· Web viewThis work is a combination of original work and a derivative of Prealgebra Workbook, Beginning Algebra Workbook, and Intermediate Algebra Workbook by Tyler Wallace which

Name: __________________________________________________

Big Bend Community College

Emporium ModelMath Courses

Workbook

A workbook to supplement video lectures and online homework by:

Tyler WallaceSalah Abed

Sarah AdamsMariah HelvyApril Mayer

Michele Sherwood

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This project was made possible in part by a federal STEM-HSI grant under Title III part F and by the generous support of Big Bend Community College and the Math Department.

Copyright 2012, Some Rights Reserved CC-BY-NC-SA. This work is a combination of original work and a derivative of Prealgebra Workbook, Beginning Algebra Workbook, and Intermediate Algebra Workbook by Tyler Wallace which all hold a CC-BY License. Cover art by Sarah Adams with CC-BY-NC-SA license.

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Table of Contents

Unit 1: Integers and Algebraic Expressions....................................................................................8

1.1 Whole Numbers and Decimals.....................................................................................91.2 Operations with Integers............................................................................................191.3 Exponents and Order of Operations...........................................................................251.4 Simplify Algebraic Expressions...................................................................................29

Unit 2: Fractions...........................................................................................................................35

2.1 Prime Factorization....................................................................................................362.2 Reduce Fractions........................................................................................................392.3 Multiply and Divide Fractions.....................................................................................452.4 Least Common Multiple.............................................................................................532.5 Add and Subtract Fractions........................................................................................572.6 Order of Operations with Fractions............................................................................602.7 Mixed Numbers..........................................................................................................63

Unit 3: Linear Equations...............................................................................................................66

3.1 One Step Equations....................................................................................................673.2 Two Step Equations....................................................................................................713.3 General Linear Equations...........................................................................................733.4 Equations with Decimals and Fractions......................................................................76

Unit 4: Stats, Graphing, Proportions, and Percents.....................................................................78

4.1 Averages.....................................................................................................................794.2 Probability and Plotting Points...................................................................................844.3 Rates and Unit Rates..................................................................................................884.4 Proportions and Applications.....................................................................................904.5 Introduction to Percents............................................................................................924.6 Translate Percents and Applications..........................................................................944.7 Percents as Proportions and Applications..................................................................99

Unit 5: Geometry and Intro to Polynomials...............................................................................102

Conversion Factors.........................................................................................................1035.1 Convert Units........................................................................................................... 104Geometry Formulas.......................................................................................................1075.2 Area, Volume, and Temperature..............................................................................1085.3 Pythagorean Theorem..............................................................................................1195.4 Introduction to Polynomials.....................................................................................1225.5 Scientific Notation....................................................................................................126

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Unit 6: Proficiency Exam #1.......................................................................................................128

Unit 7: Linear Equations and Applications.................................................................................129

7. 1 Order of Operations................................................................................................1307.2 Evaluate and Simplify Algebraic Expressions............................................................1347.3 Solve Linear Equations.............................................................................................1387.4 Formulas.................................................................................................................. 1427.5 Absolute Value Equations........................................................................................1467.6 Word Problems........................................................................................................1497.7 Age Problems...........................................................................................................1547.8 Distance, Rate, Time Problems.................................................................................1577.9 Inequalities...............................................................................................................160

Unit 8: Graphing Linear Equations.............................................................................................165

8.1 Slope........................................................................................................................ 1668.2 Equations of Lines....................................................................................................1678.3 Parallel and Perpendicular Lines..............................................................................174

Unit 9: Polynomials....................................................................................................................177

9.1 Exponents.................................................................................................................1789.2 Scientific Notation....................................................................................................1849.3 Add, Subtract, Multiply Polynomials........................................................................1899.4 Polynomial Long Division..........................................................................................197

Unit 10: Factoring...................................................................................................................... 201

10.1 Factor Common Factors and Grouping..................................................................20210.2 Factor Trinomials ...................................................................................................20710.3 Factoring Tricks......................................................................................................21110.4 Factoring Strategy..................................................................................................21810.5 Solving Equations by Factoring...............................................................................219

Unit 11: Rational Expressions.....................................................................................................224

11.1 Reduce Rational Expressions..................................................................................22511.2 Multiply and Divide Rational Expressions...............................................................22811.3 Add and Subtract Rational Expressions..................................................................231Conversion Factors.........................................................................................................23711.4 Dimensional Analysis..............................................................................................238

Unit 12: Proficiency Exam #2.....................................................................................................241

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Unit 13: Compound Inequalities................................................................................................242

13.1 Review Inequalities................................................................................................24313.2 Compound Inequalities..........................................................................................24513.3 Absolute Value Inequalities....................................................................................250

Unit 14: Systems of Equations...................................................................................................253

14.1 Systems by Graphing..............................................................................................25414.2 Systems by Substitution and Addition/Elimination................................................25614.3 Systems with Three Variables................................................................................27014.4 Applications of Systems..........................................................................................274

Unit 15: Radicals........................................................................................................................ 286

15.1 Simplify Radicals.....................................................................................................28715.2 Add, Subtract, and Multiply Radicals......................................................................29115.3 Rationalize Denominators......................................................................................29815.4 Rational Exponents.................................................................................................30215.5 Radicals of Mixed Index..........................................................................................30615.6 Complex Numbers..................................................................................................309

Unit 16: Quadratics, Rational Equations, and Applications........................................................315

16.1 Complete the Square.............................................................................................31616.2 Quadratic Formula.................................................................................................32116.3 Equations with Radicals..........................................................................................32516.4 Equations with Exponents......................................................................................32916.5 Rectangle Problems................................................................................................33316.6 Rational Equations.................................................................................................33816.7 Work Problems.......................................................................................................34116.8 Distance and Revenue Problems............................................................................34316.9 Compound Fractions..............................................................................................350

Unit 17: Functions......................................................................................................................354

17.1 Evaluate Functions.................................................................................................35517.2 Operations on Functions........................................................................................35917.3 Inverse Functions...................................................................................................36717.4 Graphs of Quadratic Functions...............................................................................37217.5 Exponential Equations............................................................................................37417.6 Compound Interest................................................................................................37717.7 Logarithms............................................................................................................. 379

Unit 18: Proficiency Exam #3.....................................................................................................382

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Unit 1:

Integers and Algebraic Expressions

To work through the unit you should:

1. Watch a video, as you watch, fill out the workbook (top and example sections).2. Complete Q1 and Q2 in WAMAP, put your work in the bottom sections of the page.3. Repeat #1 and #2 with each page until you reach . 4. Complete the homework assignment on your own paper.5. Repeat #1-#4 until you reach the end of the unit.6. Complete the practice test on your own paper.7. Take the unit exam.

