 212 211 210 29 28 27 26 25

Example 3 Solve an inequality using division

24x 36


Your Notes


DIVISION PROPERTY OF INEQUALITY

Words Dividing each side of an inequality by a positive number produces an equivalent inequality .

Dividing each side of an inequality by a negative number and reversing the direction of the inequality symbol produces an equivalent inequality.

Algebra If a 0, then a } c b } c .

If a > b and c > 0, then a } c > b } c .

If a > b and c < 0, then a } c Use division property of inequality:Divide each side by 24 and reverse the inequality symbol.

x > 29 Simplify.


213 212 211 210 29 28 27 26 25

Example 3 Solve an inequality using division

24x

24

36

24


Your Notes


Homework

Pizza Party You have a budget of $45 to buy pizza for a student council meeting. Pizzas cost $7.50 each. Write and solve an inequality to find the possible numbers of pizzas that you can buy.

Solution

Price per pizza (dollars per pizza)

p Number of pizzas (pizzas) ≤

Budget amount (dollars)

p p ≤

Write inequality.

p ≤ Divide each side by .

You can buy at most pizzas.

Example 4 Solve a real-world problem

5. 29k < 36 6. 10n ≥ 140

26 25 24 23 22 21

12 14 16 18 20 22

7. In Example 4, suppose that you had a budget of $50 and each pizza costs $8. Write and solve an inequality to find the possible numbers of pizzas that you can buy.



Your Notes


Homework

Pizza Party You have a budget of $45 to buy pizza for a student council meeting. Pizzas cost $7.50 each. Write and solve an inequality to find the possible numbers of pizzas that you can buy.

Solution

Price per pizza (dollars per pizza)

p Number of pizzas (pizzas) ≤

Budget amount (dollars)

7.50 p p ≤ 45

7.50 p p ≤ 45 Write inequality.

p ≤ 6 Divide each side by 7.50 .

You can buy at most 6 pizzas.

Example 4 Solve a real-world problem

5. 29k < 36 6. 10n ≥ 140

k > 24 n ≥ 14

26 25 24 23 22 21

12 14 16 18 20 22

7. In Example 4, suppose that you had a budget of $50 and each pizza costs $8. Write and solve an inequality to find the possible numbers of pizzas that you can buy.

8 p p ≤ 50; p ≤ 6.25; You can buy at most 6 pizzas.



Your Notes


6.3 Solve Multi-Step InequalitiesGoal p Solve multi-step inequalities.

Solve 4x 1 6 ≥ 54. Graph your solution.

Solution 4x 1 6 ≥ 54 Write original inequality.

4x ≥ 48 Subtract from each side.

Divide each side by .


8 14131211109 15 16

Example 1 Solve a two-step inequalityYour Notes

Solve 2 1 } 3 (x 1 21) < 2.

Solution

2 1 } 3 (x 1 21) < 2 Write original inequality.

2 1 } 3 x 2 < 2 Distributive property

2 1 } 3 x < Add to each side.

Multiply each side by . the inequality symbol.


228 227 226 225 224 223 222 221 220

Example 2 Solve a multi-step inequality



6.3 Solve Multi-Step InequalitiesGoal p Solve multi-step inequalities.

Solve 4x 1 6 ≥ 54. Graph your solution.

Solution 4x 1 6 ≥ 54 Write original inequality.

4x ≥ 48 Subtract 6 from each side.

x ≥ 12 Divide each side by 4 .

The solutions are all real numbers greater than or equal to 12 .

8 14131211109 15 16

Example 1 Solve a two-step inequalityYour Notes

Solve 2 1 } 3 (x 1 21) < 2.

Solution

2 1 } 3 (x 1 21) < 2 Write original inequality.

2 1 } 3 x 2 7 < 2 Distributive property

2 1 } 3 x < 9 Add 7 to each side.

x > 227 Multiply each side by 23 . Reverse the inequality symbol.


228 227 226 225 224 223 222 221 220

Example 2 Solve a multi-step inequality



1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0

28 2324252627

3 54210


Solve the inequality, if possible.

a. 8x 1 3 > 2(4x 1 1)

b. 3(8b 2 1) ≤ 24b 2 4

Solutiona. 8x 1 3 > 2(4x 1 1) Write original inequality.

8x 1 3 > Distributive property

Subtract from each side.

are solutions because is .

b. 3(8b 2 1) ≤ 24b 2 4 Write original inequality.

≤ 24b 2 4 Distributive property

Subtract from each side.

There are because is .

Example 3 Identify the number of solutions of an inequality


Your Notes


1. 25w 2 2 ≥ 23 2. 2(y 2 2.2) > 0

w ≤ 25 y > 2.2

28 2324252627

3 54210


Solve the inequality, if possible.

a. 8x 1 3 > 2(4x 1 1)

b. 3(8b 2 1) ≤ 24b 2 4

Solutiona. 8x 1 3 > 2(4x 1 1) Write original inequality.

