Chapter 1 Equations and Inequalities. 2 Minute Vocabulary Activity Take 2 minutes to define and/or...

Chapter 1Equations and Inequalities

2 Minute Vocabulary Activity•Take 2 minutes to define and/or give an

example of each vocabulary word below…VariableVariable CoefficientCoefficient

Monomial Monomial ConstantConstantDegreeDegree Order of operationsOrder of operations

TermTerm Like termsLike termsBinomialBinomial PolynomialPolynomial

TrinomialTrinomial

1.1

How do you use the order of operations to correctly evaluate expressions?•Example 1:

▫Evaluate (x – y)3 + 3 if x = 1 and y = 4

•Example 2:▫Evaluate 8xy + z3 if x = 5, y = -2, and z =

-1 y2 + 5

•Example 3:▫Find the area of a trapezoid with base

lengths of 13 meters and 25 meters and a height of 8 meters. A= ½ h (b1 + b2)

What are the different types of numbers?•Real Numbers

▫The numbers used in everyday life, each corresponding to exactly one point on the number line.

•Rational Numbers▫A real number that can be expressed as a

fraction (ratio). The decimal form is either terminating or repeating.

• Irrational Numbers▫Any real number that is not rational.

•Not Real Numbers▫The square root of a negative number

1.2

Definitions•Natural Numbers (N): counting numbers 1,

2, 3…•Whole Numbers(W): natural numbers plus 0• Integers(Z): whole numbers plus the opposite

of any natural number• Irrational Numbers(I): any number with or

√ where the number under the √ is not a perfect square

•Not real Numbers: any √ where the number under the √ is negative

Practice

Example 1:Name the sets to which each number

belongsa. √6b. 5c. -2

3

The Reminders from Algebra I• Properties that you MUST know…

• Commutative: (order changes)▫ (+) a + b = b + a (●) a•b = b•a

• Associative: (groups change but order doesn’t)▫ (+) (a + b) + c=a + (b + c) (●) (a • b)•c= a • (b •c)

• Identity: (after adding or multiplying # is same)▫ (+) a + 0 = a (●) a • 1 = a

• Inverse: (add or multiply the # to cancel)▫ (+) a + (-a) = 0 (●) a • 1/a = 1

• Distributive: (multiply # outside by all inside)▫ a(b+c) = ab +ac

More Practice

•Example 2Name the property

a. ( -8 + 8) + 15 = 0 + 15

b. ( 5 + 7) + 8 = 8 + (5 + 7)

c. ¼ (4x) = x

Verbal Expressions to Algebraic Expressions•1. Write an algebraic expression to

represent each verbal expressiona. three times the square of a

number

b. twice the sum of a number and 3

c. the cube of a number increased by 4 times the same number

1.3

Algebraic to Verbal Sentence

•2. Write a verbal sentence to represent each equation.

a. n + (-8) = -9

b. g – 5 = -2

c. 2c = c2 - 4

Solving Equations Practice #3

a. a + 4.39 = 76

b. -3d = 18 5

c. 2(2x + 3) – 3(4x – 5) = 22

d. -10x + 3(4x – 2) = 6

Apply the properties of Equality

•If what is the value of 3n - 3

•If what is the value of 5y - 6

5

983 n

3

825 y

Solve for a Variable

•The formula for the surface area S of a cone is where l is the slant height of the cone and r is the radius of the bas. Solve the formula for l.

2rrlS

Write an Equation

•Josh spent $425 of his $1685 budget for home improvements. He would like to replace six interior doors next. What can he afford to spend on each door?

16

Absolute Value

•For any real number a, if a is positive or zero, the absolute value of a is a. If a is negative, the absolute value of a is the opposite of a.

|a|= a if a >0 |a|= -a if a < 0

1.4

Work in pairs (speed-date activity)

Evaluate an Expression with Absolute Value

a. 1.4 + |5y – 7| if y = -3

b. |4x + 3| - 3 ½ if x = -2

Solve an Absolute Value Equationc. |x – 18| = 5

d. 9 = |x + 12|

e. 8 = |y + 5|

f. |5x – 6 | + 9 = 0

Solve an Absolute Value Equationg. |x + 6| = 3x – 2

h. 2|x + 1| - x = 3x – 4

i. -2|3a – 2| = 6

j. 3|2x + 2| - 2x = x + 3

Remember those Algebra 1 Properties?•When solving inequalities the properties

all work the same as with equations except…

When you multiply or divide by a negative number you must flip the inequality symbols

Ex: -12x > 96

-12x > 96 -12 -12

x < -8

1.5

Set-Builder Notation- How to write you answers

• The solution set of an inequality▫ Example

-0.25y > 2-0.25y > 2-0.25 -0.25

y < -8

{y | y < -8}*read the set of all y such that y is less than

or equal to negative 8

•Example 19

4m

m

-1 0

49 mm

410 m

10

4m

5

2m

5

2m

•Remember < and > use open dots •Remember ≤ and ≥ use closed dots

Solve the inequality and graph the solution set

Example 2. 7x – 5 > 6x + 4

9

5455

45

)6(46)6(57

4657

x

x

x

xxxx

xx

7 86 1511109 12 13 14

{x| x > 9}

Your Turn

The solution to an “AND” inequality is the intersection of their graphs (what they share)

10 < 3y-2 < 19

1.6

2 31 10654 7 8 9

AND Special Cases• x > 5 and x < 1

• No intersection• No Solution

• x > 2 and x > 0

• {x| x > 2}

1 5 -1 20 1

The solution of an “OR” inequality is the union of their graphs (graph both and keep everything)

x+5>7 or x+2<-2

-5 -4-6 3-1-2-3 0 1 2

OR Special Cases• x>3 or x<7

All Real Numbers ARN

• x>2 or x > 5

{x | x>2}

3 7 2 5

Absolute Value Inequalities

•Rules:▫If |a| < b or |a| < b then it is an AND▫If |a| > b or |a| > b the it is an OR

Less thAN ------ AND GreatOR ------ OR

Example

•|3x-6|<12

54 6-3 21 30-1-2