9

1.1a Whole Numbers and Decimals – Rounding Whole Numbers

Whole numbers are:

Place Value:

4, 2 8 7, 1 9 2

Rounding: Look at the _____________ digit. Round up if it is __________ and round down if it is __________

Example 1:Round 5,459,246

To the nearest thousand

Example 2:Round 5,459,246

To the nearest hundred-thousand

Q 1: Q 2:

10

1.1b Whole Numbers and Decimals – Rounding Decimals

Decimals are ___________ of the ______________

Place Value:

8. 1 7 2 6 9 3

Example 1:Round 4.01276

To the nearest thousandth

Example 2:Round 4.01276

To the nearest hundredth

Q 1: Q 2:

11

1.1c Whole Numbers and Decimals – Add Whole Numbers

To add we _________________ place values and work __________ to _____________

Example 1:458+321 Example 2:716+485

Q 1: Q 2:

1.1d Whole Numbers and Decimals – Add Decimals

12

When adding decimals we must _________________ the ___________________

Place decimal:

Example 1:4.21+8.962

Example 2:0.523+0.08

Q 1: Q 2:

1.1e Whole Numbers and Decimals – Subtract Whole Numbers

13

To subtract we __________________ place value and work ___________ to ______________

Example 1:967−341

Example 2:5037−2419

Q 1: Q 2:

1.1f Whole Numbers and Decimals – Subtract Decimals

14

When subtracting decimals we must _________________ the ___________________

Important: We may need additional ________________ to line up!

Place decimal:

Example 1:3.4−1.29

Example 2:4.03−0.051

Q 1: Q 2:

1.1g Whole Numbers and Decimals – Multiply Whole Numbers

15

Multiply __________ the digits together

Use _____ to hold place value

After multiplying we _________

Different ways show “multiply”:

Example 1:23 ∙56

Example 2:167(48)

Q 1: Q 2:

1.1h Whole Numbers and Decimals – Multiply Decimals

16

Place decimal:

Example 1:4.2 ∙1.8

Example 2:2.6(3.52)

Q 1: Q 2:

1.1i Whole Numbers and Decimals – Divide Whole Numbers

17

Long division places the ____________ number in front!

The leftovers:

Different ways to show “divide”:

Example 1:452÷13

Example 2:1202424

Q 1: Q 2:

1.1j Whole Numbers and Decimals – Divide Decimals

18

No decimals in the ________________ or ________________

Move the ________________ in both the _______________ and _____________________

If you run out of digits you can _______________________

Place decimal:

Example 1:2.5682.4

Example 2:19.5÷25

Q 1: Q 2:

You have completed the videos for 1.1 Whole Numbers and Decimals. On your own paper complete the homework assignment.

1.2a Operations with Integers – Integers and Absolute Value

19

Integers:

Opposite means a _________________ over ____________________ on the ________________________

The symbol ____ means _________________, __________________ AND __________________ !

Absolute Value is the _______________ from zero. It is always ________________

Example 1:−(−6)

Example 2:−(3)

Example 3:|−8|

Example 4:¿4∨¿

Example 5:−¿−7∨¿

Example 6:−¿2∨¿

Q 1: Q 2:

Q 3: Q 4:

Q 5: Q 6:

1.2b Operations with Integers – Multiply and Divide with Different Signs

20

A pattern to multiplying: 2 ∙2=¿2 ∙1=¿2 ∙0=¿2 ∙ (−1 )=¿

Multiplying and dividing with a negative and a positive (or positive and negative) is a _________________

To remember: When _________ things happen to _________ people it is a _________ thing


Example 1:−54÷9

Example 2:3(−8)

Q 1: Q 2:

1.2c Operations with Integers – Multiply and Divide with the Same Sign

21

A pattern to multiplying: −2 ∙2=¿−2 ∙1=¿−2 ∙0=¿

−2 ∙ (−1 )=¿

Multiplying and dividing a negative times a negative is a ________________


Example 1:−7(−4)

Example 2:−15−3

Q 1: Q 2:

1.2d Operations with Integers – Add with the Same Sign

22

Adding and Subtracting Integers: Keep the ___________ with the ______________ after it

Double signs: 3+(−4 )=¿ 3−(−4 )=¿

Visualize: −2+(−3 ):

Add with the same sign:

Example 1:−7+(−4)

Example 2:(−6 )+(−8)

Q 1: Q 2:

1.2e Operations with Integers – Add with Different Signs

23

Visualize: −2+4

Visualize: 1+(−3)

Adding with different signs:

Example 1:5+(−2)

Example 2:−9−(−4)

Q 1: Q 2:

1.2f Operations with Integers – Add and Subtract Several Integers

24

When adding and subtracting many integers we work __________ to ____________

Example 1:−5+(−2 )−(−6 )−4+8

Example 2:4−8+(−3 )−(−1 )+3

Q 1: Q 2:

You have completed the videos for 1.2 Operations with Integers. On your own paper complete the homework assignment.

1.3a Exponents and Order of Operations – Exponents

25

Exponents are ______________________________

53=¿

Example 1:25

Example 2:72

Q 1: Q 2:

1.3b Exponents and Order of Operations – Exponents on Negatives

26

Exponents only effect what they are ________________.