8x 1 3 > 8x 1 2 Distributive property

3 > 2 Subtract 8x from each side.

All real numbers are solutions because 3 > 2 is true .

b. 3(8b 2 1) ≤ 24b 2 4 Write original inequality.

24b 2 3 ≤ 24b 2 4 Distributive property

23 ≤ 24 Subtract 24b from each side.

There are no solutions because 23 ≤ 24 is false .

Example 3 Identify the number of solutions of an inequality


Your Notes


Homework

3. 18 1 4w ≥ 1 } 2 (8w 1 36) 4. 22(3z 2 1) < 1 2 6z

Checkpoint Solve the inequality, if possible.

Cell Phone Your cell phone plan is $35 a month for 1000 minutes. You are charged $.25 per minute for any additional minutes. What are the possible numbers of additional minutes you can use if you want to spend no more than $50 on your monthly cell phone bill?

SolutionThe amount spent on the monthly plan plus additional minutes must be less than or equal to your monthly budget. Let m be the number of additional minutes that you use.

Price per minute

(dollars/min) p

Number of minutes

(minutes) 1

Monthly fee

(dollars) ≤

Monthly budget (dollars)

p m 1 ≤

≤ Write inequality.

m ≤ Subtract from each side.

m ≤ Divide each side by .

You can use an additional per month to keep within your monthly cell phone budget.



Your Notes


Homework

3. 18 1 4w ≥ 1 } 2 (8w 1 36) 4. 22(3z 2 1) < 1 2 6z

All real numbers are No solutionsolutions.

Checkpoint Solve the inequality, if possible.

Cell Phone Your cell phone plan is $35 a month for 1000 minutes. You are charged $.25 per minute for any additional minutes. What are the possible numbers of additional minutes you can use if you want to spend no more than $50 on your monthly cell phone bill?

SolutionThe amount spent on the monthly plan plus additional minutes must be less than or equal to your monthly budget. Let m be the number of additional minutes that you use.

Price per minute

(dollars/min) p

Number of minutes

(minutes) 1

Monthly fee

(dollars) ≤

Monthly budget (dollars)

0.25 p m 1 35 ≤ 50

0.25 p m 1 35 ≤ 50 Write inequality.

0.25 m ≤ 15 Subtract 35 from each side.

m ≤ 60 Divide each side by 0.25 .

You can use an additional 60 minutes or fewer per month to keep within your monthly cell phone budget.



Your Notes


Solve Linear Inequalities by Graphing

Your Notes

Goal p Use graphs to solve linear inequalities.

SOLVING LINEAR INEQUALITIES GRAPHICALLY

Step 1 Write the inequality as: ax 1 b , 0 , ax 1 b 0, , or .

Step 2 Write the related equation y 5 1 .

Step 3 Graph the equation y 5 .

• The solutions of ax 1 b 0 are the of the points on the graph of y 5 that lie

the x-axis.

• The solutions of ax 1 b 0 are the of the points on the graph of

y 5 that lie .

• If the inequality symbol is or , the of the graph is also a solution.

Example 1 Solve an inequality graphically

Solve 6x + 4 . 10 graphically.

Solution

Step 1 Write the inequality as

ax 1 b 0

Step 2 Write the function.

y 5

Step 3 Graph y 5 .

From the inequality symbol and the graph’s x-intercept of , x .

CHECK

Choose a number and substitute.

6( ) 1 4 . 10 . 10

The solution checks.158 6.3 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

y

x

11

Focus on GraphingUse after Lesson 6.3


Solve Linear Inequalities by Graphing

Your Notes

Goal p Use graphs to solve linear inequalities.

SOLVING LINEAR INEQUALITIES GRAPHICALLY

Step 1 Write the inequality as: ax 1 b , 0 , ax 1 b # 0, ax 1 b . 0 , or ax 1 b $ 0 .

Step 2 Write the related equation y 5 ax 1 b .

Step 3 Graph the equation y 5 ax 1 b .

• The solutions of ax 1 b $ 0 are the x-coordinates of the points on the graph of y 5 ax 1 b that lie above the x-axis.

• The solutions of ax 1 b # 0 are the x-coordinates of the points on the graph of y 5 ax 1 b that lie below the x-axis .

• If the inequality symbol is # or $ , the x-intercept of the graph is also a solution.

Example 1 Solve an inequality graphically

Solve 6x + 4 . 10 graphically.

Solution

Step 1 Write the inequality as

ax 1 b . 0 6x 2 6 . 0

Step 2 Write the related function.

y 5 6x 2 6

Step 3 Graph y 5 6x 2 6 .