(−5 )2=¿ −52=¿

Example 1:

−34

Example 2:

(−2 )6

Q 1: Q 2:

1.3c Exponents and Order of Operations – Order of Operations

27

Why we need an order: 2+3(4) 2+3(4)

The order:

Example 1:23+5(4−7)

Example 2:24÷6 ∙2−32(7−9)

Q 1: Q 2:

1.3d Exponents and Order of Operations – Order with Absolute Value

28

Absolute values works just like ______________ but makes the number inside __________

after it has been _______________

Example 1:3−2∨7−42∨¿

Example 2:|4 (2 )−6|3−42

Q 1: Q 2:

You have completed the videos for 1.3 Exponents and Order of Operations. On your own paper complete the homework assignment.

1.4a Simplify Algebraic Expressions – Substitute a Value

29

I have a ______________ eggs means I have _____ eggs.

Variables are ____________ that represent ______________ amounts

If we know the amount we can ____________ it in an expression

Whenever we make a substitution or __________________ put it in ____________________

Example 1:Evaluate: −x2−7 x−12

When x=−4

Example 2:Evaluate: b2−4 ac

When a=2 , b=−3 ,∧c=−5

Q 1: Q 2:

1.4b Simplify Algebraic Expressions – Is it a Solution?

30

An equation is made up of two ______________ expressions

A solution is the value of the ______________ that makes the equation ________________

Example 1:Is x=3 the solution to

−2 x+7=1 ?

Example 2:Is x=−3 the solution to

2 x−5=7 x+5?

Q 1: Q 2:

1.4c Simplify Algebraic Expressions – Combine Like Terms

31

John has 5 cats and 3 dogs. Sue has 2 cats and 1 dog. Together they have ___ cats and ___ dogs.

Terms are __________ and ____________ that are ________________ together

Like terms are terms that have matching _______________ and _______________

Combine like terms: ___________ the coefficients or ______________ from __________________

Example 1:9 x+2 y−7 x−5 y+2x

Example 2:5 x2−2x−9+4 x−7 x2+6

Q 1: Q 2:

1.4d Simplify Algebraic Expressions – Distributive Property

32

Multiplication is __________________

3 (2x+5 )=¿

Distributive Property: 3(2 x+5)

Example 1:−4(2 x+5 y−7)

Example 2:7(9x2−7 x+8)

Q 1: Q 2:

1.4e Simplify Algebraic Expressions – Distribute and Combine Like Terms

33

Order of operations states we ________________ before we ____________________

Therefore we will _________________ first and then _________________________ second

Example 1:

5 (2x+6 y−2 )−4(x+3−6 y)

Example 2:

2 (4 x2−6x+1 )−(x2+5 x+3)

Q 1: Q 2:

1.4f Simplify Algebraic Expressions – Perimeter Problems

34

Perimeter:

Find the perimeter by ________________ the sides together

Example 1: Example 2:

Q 1: Q 2:

You have completed the videos for 1.4 Simplify Algebraic Expressions. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 1: Integers and Algebraic Expressions. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

35

5 x−4

6−4 x

3

1−x

7−2x

3+x−9−2x

8−4 x 3 x+7

Unit 2:

Fractions



36

2.1a Prime Factorization – Prime and Composite

Prime numbers are divisible by _____ and ___________

Examples of Primes:

Composite numbers are divisible by ______________________

Example 1:Prime or Composite:

89

Example 2:Prime or Composite:

147

Q 1: Q 2:

37

2.1b Prime Factorization – Divisibility Tests

A number is divisible by a smaller number if the small number ___________________ into the number

Divisibility tests:

2:

3:

5:

7, 11, 13, 17, 19:

Example 1:2730 is divisible by which prime numbers?

Example 2:133 is divisible by which numbers?

Q 1: Q 2:

38

2.1c Prime Factorization – Prime Factorization

Prime Factorization: a _______________ of __________________ numbers

To find a prime factorization we divide by _________________

Example 1:Find the prime factorization of

360

Example 2:Find the prime factorization of

1224

Q 1: Q 2:

You have completed the videos for 2.1 Prime Factorization. On your own paper complete the homework assignment.

39

2.2a Reduce Fractions – Introduction to Fractions

Fraction is a ______________ of a ________________

Example: 45 where the 4 is the ________, called the _______________ and the 5 is the ________ called the

____________

Example 1:What fraction is shaded?

Example 2:What fraction is shaded?

Q 1: Q 2:

40

http://www.clipartstation.com/clipart_indexer4/index/view?id=21357

http://www.clipartstation.com/clipart_indexer4/index/view?id=21184

2.2b Reduce Fractions – Convert Fractions to Decimals

The fraction bar represents _______________________

To convert a fraction to a decimal we _________________

To help with this we will use a ____________________

If the decimal repeats we will use a ___________

Example 1:

Convert to decimal: 732

Example 2:

Convert to decimal: 3299

Q 1: Q 2:

41

2.2c Reduce Fractions – Equivalent Fractions

Equivalent fractions:

To find an equivalent fraction ______________________________ the _____________________________

and __________________________ by the ______________________________

Example 1:Find three equivalent fractions:

37

Example 2:Find three equivalent fractions:

43

Q 1: Q 2:

42

2.2d Reduce Fractions – Reduce with Prime Factorizations

Reduced Fraction: The __________________ and ___________________ have no common _____________

To reduce we find the ___________________________ and divide out ________________________

Example 1:2436

Example 2:10570

Q 1: Q 2:

43

2.2e Reduce Fractions - Reduce

Sometimes we can _________ the common factors and ___________________________

Example 1:2436

Example 2:10570

Q 1: Q 2:

44

2.2f Reduce Fractions – Reduce with Variables

When reducing with variables, ___________________ the variables that are in _______________

With exponents it may help to _____________________

Example 1:4 x2 yz10x y3

Example 2:27a3bc9a2b2c

Q 1: Q 2:

You have completed the videos for 2.2 Reduce Fractions. On your own paper complete the homework assignment.