From the inequality symbol . and the graph’s x-intercept of 1 , x . 1 .

CHECK

Choose a number . 1 and substitute.

6( 2 ) 1 4 . 10 16 . 10

The solution checks.158 6.3 Focus on Graphing • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

y

x

11

(1, 0)

Focus on GraphingUse after Lesson 6.3



You can rent a video game for $3.99 for a given number of days. Each additional day costs $1.59. Your budget is $15. How many additional days can you afford?

Solution

Step 1 Write a model. Then write an .

Rate for additional days

•

Additional days

1

Cost of initial rental

#

Amount budgeted

• 1 3.99 #

Write the inequality in the form .

1 3.99 # 15

Step 2 Write the related equation y = .

Step 3 Graph the related equation on a graphing calculator. Draw the line you see on the calculator.

The inequality symbol is and the x-intercept is about . Because you cannot rent a game for part of a day, round x to . To stay within your budget, you can afford to rent up to additional days.

Use this viewing window:Xmin 5 21Xmax 5 9Xsci 5 1Ymin 5 212Ymax 5 3Ysci 5 1

1. 0.5x 1 9 $ 17

2. A used bookstore sells a given number of books

for $8.99 and each additional book for $3.69. How many additional books can you buy if you have $25? Use a graphing calculator.

Checkpoint Solve the inequalities graphically.

2

2

y

x


Homework

Your Notes



You can rent a video game for $3.99 for a given number of days. Each additional day costs $1.59. Your budget is $15. How many additional days can you afford?

Solution

Step 1 Write a verbal model. Then write an inequality .

Rate for additional days

•

Additional days

1

Cost of initial rental

#

Amount budgeted

1.59 • x 1 3.99 # 15

Write the inequality in the form ax + b # 0.

1.59x 1 3.99 # 15 1.59x 2 11.01 # 0

Step 2 Write the related equation y = 1.59x 2 11.01.

Step 3 Graph the related equation on a graphing calculator. Draw the line you see on the calculator.

The inequality symbol is # and the x-intercept is about 6.9 . Because you cannot rent a game for part of a day, round x down to 6 . To stay within your budget, you can afford to rent up to 6 additional days.

Use this viewing window:Xmin 5 21Xmax 5 9Xsci 5 1Ymin 5 212Ymax 5 3Ysci 5 1

1. 0.5x 1 9 $ 17

x $ 16

2. A used bookstore sells a given number of books

for $8.99 and each additional book for $3.69. How many additional books can you buy if you have $25? Use a graphing calculator.

4 additional books

Checkpoint Solve the inequalities graphically.

X 6.9 Y 0.039

2

2

y

x

(16, 0)


Homework

Your Notes


6.4 Solve Compound InequalitiesGoal p Solve and graph compound inequalities.

VOCABULARY

Compound inequality

Translate the verbal phrase into an inequality. Then graph the inequality.

a. All real numbers that are greater than or equal to 22 and less than 2.

b. All real numbers that are less than or equal to 3 or greater than 6.

c. All real numbers that are greater than 28 and less than or equal to 23.

Solution

a. 22 x 2

24 210212223 3 4

b. x 3 or x 6

654 7210 3 8

c. 28 x 23

210 272829 242526 2223

Example 1 Write and graph compound inequalities

Your Notes



6.4 Solve Compound InequalitiesGoal p Solve and graph compound inequalities.

VOCABULARY

Compound inequality A compound inequality consists of two separate inequalities joined by and or or.

Translate the verbal phrase into an inequality. Then graph the inequality.

a. All real numbers that are greater than or equal to 22 and less than 2.

b. All real numbers that are less than or equal to 3 or greater than 6.

c. All real numbers that are greater than 28 and less than or equal to 23.

Solution

a. 22 ≤ x < 2

24 210212223 3 4

b. x ≤ 3 or x > 6

654 7210 3 8

c. 28 < x ≤ 23

210 272829 242526 2223

Example 1 Write and graph compound inequalities

Your Notes



Solve 15 ≤ 3x 2 3 < 24. Graph your solution.

SolutionSeparate the compound inequality into two inequalities. Then solve each inequality separately.

15 ≤ 3x 2 3 and 3x 2 3 < 24 Write two inequalities.

≤ 3x and 3x < Add to each expression.

≤ x and x < Divide each expression by .

The compound inequality can be written as . The solutions are all real numbers

and .

987 10 1143 5 6

Example 2 Solve a compound inequality with and

Solve 15 < 27x 1 1 < 50. Graph your solution.

Solution 15 < 27x 1 1 < 50 Write original

inequality.

< 27x < Subtract from each expression.

> x > Divide each expression by and

.

The solutions are all real numbers and .