45

2.3a Multiply and Divide Fractions – Multiply with No Reducing

Multiply the ____________________ together and multiply the _______________________ together

Example 1:47

∙ 53

Example 2:16

∙ 54

Q 1: Q 2:

46

2.3b Multiply and Divide Fractions – Multiply with Reducing

Consider: 49

∙ 65=¿

But if we factor each: 49

∙ 65=¿

When _____________ we can _____________ a common factor from the ____________ and ____________

Example 1:635

∙ 1415

Example 2:614

∙ 3513

Q 1: Q 2:

47

2.3c Multiply and Divide Fractions – Multiply with Variables

With exponents on variables it may help to _______________________

Remember, repeated ________________ is done with ____________________

Example 1:6 x2 y7

∙ 14 y3 x

Example 2:30a3b2

∙ 21ab10

Q 1: Q 2:

48

2.3d Multiply and Divide Fractions – Multiply with Whole Numbers and Fractions

Whole numbers can be made into fractions by putting them over _____

Example 1:38

∙20

Example 2:

35 ∙ 67

Q 1: Q 2:

49

2.3e Multiply and Divide Fractions - Reciprocals

Reciprocal:

Reciprocals multiply to ____

Example 1:

Find the reciprocal of 65

Example 2:Find the reciprocal of −8

Q 1: Q 2:

50

2.3f Multiply and Divide Fractions – Divide Fractions

Divide fractions by _________________ by the __________________

Example 1:1415

÷ 356

Example 2:310

÷ 615

Q 1: Q 2:

51

2.3g Multiply and Divide Fractions – Divide with Variables

With exponents on variables it may help to _______________________

Remember, repeated ________________ is done with ____________________

Example 1:10x3 y2

÷ 1021xy

Example 2:14m3n

÷ 76m2n

Q 1: Q 2:

52

2.3h Multiply and Divide Fractions – Divide with Whole Numbers and Fractions

Whole numbers can be made into fractions by putting them over _____

Example 1:

28÷ 78

Example 2:49

÷14

Q 1: Q 2:

53

You have completed the videos for 2.3 Multiply and Divide Fractions. On your own paper complete the homework assignment.

2.4a Least Common Multiple - Multiples

Multiples are the found by ______________ by other numbers

Example 1:Find the first three multiples of 8

Example 2:Find the first three multiples of −7

Q 1: Q 2:

54

2.4b Least Common Multiple – LCM Using Mental Math

Least Common Multiple (LCM):

Multiples of 15:

Multiples of 20:

Common multiples of 15 and 20:

Least common multiple of 15 and 20:

Using mental math: Test _____________ of the _______ number: Can it be divided by the ______________

Example 1:Find the LCM of 12 and 9


Q 1: Q 2:

55

2.4c Least Common Multiple – LCM Using Prime Factorization

To find an LCM of two larger numbers:

1. Find the ______________________ of each

2. Use all the unique ___________________

3. Assign the _________________________ to each factor



Q 1: Q 2:

56

2.4d Least Common Multiple – LCM with Variables

To find the LCM with variables:

1. Use all the unique ____________________

2. Assign the _________________________ to each variable

Example 1:Find the LCM of a3b2c and a2b7d2

Example 2:Find the LCM of 6 x2 z and 8 x3 y2

Q 1: Q 2:

57

You have completed the videos for 2.4 Least Common Multiple. On your own paper complete the homework assignment.

2.5a Add and Subtract Fractions – With Common Denominator

Consider: 25+ 15

+ =To add fractions that have the same denominator: _________ numerators and _____________ denominators

When adding fractions always check to ___________ at the ___________ of the problem

Example 1:47−27

Example 2:710

+ 510

Q 1: Q 2:

58

2.5b Add and Subtract Fractions – With Different Denominators

If the denominators don’t match we will find the ________________________

Multiply ______________________ by missing factors.

Then multiply the ___________________ by the ___________ factors

If you do not know the LCD you can always ______________ the two _______________

Example 1:53+ 49

Example 2:34−56

Q 1: Q 2:

59

2.5c Add and Subtract Fractions – With Different Large Denominators

We may have to use _________________ to find the LCD.

To build up to the LCD we multiply by any _____________________ factors

Example 1:724

+ 1136

Example 2:554

− 790

Q 1: Q 2:

60

You have completed the videos for 2.5 Add and Subtract Fractions. On your own paper complete the homework assignment.

2.6a Order of Operations with Fractions – Exponents on Fractions

Exponents mean ______________________________

( 23 )2

=¿

Or we could put the exponent on the __________________ and ____________________

Example 1:

( 32 )3

Example 2:

( 74 )2

Q 1: Q 2:

61

2.6b Order of Operations with Fractions – Order of Operations with Fractions

The order

1.

2.

3.

4.

You may need some _________________

Example 1:910

÷ 125

+( 52 )2

∙ 130

Example 2:

( 85 )2

− 910 |73−9

2|

62

Q 1: Q 2:

63

You have completed the videos for 2.6 Order of Operations with Fractions. On your own paper complete the homework assignment.

2.7a Mixed Numbers – Mixed Numbers and Conversions

Mixed number:

Change a mixed number to a fraction: ___________ the whole and ____________

and ___________ the __________________

Change a fraction to a mixed number: _______________, the remainder is the new _________________

Example 1:

Convert 5911 to a fraction

Example 2:

Convert 7312 to a mixed number

Q 1: Q 2:

64

2.7b Mixed Numbers – Add and Subtract Mixed Numbers

To do math with mixed numbers it is easiest to _____________ to a _________________

When you have your answer, ________________________

Example 1:

5 25+7 310

Example 2:

−2 13+6 49

Q 1: Q 2:

65

2.7c Mixed Numbers – Multiply and Divide Mixed Numbers

To do math with mixed numbers it is easiest to _____________ to a _________________

When you have your answer, _______________________

Example 1:

2 45

∙3 47

Example 2:

5 13

÷2 16

Q 1: Q 2:

You have completed the videos for 2.7 Mixed Numbers. On your own paper complete the homework assignment.

66

Congratulations! You made it through the material for Unit 2: Fractions. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

Unit 3:

Linear Equations



67

3.1a One Step Equations – Addition Principle

A _______________ to an equation is the value for the ____________ that makes the equation ___________

We can _________ anything to __________________ of the equation

Addition Principle: To move a negative term we do the opposite and _________ it to ______________

Very Important to __________ your work!

Example 1:x−9=4

Example 2:−3=−5+x

Q 1: Q 2:

68

3.1b One Step Equations – Subtraction Principle


Subtraction Principle: To move a positive term we do the opposite and __________ it from ______________


Example 1: x+8=−4

Example 2:3=7+x

Q 1: Q 2:

69

3.1c One Step Equations – Division Principle


Division Principle: To undo multiplication of factors we do the opposite and _______________ it

from ______________


Example 1:7 x=147

Example 2:−8 x=72

Q 1: Q 2:

70

3.1d One Step Equations – Multiplication Principle

We can ____________ anything to __________________ of the equation

Multiplication Principle: To clear division we do the opposite and _______________ it by ______________


Example 1:x7=−4

Example 2:

5= x−2

Q 1: Q 2:

You have completed the videos for 3.1 One Step Equations. On your own paper complete the homework assignment.

71

3.2a Two Step Equations – Two Steps

Simplifying we use order of operations and we _____________________ before we ____________________

Solving we work __________________ and we _____________________ before we ____________________

Example 1:5 x−7=8

Example 2:−9=−5−2x

Q 1: Q 2:

72

3.2b Two Step Equations – Negative Variables

If there is no number in front of a variable, we assume there is a _____ in front

This means −x is the same as _________

Example 1:−x+8=5

Example 2:−4=−6−x

Q 1: Q 2:

You have completed the videos for 3.2 Two Step Equations. On your own paper complete the homework assignment.

73

3.3a General Linear Equations – Variable on Both Sides

When solving an equation we want the variable on _______________________________

If the variable is on both sides we will ________ the _____________ one by _______________________

Example 1:5 x+7=9x−2

Example 2:−6 x+1=2 x−12

Q 1: Q 2:

74

3.3b General Linear Equations – Combine Like Terms

Before we solve we must _______________ the __________ and ___________ sides

One way to do this is ________________________

Example 1:5 x−3−2x=7+8 x−1

Example 2:4+x−2=−3 x+8+2x

Q 1: Q 2:

75

3.3c General Linear Equations – Distribute and Combine

Before we solve we must _______________ the __________ and ___________ sides

One way to do this is ________________________

Example 1:2 (3x−1 )=4 x+6−x

Example 2:3 (2x+1 )−9 x=4 (x+6 )−20

Q 1: Q 2:

You have completed the videos for 3.3 General Linear Equations. On your own paper complete the homework assignment.

76

3.4a Equations with Decimals and Fractions – Decimals

When solving with decimals the pattern of solving is ____________________

A ______________ may be helpful to speed up calculations

Example 1:3.2 x+7.11=−19.77

Example 2:2.1 ( x−4.3 )=5.7 x−9.19−3.8x

Q 1: Q 2:

77

3.4b Equations with Decimals and Fractions – Clear Fractions with LCD

If the equation has fractions, we can clear the fractions by ____________________ each term by the _____

After multiplying we can _________________ to get an equation with no __________________

Example 1:56

x−13=72

Example 2:38

x+ 34=−5+7

2x

Q 1: Q 2:

You have completed the videos for 3.4 Equations with Decimals and Fractions. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 3: Linear Equations. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

78

Unit 4:

Stats, Graphing, Proportions and Percents

(You may use a calculator on this unit)



79

4.1a Averages - Mean

Mean: The average if all items were the __________________ or spread out __________________

To calculate the mean:

Example 1:Find the Mean:

5 ,8 ,6 ,7 ,9 , 4 ,8 ,10

Example 2:Find the Mean:

23 ,26 ,27 ,21 ,26 ,22 ,73 ,24 ,23

Q 1: Q 2:

80

4.1b Averages – Missing Value

The mean is when all the items are the _________________ or spread out _____________________

If we know the mean and are missing a value, calculate the _________________ using the _______________

Example 1:On three tests a student earns 83%, 71%, and 81%. What must she earn on her fourth test to raise her average of the four tests up to 80%?

Example 2:Another student has a goal of 90% on his four tests. On the first three tests he earned 92%, 75%, and 89%. Is it possible for him to reach his goal of 90%? What score would he have to earn?

Q 1: Q 2:

81

4.1c Averages – Weighted Mean

Weighted average: Values that occur _______________ have a larger _____________ on the average (mean)

To calculate the total we ________________ the _____________________ by the ___________________

Example 1:In a survey, students were asked how many siblings they had. The results are below. Calculate the average number of siblings of the survey responders.

Siblings Responses0 81 382 213 154 2

Example 2:Grade Point Average (GPA) is calculated as a weighted average. The credits of a course are considered the “frequency” of the course. In this way, classes that are more credits have a larger effect on grade than classes with fewer credits. Calculate the GPA of the following report card:

Class Credits GradeEnglish 4 3.2Math 5 4.0History 3 2.8PE 1 0.7

Q 1: Q 2:

82

4.1d Averages - Median

Median: The average at which ___________ the data is _______________ and __________ is ____________

To calculate the median:

If two values are in the middle:

Example 1:Find the Median:5 ,8 ,6 ,7 ,9 , 4 ,8 ,10

Example 2:Find the Median:

23 ,26 ,27 ,21 ,26 ,22 ,73 ,24 ,23

Q 1: Q 2:

83

4.1e Averages - Mode

Mode: the average or value that occurs _______________________

It is possible to have _______________ modes or ____ mode

Example 1:Find the Mode:

5 ,8 ,6 ,7 ,9 , 4 ,8 ,10

Example 2:Find the Mode:

23 ,26 ,27 ,21 ,26 ,22 ,73 ,24 ,23

Q 1: Q 2:

You have completed the videos for 4.1 Averages. On your own paper complete the homework assignment.

84

4.2a Probability and Plotting Points – Basic Probability

Probability:

Basic Probability Fraction:

Example 1:A bag contains 3 blue marbles, 2 red marbles and 1 green marble. If you were to draw one marble at random, what is the probability of drawing...

1) A blue marble?

2) A red marble?

3) A black marble?

Example 2:If you roll a standard six-sided die, what is the probability you roll…

1) A three?