29 262728 25 222324 21



Your Notes


Solve 15 ≤ 3x 2 3 < 24. Graph your solution.

SolutionSeparate the compound inequality into two inequalities. Then solve each inequality separately.

15 ≤ 3x 2 3 and 3x 2 3 < 24 Write two inequalities.

18 ≤ 3x and 3x < 27 Add 3 to each expression.

6 ≤ x and x < 9 Divide each expression by 3 .

The compound inequality can be written as 6 ≤ x < 9 . The solutions are all real numbers greater than or equal to 6 and less than 9 .

987 10 1143 5 6


Solve 15 < 27x 1 1 < 50. Graph your solution.

Solution 15 < 27x 1 1 < 50 Write original

inequality.

14 < 27x < 49 Subtract 1 from each expression.

22 > x > 27 Divide each expression by 27 and reverse both inequality symbols .

The solutions are all real numbers greater than 27 and less than 22 .

29 262728 25 222324 21



Your Notes


Solve 5x 1 6 ≤ 29 or 2x 2 8 > 12. Graph your solution.

Solution5x 1 6 ≤ 29 or 2x 2 8 > 12 Write original

inequality.

5x ≤ or 2x > Use addition or subtraction property of inequality.

x ≤ or x > Use division property of inequality.

The solutions are all real numbers or .

26 24 22 0 12108642

Example 4 Solve a compound inequality with or

1. 23 ≤ 22x 1 1 < 11

26 25 24 23 22 21 3210

2. 9x 1 1 < 217 or 7x 2 12 > 9

24 23 22 21 543210


Homework


Your Notes


Solve 5x 1 6 ≤ 29 or 2x 2 8 > 12. Graph your solution.

Solution5x 1 6 ≤ 29 or 2x 2 8 > 12 Write original

inequality.

5x ≤ 215 or 2x > 20 Use addition or subtraction property of inequality.

x ≤ 23 or x > 10 Use division property of inequality.

The solutions are all real numbers less than or equal to 23 or greater than 10.

26 24 22

23

0 12108642

Example 4 Solve a compound inequality with or

1. 23 ≤ 22x 1 1 < 11

2 ≥ x > 25

26 25 24 23 22 21 3210

2. 9x 1 1 < 217 or 7x 2 12 > 9

x < 22 or x > 3

24 23 22 21 543210


Homework


Your Notes


6.5 Solve Absolute Value EquationsGoal p Solve absolute value equations.

VOCABULARY

Absolute value equation

Absolute deviation

SOLVING AN ABSOLUTE VALUE EQUATION

The equation ⏐ax 1 b⏐5 c where c ≥ 0 is equivalent to the statement or .

Solve ⏐x 2 9⏐5 2.

Solution

⏐x 2 9⏐5 2 Write original equation.

x 2 9 5 2 or x 2 9 5 22 Rewrite as two equations.

x 5 or x 5 Add to each side.

The solutions are and . Check your solution.

CHECK

⏐x 2 9⏐5 2 ⏐x 2 9⏐5 2 Write original equation.

⏐ 2 9⏐5 2 ⏐ 2 9⏐5 2 Substitute for x.

⏐ ⏐5 2 ⏐ ⏐5 2 Subtract.

✓ ✓ Simplify. Solution checks.

Example 1 Solve an absolute value equation


Your Notes


6.5 Solve Absolute Value EquationsGoal p Solve absolute value equations.

VOCABULARY

Absolute value equation An equation that contains an absolute value expression

Absolute deviation The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value.

SOLVING AN ABSOLUTE VALUE EQUATION

The equation ⏐ax 1 b⏐5 c where c ≥ 0 is equivalent to the statement ax 1 b 5 c or ax 1 b 5 2c.

Solve ⏐x 2 9⏐5 2.

Solution

⏐x 2 9⏐5 2 Write original equation.

x 2 9 5 2 or x 2 9 5 22 Rewrite as two equations.

x 5 11 or x 5 7 Add 9 to each side.

The solutions are 11 and 7 . Check your solution.

CHECK

⏐x 2 9⏐5 2 ⏐x 2 9⏐5 2 Write original equation.

⏐ 11 2 9⏐5 2 ⏐ 7 2 9⏐5 2 Substitute for x.

⏐ 2 ⏐5 2 ⏐ 22 ⏐5 2 Subtract.

2 5 2 ✓ 2 5 2 ✓ Simplify. Solution checks.

Example 1 Solve an absolute value equation


Your Notes


Solve 4⏐2x 1 8⏐1 6 5 30.

Solution

First, rewrite the equation in the form .

4⏐2x 1 8⏐1 6 5 30 Write original equation.

4⏐2x 1 8⏐5 Subtract from each side.

⏐2x 1 8⏐5 Divide each side by .

Next, solve the absolute value equation.

⏐2x 1 8⏐ 5 Write absolute value equation.

2x 1 8 5 or 2x 1 8 5 Rewrite as two equations.

2x 5 or 2x 5 Subtract from each side.

x 5 or x 5 Divide each side by .

Example 2 Rewrite an absolute value equation

1. ⏐x 1 6⏐5 11 2. 3⏐5x 2 10⏐1 6 5 21

Checkpoint Solve the equation.

Remember to check your solutions in the original equation for accuracy.


Your Notes


Solve 4⏐2x 1 8⏐1 6 5 30.

Solution

First, rewrite the equation in the form ⏐ax 1 b⏐ 5 c .

4⏐2x 1 8⏐1 6 5 30 Write original equation.

4⏐2x 1 8⏐5 24 Subtract 6 from each side.

⏐2x 1 8⏐5 6 Divide each side by 4 .

Next, solve the absolute value equation.

⏐2x 1 8⏐ 5 6 Write absolute value equation.

2x 1 8 5 6 or 2x 1 8 5 26 Rewrite as two equations.

2x 5 22 or 2x 5 214 Subtract 8 from each side.

x 5 21 or x 5 27 Divide each side by 2 .

Example 2 Rewrite an absolute value equation

1. ⏐x 1 6⏐5 11 2. 3⏐5x 2 10⏐1 6 5 21

5 and 217 3 and 1

Checkpoint Solve the equation.

Remember to check your solutions in the original equation for accuracy.


Your Notes


Solve ⏐7x 2 3⏐1 8 5 5, if possible.

Solution

⏐7x 2 3⏐1 8 5 5 Write original equation.

⏐7x 2 3⏐5 Subtract from each side.

The absolute value of a number is never . So, there are no solutions.

Example 3 Decide if an equation has no solutions

The absolute deviation of x from 10 is 1.8. Find the values of x that satisfy this requirement.

SolutionAbsolute deviation 5⏐x 2 given value⏐

5⏐x 2 ⏐

Write original equation.

5 x 2 or 5 x 2 Rewrite as two equations.

5 x or 5 x Add to each side.

So, x is or .

Example 4 Use absolute deviation

Homework

3. Find the values of x that satisfy the definition of absolute value for a given value of 213.6 and an absolute deviation of 2.8.



Your Notes


Solve ⏐7x 2 3⏐1 8 5 5, if possible.

Solution

⏐7x 2 3⏐1 8 5 5 Write original equation.

⏐7x 2 3⏐5 23 Subtract 8 from each side.

The absolute value of a number is never negative . So, there are no solutions.

Example 3 Decide if an equation has no solutions

The absolute deviation of x from 10 is 1.8. Find the values of x that satisfy this requirement.

SolutionAbsolute deviation 5⏐x 2 given value⏐

1.8 5⏐x 2 10 ⏐

1.8 5 ⏐x 2 10⏐ Write original equation.

1.8 5 x 2 10 or 21.8 5 x 2 10 Rewrite as two equations.

11.8 5 x or 8.2 5 x Add 10 to each side.

So, x is 11.8 or 8.2 .

Example 4 Use absolute deviation

Homework

3. Find the values of x that satisfy the definition of absolute value for a given value of 213.6 and an absolute deviation of 2.8.

210.8 and 216.4



Your Notes


Goal p Graph absolute value functions.

Graph Absolute Value Functions

THE PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS

x f(x) 5 |x|

22 |22| 5 221 |21| 5

0

1

2

y

x

11

f(x) |x|

166 6.5 Focus on Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Example 1 Graph g(x) 5 |x 2 h| and g(x) 5 |x| 1 k

Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.a. g(x) 5 |x 2 4| b. g(x) 5 |x| 1 3

Step 1 Make a table of values.

a. b.

x 1 2 3 4 5 x 22 21 0 1 2g(x) 3 g(x) 5

Step 2 Graph the function.

a. b. y

x

11

y

x

11

Step 3 Compare graphs of g and f.

The graph of g(x) 5 |x 2 4| is units of f(x) 5 |x|.

The graph of g(x) 5 |x| 1 3 is units f(x) 5 |x|.

The graphs of g(x) are tranlations of the graph f(x) 5 |x|.Translations can be either horizontal or vertical.


Your Notes


Goal p Graph absolute value functions.

Graph Absolute Value Functions

THE PARENT FUNCTION FOR ABSOLUTE VALUE FUNCTIONS

x f(x) 5 |x|

22 |22| 5 221 |21| 5 1 0 |0| 5 0

1 |1| 5 1

2 |2| 5 2

y

x

11

f(x) |x|

166 6.5 Focus on Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.