2) An even number?

3) A number smaller than three?

4) A seven?

Q 1: Q 2:

85

4.2b Probability and Plotting Points – Compound Events

The probability of this OR that: we __________ the individual probabilities

The probability of this AND that: we _______________ the individual probabilities

Example 1:A bag contains 3 blue marbles, 2 red marbles and 1 green marble. If you were to draw one marble at random, what is the probability of drawing...

1) A blue or green marble?

2) A green or red marble?

3) A blue or red or green marble?

Example 2:If you roll a standard six-sided die and then draw a marble out of a bag with 7 red and 3 black marbles, what is the probability you get…

1) A three and a black?

2) An even and a red?

Q 1: Q 2:

86

4.2c Probability and Plotting Points – Give Coordinate

Coordinate Plane:

x-axis:

y-axis:

Origin:

Coordinate Point:

Example 1:Give the coordinates of points A, B, and C

Example 2:Give the coordinates of points E, F, and G

Q 1: Q 2:

87

A

B

C

4.2d – Probability and Plotting Points – Plot Points

To plot a point start at the ______________________ and move ________________ then ____________

Negatives move the point _________________________________

Example 1:Plot the points:

A (2 ,−4 ) ,B (−3 ,−1 ) ,C (0,2 ) , D(−3,4 )

Example 2:Plot the points:

E (1,2 ) ,F (−1,0 ) ,G(0,0)

Q 1: Q 2:

You have completed the videos for 4.2 Probability and Plotting Points. On your own paper complete the homework assignment.

88

4.3a Rates and Unit Rates – Find Rates

Rate: Amount per ____________

To set up, the word ___________ is the ________________

Example 1:Rafael made $22,512 last year. What is his rate of pay per month?

Example 2:Giovanni covered his 2,500 square foot yard with 700 ounces of fertilizer. What is the rate of coverage the fertilizer can cover in ounces per square foot?

Q 1: Q 2:

89

4.3b Rates and Unit Rates – Find Unit Rate

Unit Rate: Rate of _________________ per ________________________

Use unit rates to identify the ____________________

Example 1:A 20 ounce bottle of soda sells for $1.99. What is the unit price?

Example 2:Lemon juice comes in a 24 oz bottle and a 32 oz bottle. The 24 oz bottle sells for $1.98 and the 32 oz bottle sells for $2.98. Which is the better deal and what is the unit price?

Q 1: Q 2:

You have completed the videos for 4.3 Rates and Unit Rates. On your own paper complete the homework assignment.

90

4.4a Proportions and Applications – Solving Proportions

To solve a proportion we ________________ both sides by the ___________________

The quick method: Multiply by the __________________

Example 1:7x=65

Example 2:85= x3

Q 1: Q 2:

91

4.4b Proportions and Applications – Proportion Applications

When solving applications we must first identify what we are ____________________

Clearly label the _____________________ and __________________________ of the proportion

Example 1:A 65 inch tall man wants to determine how tall a large tree is. He noticed at a certain time his shadow was 14 inches long. When he measured the shadow of the tree he found it was 48 inches long. How tall is the tree?

Example 2:A manufacturer knows that out of every 300 parts the company ships, on average 18 are defective. If the company ships 5800 parts in a day, how many will be defective?

Q 1: Q 2:

You have completed the videos for 4.4 Proportions and Applications. On your own paper complete the homework assignment.

92

4.5a Introduction to Percents – Convert Percents and Decimals

Percent:

To convert a decimal to a percent: Multiply by ______ or move the decimal ____________ to the _________

To convert a percent to a decimal: Divide by ______ or move the decimal ____________ to the _________

Example 1:Convert 0.582 to a percent

Example 2:Convert 145.6% to a decimal

Q 1: Q 2:

93

4.5b Introduction to Percents – Convert Percents and Fractions

To convert a fraction to a percent: First _______________ then convert the ____________ to a ___________

To convert a percent to a fraction: Put the percent over __________ and _____________________

Example 1:

Convert 1720 to a percent

Example 2:Convert 32% to a fraction

Q 1: Q 2:

You have completed the videos for 4.5 Introduction to Percents. On your own paper complete the homework assignment.

94

4.6a Translate Percents and Applications – Translate and Solve

Key words to translate:

What

Is

Of

Percent

Example 1:What is 70% of 40?

Example 2:45% of what is 70?

Q 1: Q 2:

95

4.6b Translate Percents and Applications - Discount

Discount Equation:

Example 1:A computer that normally costs $549 is on sale at 22% off. What is the new price of the computer?

Example 2:A desk that normally sells for $224 is on sale for $188.16. What is the percent discount?

Q 1: Q 2:

96

4.6c Translate Percents and Applications – Sales Tax

Sales Tax Equation:

Example 1:What is the sales tax on a $499 television if the tax rate is 8.5%? What would you pay for the TV?

Example 2:Marc purchased a $390 table and paid $25.35 in sales tax. What is the tax rate?

Q 1: Q 2:

97

4.6d Translate Percents and Applications – Commission

Commission Equation:

Example 1:A cell phone company pays a 14% commission to its representatives. If an employee sells $23,000 worth of product in a month, how much will she be paid?

Example 2:A real estate agent sold $845,000 worth of properties and earned $16,900 in his commission. What is his commission rate?

Q 1: Q 2:

98

4.6e Translate Percents and Applications – Simple Interest

Interest:

Simple Interest Equation:

Time must be in ______________

Example 1:A bank account pays 2.1% simple interest on a certain account. If you invest $3500 for 4 years, how much will you earn in interest?

Example 2:A bank gives a loan with 4.5% simple interest for 9 months on a $12,000 loan. How much is owed back to the bank at the end of the loan?

Q 1: Q 2:

You have completed the videos for 4.6 Translate Percents and Applications. On your own paper complete the homework assignment.

99

4.7a Percents as Proportions and Applications – Translate and Solve

Percent Proportion:

Example 1:What percent of 25 is 16?

Example 2:14 is 60% of what?