Example 1 Graph g(x) 5 |x 2 h| and g(x) 5 |x| 1 k

Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.a. g(x) 5 |x 2 4| b. g(x) 5 |x| 1 3


a. b.

x 1 2 3 4 5 x 22 21 0 1 2g(x) 3 2 1 0 1 g(x) 5 4 3 4 5


a. b. y

x

11

y

x

11


The graph of g(x) 5 |x 2 4| is 4 units to the right of f(x) 5 |x|.

The graph of g(x) 5 |x| 1 3 is 3 units above f(x) 5 |x|.

The graphs of g(x) are tranlations of the graph f(x) 5 |x|.Translations can be either horizontal or vertical.


Your Notes


Example 2 Graph g(x) = a|x|

Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.

a. g(x) 5 22 |x| b. g(x) 5 0.75 |x|


a. x 22 21 0 1 2g(x) 24

b. x 24 22 0 2 4g(x) 3


a. y

x

11

b. y

x

11


a. The graph of g(x) 5 −2|x| opens and is than the graph of f(x) = |x|.

b. The graph of g(x) 5 0.75|x| opens and is than the graph of f(x) 5 |x|.

Graphs can be a vertical stretch or a vertical shrink of the graph of f(x) 5 |x|.

Copyright © Holt McDougal. All rights reserved. 6.5 Focus on Functions • Algebra 1 Notetaking Guide 167

Checkpoint Graph the functions. Compare the graph with the graph of f(x) 5 |x|.

1. g(x) 5 |x 1 2| 2. g(x) 5 |x| 2 0.5 3. g(x) 5 3|x|y

x

11

y

x

11

y

x

11

Homework

Your Notes


Example 2 Graph g(x) = a|x|

Graph the functions. Compare the graphs with the graph of f(x) 5 |x|.

a. g(x) 5 22 |x| b. g(x) 5 0.75 |x|


a. x 22 21 0 1 2g(x) 24 22 0 22 24

b. x 24 22 0 2 4g(x) 3 1.5 0 1.5 3


a. y

x

11

b. y

x

11


a. The graph of g(x) 5 −2|x| opens down and is narrower than the graph of f(x) = |x|.

b. The graph of g(x) 5 0.75|x| opens up and is wider than the graph of f(x) 5 |x|.

Graphs can be a vertical stretch or a vertical shrink of the graph of f(x) 5 |x|.

Copyright © Holt McDougal. All rights reserved. 6.5 Focus on Functions • Algebra 1 Notetaking Guide 167

Checkpoint Graph the functions. Compare the graph with the graph of f(x) 5 |x|.

1. g(x) 5 |x 1 2| 2. g(x) 5 |x| 2 0.5 3. g(x) 5 3|x|y

x

11

y

x

11

y

x

11

Homework

Your Notes

opens up; 2 units to the left

opens up; narrower

opens up; 0.5 unit down


6.6 Solve Absolute ValueInequalitiesGoal p Solve absolute value inequalities.

Solve the inequality. Graph your solution.

a. ⏐x⏐ ≤ 9 b. ⏐x⏐ > 1 } 4

Solutiona. The distance between x and 0 is less than or equal

to 9. So, ≤ x ≤ . The solutions are all real numbers and

.

212 630232629 9 12

b. The distance between x and 0 is greater than 1 } 4 .

So, x > or x < . The solutions are all real

numbers or

.

1021

Example 1 Solve an absolute value inequality

SOLVING ABSOLUTE VALUE INEQUALITIES

• The inequality ⏐ax 1 b⏐< c where c > 0 is equivalent to the compound inequality .

• The inequality ⏐ax 1 b⏐> c where c > 0 is equivalent to the compound inequality or

.

Note that < can be replaced by ≤ and > can be replaced by ≥.


Your Notes


6.6 Solve Absolute ValueInequalitiesGoal p Solve absolute value inequalities.

Solve the inequality. Graph your solution.

a. ⏐x⏐ ≤ 9 b. ⏐x⏐ > 1 } 4

Solutiona. The distance between x and 0 is less than or equal

to 9. So, 29 ≤ x ≤ 9 . The solutions are all real numbers less than or equal to 9 and greater than or equal to 29 .

212 630232629 9 12

b. The distance between x and 0 is greater than 1 } 4 .

So, x > 1 } 4 or x < 2 1 } 4 . The solutions are all real

numbers greater than 1 } 4

or less than 2 1 } 4 .

1021


SOLVING ABSOLUTE VALUE INEQUALITIES

• The inequality ⏐ax 1 b⏐< c where c > 0 is equivalent to the compound inequality 2c < ax 1 b < c.

• The inequality ⏐ax 1 b⏐> c where c > 0 is equivalent to the compound inequality ax 1 b < 2c or ax 1 b > c.

Note that < can be replaced by ≤ and > can be replaced by ≥.


Your Notes


Solve ⏐2x 2 7⏐< 9. Graph your solution.