Q 1: Q 2:

100

4.7b Percents as Proportions and Applications – General Applications

“OF” represents ___________________

“IS” represents ____________________

Example 1:Among male smokers, the lifetime risk of developing lung cancer is 17.2%. According to the Washington State Department of Health, in 2011 the state had 760,000 smokers. How many are at risk of developing lung cancer in their lifetime?

Example 2:In 2010, women made up 58% of Big Bend Community College’s students. If there were 1688 women enrolled in 2010, how many students were there total?

Q 1: Q 2:

101

4.7c Percents as Proportions and Applications – Percent Increase/Decrease

Percent Increase/Decrease Proportions:

Example 1:The price of a sofa was $299. During a weekend sale the price was dropped to $179. What was the percent decrease?

Example 2:The population of a small town was 12,345 in 1990. By 2000, the population was 31,416. What was the percent increase?

Q 1: Q 2:

You have completed the videos for 4.7 Percents as Proportions and Applications. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 4: Stats, Graphing, Proportions, and Percents. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

102

Unit 5:

Geometry and Intro to Polynomials

(You may use a calculator on this unit)



103

104

5.1a Convert Units – One Step Conversions

Consider: ¿

Divide out units by placing them in the __________________ part of the fraction

Conversion factor: Same _________ in numerator and denominator, but different _________

Dimensional analysis: Multiply by a _____________________ to convert units

Example 1:17.2 in = _____ cm

Example 2:88 lbs = _____kg

Q 1: Q 2:

5.1b Convert Units – Multi-Step Conversions

105

If we do not have the correct conversion factor we can convert using _______________ conversion factors

Example 1:365 g = _____ lbs

Example 2:5 gal = _____ cups

Q 1: Q 2:

5.1c Convert Units – Dual Unit Conversions

106

Dual Unit:

“Per” is the _________________________

With dual units we convert ___________________________

Example 1:45 oz per min = _____ lbs per hr

Example 2:35 mi per hr = _____ ft per sec

Q 1: Q 2:

You have completed the videos for 5.1 Convert Units. On your own paper complete the homework assignment.

107

108

5.2a Area, Volume, and Temperature – Area of Rectangle and Parallelogram

Area of a rectangle:

Area of a parallelogram:

Important: height must meet the base at a perfect ___________ angle

Example 1:Find the area:


Q 1: Q 2:

109

15 in

8 in 7 cm

21 cm

9 cm

5.2b Area, Volume, and Temperature – Area of Triangle

Area of a triangle:




Q 1: Q 2:

110

8 ft

5 ft

6 m

5 m 3 m

4.3 m

5.2c Area, Volume, and Temperature – Area of Trapezoid

Area of a trapezoid:

The bases (a and b) must be __________________

The _____________ connects the two _________________




Q 1: Q 2:

111

6 cm

12 cm

5 cm5 cm 4 cm

4 in

2 in

9 in

5 in

5.2d Area, Volume, and Temperature – Area of Circle

Diameter:

Radius:

π=¿

Area of a Circle:



Q 1: Q 2:

112

4 ft

12 cm

5.2e Area, Volume, and Temperature – Composition of Shapes

If we do not have a formula for a shape we must ___________________________

Shapes attached to each other are _________________

Shapes cut out are _____________________



Q 1: Q 2:

113

9 in

4 in6 in 2 mm

5.2f Area, Volume, and Temperature – Volume of a Rectangular Solid

Volume of rectangular solid:

Example 1:Find the volume:

Example 2:On Southwest Airlines, the maximum size of a carry-on bag is a length of 24 inches, a width of 10 inches and a height of 16 inches. How much is packed in this maximum sized bag?

Q 1: Q 2:

114

247 ft

32 ft

143 ft

5.2g Area, Volume, and Temperature – Volume of a Cylinder

Volume of a cylinder:


Example 2:A can of soda is about 5 inches tall with a diameter of 2.5 inches. How much soda can fit into the can?

Q 1: Q 2:

115

8 ft

6 ft

5.2h Area, Volume, and Temperature – Volume of a Cone

Volume of a cone:


Example 2:How much ice cream can fit inside a cone that is 5 cm tall and 3 cm wide?

Q 1: Q 2:

116

24 ft2 ft

5.2i Area, Volume, and Temperature – Volume of a Sphere

Volume of a sphere:


Example 2:Find the volume of the “planet” Pluto if the radius is approximately 1195 km.

Q 1: Q 2:

117

9 ft

5.2j Area, Volume, and Temperature – Composition of Solids

If we do not have a formula for a shape we must ___________________________

Example 1:Find the volume

Q 1:

118

4 ft

10 ft

5.2k Area, Volume, and Temperature – Convert Temperature

Formulas to convert temperature:

C=¿ F=¿

Example 1:98.6ᵒ F is body temperature. Find body temperature in degrees Celsius.

Example 2:100ᵒ C is the boiling point of water. Find at what temperature water boils in degrees Fahrenheit.

Q 1: Q 2:

You have completed the videos for 5.2 Area, Volume, and Temperature. On your own paper complete the homework assignment.

119

5.3a Pythagorean Theorem – Square Roots

Square root asks the question what number ______________ is the inside number?

√25=¿

On calculator use the _____ button (may have to use the ______ or ______ button first)

Example 1:√225

Example 2:√456

Q 1: Q 2:

120

5.3b Pythagorean Theorem – Find Missing Side

Name the sides of the right triangle:

Pythagorean Theorem:

C is always the ______________________

When finding a leg with the Pythagorean theorem, first _________________ then _______________

Example 1:Find the missing side:

Example 2:Find the missing side:

Q 1: Q 2:

121

2 yds

7 yds

4 cm

8 cm

5.3c Pythagorean Theorem – Applications

With applications it is always helpful to __________________________________

Example 1:The base of a ladder is four feet from a building. The top of the ladder is eight feet up the building. How long is the ladder?

Example 2:A young boy is flying a kite. He let out 21 meters of string until the kite was flying over the head of his sister who was 9 meters away. How high is the kite?

Q 1: Q 2:

You have completed the videos for 5.3 Pythagorean Theorem. On your own paper complete the homework assignment.