Solution

⏐2x 2 7⏐< 9 Write original inequality.

< 2x 2 7 < Rewrite as compound inequality.

Add to each expression.

Divide each expression by .

The solutions are all real numbers and . Check several solutions in the original inequality.

864202224 10


Solve ⏐x 1 8⏐2 4 ≥ 2. Graph your solution.

Solution

⏐x 1 8⏐2 4 ≥ 2 Write original inequality.

⏐x 1 8⏐≥ Add to each side.

x 1 8 ≥ or x 1 8 ≤ Rewrite as compound inequality.

x ≥ or x ≤ Subtract from each side.

The solutions are all real numbers or .

214 22242628210212



Your Notes


Solve ⏐2x 2 7⏐< 9. Graph your solution.

Solution

⏐2x 2 7⏐< 9 Write original inequality.

29 < 2x 2 7 < 9 Rewrite as compound inequality.

22 < 2x < 16 Add 7 to each expression.

21 < x < 8 Divide each expression by 2 .

The solutions are all real numbers greater than 21 and less than 8 . Check several solutions in the original inequality.

864202224 10


Solve ⏐x 1 8⏐2 4 ≥ 2. Graph your solution.

Solution

⏐x 1 8⏐2 4 ≥ 2 Write original inequality.

⏐x 1 8⏐≥ 6 Add 4 to each side.

x 1 8 ≥ 6 or x 1 8 ≤ 26 Rewrite as compound inequality.

x ≥ 22 or x ≤ 214 Subtract 8 from each side.

The solutions are all real numbers greater than or equal to 22 or less than or equal to 214 .

214 22242628210212



Your Notes


Homework

SOLVING INEQUALITIES

One-Step and Multi-Step Inequalities

• Follow the steps for solving an equation, but the inequality symbol when

.

Compound Inequalities

• If necessary, rewrite the inequality as two separate inequalities. Then solve each inequality separately. Include or in the solution.

Absolute Value Inequalities

• If necessary, isolate the absolute value expression on one side of the inequality. Rewrite the absolute value inequality as a . Then solve the compound inequality.

1. 3⏐x 2 6⏐ > 9 2. ⏐6x 2 11⏐ ≤ 7

23 0 3 6 9 12

10 2 3 4 5

3. 22⏐6x 2 1⏐1 5 < 3 121 0



Your Notes


Homework

SOLVING INEQUALITIES

One-Step and Multi-Step Inequalities

• Follow the steps for solving an equation, but reverse the inequality symbol when multiplying or dividing by a negative number .

Compound Inequalities

• If necessary, rewrite the inequality as two separate inequalities. Then solve each inequality separately. Include and or or in the solution.

Absolute Value Inequalities

• If necessary, isolate the absolute value expression on one side of the inequality. Rewrite the absolute value inequality as a compound inequality . Then solve the compound inequality.

1. 3⏐x 2 6⏐ > 9 2. ⏐6x 2 11⏐ ≤ 7

x > 9 or x < 3 2 } 3 ≤ x ≤ 3

23 0 3 6 9 12

10 2 3 4 5

23

3. 22⏐6x 2 1⏐1 5 < 3 121 0

x > 1 } 3 or x < 0



Your Notes


6.7

VOCABULARY

Linear inequality in two variables

Graph of an inequality in two variables

Goal p Graph linear inequalities in two variables.

Graph Linear Inequalitiesin Two Variables

Tell whether the ordered pair is a solution of 3x 2 4y > 9.

a. (2, 0) b. (2, 21)

Solutiona. Test (2, 0):

3x 2 4y > 9 Write inequality.

3( ) 2 4( ) > 9 Substitute for x and for y.

> 9 Simplify.

(2, 0) a solution.

b. Test (2, 21):


3( ) 2 4( ) > 9 Substitute for x and for y.

> 9 Simplify.

(2, 21) a solution.

Example 1 Check solutions of a linear inequality


Your Notes


6.7

VOCABULARY

Linear inequality in two variables A linear inequality in two variables is the result of replacing the 5 sign in a linear equation with <, ≤, >, or ≥.

Graph of an inequality in two variables The set of points that represent all solutions of the inequality

Goal p Graph linear inequalities in two variables.

Graph Linear Inequalitiesin Two Variables

Tell whether the ordered pair is a solution of 3x 2 4y > 9.

a. (2, 0) b. (2, 21)

Solutiona. Test (2, 0):


3( 2 ) 2 4( 0 ) > 9 Substitute 2 for x and 0 for y.

6 > 9 Simplify.

(2, 0) is not a solution.

b. Test (2, 21):


3( 2 ) 2 4( 21 ) > 9 Substitute 2 for x and 21 for y.

10 > 9 Simplify.

(2, 21) is a solution.

Example 1 Check solutions of a linear inequality


Your Notes


GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES

Step 1 Graph the boundary line. Use a line for < or >, and use a line for ≤ or ≥.

Step 2 Test a point not on by checking whether the ordered pair is a solution of the inequality.

Step 3 Shade the containing the point if the ordered pair a solution of the inequality. Shade the if the ordered pair a solution.

Graph the inequality y < 2 1 } 2 x 1 4.

Solution

1. Graph the equation y 5 2 1 } 2 x 1 4. The inequality is

<, so use a line.

2. Test (0, 0) in y < 2 1 } 2 x 1 4.

< 2 1 } 2 ( ) 1 4

<

3. the half-plane that (0, 0) because (0, 0) a solution of the inequality.

x

y

2

6

10

22222

26

6 10

Example 2 Graph a linear inequality in two variables


Your Notes


GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES

Step 1 Graph the boundary line. Use a dashed line for < or >, and use a solid line for ≤ or ≥.

Step 2 Test a point not on the boundary line by checking whether the ordered pair is a solution of the inequality.

Step 3 Shade the half-plane containing the point if the ordered pair is a solution of the inequality. Shade the other half-plane if the ordered pair is not a solution.

Graph the inequality y < 2 1 } 2 x 1 4.

Solution

1. Graph the equation y 5 2 1 } 2 x 1 4. The inequality is

<, so use a dashed line.

2. Test (0, 0) in y < 2 1 } 2 x 1 4.

0 < 2 1 } 2 ( 0 ) 1 4

0 < 4

3. Shade the half-plane that contains (0, 0) because (0, 0) is a solution of the inequality.

x

y

2

6

10

22222

26

6 10

Example 2 Graph a linear inequality in two variables


Your Notes



Graph the inequality x ≥ 4.

Solution 1. Graph the equation x 5 4. The inequality is ≥,

so use a line.

2. Test (0, 3) in x ≥ 4. You only substitute the because the inequality does

not have the variable . ≥ 4

3. the half-plane that (0, 3), because (0, 3) a solution of the inequality.

x

y

1

3

12121

23

3 5

Example 3 Graph a linear inequality in one variable

1. 2y 1 4x > 8 2. y < 2

x

y

1

3

5

1212321

23

3 5

x

y

1

3

1212321

23

3

Checkpoint Graph the inequality.

Your Notes



Graph the inequality x ≥ 4.

Solution 1. Graph the equation x 5 4. The inequality is ≥,

so use a solid line.

2. Test (0, 3) in x ≥ 4. You only substitute the x-coordinate because the inequality does not have the variable y .

0 ≥ 4

3. Shade the half-plane that does not contain (0, 3), because (0, 3) is not a solution of the inequality.

x

y

1

3

12121

23

3 5

Example 3 Graph a linear inequality in one variable

1. 2y 1 4x > 8 2. y < 2

x

y

1

3

5

1212321

23

3 5

x

y

1

3

1212321

23

3

Checkpoint Graph the inequality.

Your Notes


Example 4 Write an inequality represented by a graph

Write an inequality represented by the graph.y

x

11

Solution

Step 1 Note that the of the boundary line is

and the -intercept is .

Step 2 Write an equation in slope-intercept form

y 5 . Substitute for and

for .

Step 3 Write an inequality. The boundary line is , so replace 5 with either or . The graph is shaded the boundary line, so choose . The inequality represented by the graph is

3. 4.

y

x11

y

x

11

Checkpoint Write an inequality represented by the graph.


Homework

Your Notes


Example 4 Write an inequality represented by a graph

Write an inequality represented by the graph.y

x

11

Solution

Step 1 Note that the slope of the boundary line is 2 1 } 3

and the y -intercept is 2 .

Step 2 Write an equation in slope-intercept form

y 5 mx 1 b . Substitute 2 1 } 3 for m and 2

for b .

y 5 2 1 } 3 x 1 2

Step 3 Write an inequality. The boundary line is solid , so replace 5 with either # or $ . The graph is shaded above the boundary line, so choose $ . The inequality represented by the graph is

y $ 2 1 } 3 x 1 2

3. 4.

y

x11

y

x

11

y , 2 3 } 2 x 23 y $ x 1 1

Checkpoint Write an inequality represented by the graph.


Homework

Your Notes


Graph of an inequality

Compound inequality

Absolute deviation

Graph of a linear inequality in two variables








Graph of an inequality

2324 22 121 3 420

Compound inequality

22 ≤ x < 2

Absolute deviation

⏐x 2 given value⏐

Graph of a linear inequality in two variables

x

y

1

3

1212321

23

3 5


t 2 3 > 7 and t > 10


⏐x 1 6⏐ 5 11


y < 2x 1 4

Documents

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