122

5.4a Introduction to Polynomials – Add Polynomials

Term:

Monomial:

Binomial:

Trinomial:

Polynomial:

Adding Polynomials: _________________ the _________________________

Example 1:(4 x3−2x2+x )+(−3 x2−5x+7)

Example 2:(3ab3−2a2b+ab )+(4a2b−2ab3+4ab)

Q 1: Q 2:

123

5.4b Introduction to Polynomials – Subtract Polynomials

First ________________ the _________________.

Then ______________________________

Example 1:(3 x2−7 x+8 )−(2 x2+9x−4)

Example 2:(2 x−8 y+6 )−(−3 y−7+5 x)

Q 1: Q 2:

124

5.4c Introduction to Polynomials – Multiply Monomials

Consider a3 ∙ a2=¿

When multiplying variables we __________ the exponents

If there is no exponent, then we assume the exponent is a ___

Example 1:(4 x3 y2 z) (2 x7 y z4 )

Example 2:−7a3b2c4 ∙2ab8 c4

Q 1: Q 2:

125

5.4d Introduction to Polynomials – Multiply a Monomial by a Polynomial

Consider: 5 (3 x−6 )=¿

When multiplying a monomial by a polynomial we ___________________

Example 1:

5 x3(3 x2−4 x+2)

Example 2:

−2a3b(5ab4−6a2b7+2a4)

Q 1: Q 2:

You have completed the videos for 5.4 Introduction to Polynomials. On your own paper complete the homework assignment.

126

5.5a Scientific Notation – Convert Scientific Notation to Standard Notation

Scientific Notation: a×10b

a is greater than or equal to ____ but less than _____. This means the decimal is always after the

_____________ digit

b tells us how many times to ___________ the decimal

If b is negative, then standard notation is ____________. If b is positive, then standard notation is _________

Example 1:Convert to standard notation:

4.21×105

Example 2:Convert to standard notation:

6.2×10−3

Q 1: Q 2:

127

5.5b Scientific Notation – Convert Standard Notation to Scientific Notation

To convert to scientific notation _____________ the number of times the _______________ must move

If standard notation is small then the exponent is ____________ if it is big then the exponent is ___________

When converting we will __________ the extra ________________

Example 1:Convert to scientific notation:

48,100,000,000

Example 2:Convert to scientific notation

0.0000235

Q 1: Q 2:

128

You have completed the videos for 5.5 Scientific Notation. On your own paper complete the homework assignment.

Unit 6:

Proficiency Exam #1

To work through this unit you should:

1. Complete the practice tests on your own paper.2. Take the (three part) unit exam.

129

Unit 7:

Linear Equations and Applications



130

7.1a Order of Operations – The Order

The Order:

1.

2.

3.

4.

To remember:

Example 1:5−3(2+42)

Example 2:30÷5 (−2 )+ (4−7 )2

Q 1: Q 2:

131

7.1b Order of Operations – Lots of Parentheses

Different types of parenthesis:

Always do _____________________________ first!

Example 1:(4+2 )−[52÷ (2+3 )]

Example 2:7 {2+2 [20÷ (4+6 ) ] }

Q 1: Q 2:

132

7.1c Order of Operations – Fractions

When simplifying fractions, always simplify _________________ and _________________ first

Only reduce at after the rest has been ______________________

Example 1:(4+5 ) (2−9 )23−(22+3 )

Example 2:−42−(4+2 ∙3 )5+3 (5−4 )

Q 1: Q 2:

133

7.1d Order of Operations – Absolute Value

Absolute values works just like ______________ but makes the number inside __________

after it has been _______________

Example 1:−3∨24− (5+4 )2∨¿

Example 2:2−4∨32+(52−62 )∨¿

Q 1: Q 2:

You have completed the videos for 7.1 Order of Operations. On your own paper complete the homework assignment.

134

7.2a Evaluate and Simplify Algebraic Expressions – Substitute a Value

Replace the ________________ with what it _____________________

Whenever we make a substitution or __________________ put it in _______________________

Example 1:Evaluate 4 x2−3x+2

When x=−3

Example 2:Evaluate 4 b(2 x+3 y)

When b=−2 , x=5 , y=−7

Q 1: Q 2:

135

7.2b Evaluate and Simplify Algebraic Expressions – Combine Like Terms

Terms are _________ and _____________ that are __________________ together

Like terms are terms that have matching ______________ and _________________________

Combine like terms: _____________ the coefficients from the _______________________.

Example 1:4 x3−2x2+5 x3+2 x−4 x2−6 x

Example 2:4 y−2 x+5−6 y+7 y−9

Q 1: Q 2:

136

7.2c Evaluate and Simplify Algebraic Expressions – Distributive Property

Distributive Property: a (b+c )=¿

Example 1:−2(5 x−4 y+3)

Example 2:4 (7 x2−6 x+1)

Q 1: Q 2:

137

7.2d Evaluate and Simplify Algebraic Expressions – Distribute and Combine Like Terms

Order of operations states we ______________________ before we __________________

Therefore we will __________________ first and then ___________________________ second

Example 1:4 (3 x−7)−7 (2 x−1)

Example 2:2 (7 x−3 )−(8 x+9)

Q 1: Q 2:

You have completed the videos for 7.2 Evaluate and Simplify Algebraic Expressions. On your own paper complete the homework assignment.

138

7.3a Solve Linear Equations – Variable on Both Sides

Move the variable to one side by ____________________

Solve remaining two step equation by _______________ first and ___________________ second.

Example 1:−3 x+4=16−8 x

Example 2:2 x−7=8x−9

Q 1: Q 2:

139

7.3b Solve Linear Equations – Simplify First

The first step of solving is to __________________ each side ___________________

We can simplify by ________________ and _________________________________

Example 1:3 (2x−6 )+8=17

Example 2:12 x−5 (3x−1 )=4+3(2 x+1)

Q 1: Q 2:

140

7.3c Solve Linear Equations - Fractions

Clear fractions by ______________ by the ______

Be sure to multiply _________ term on _______ sides

Example 1:34

x−12=56

Example 2:35

x− 710

=−4+ 715

x

Q 1: Q 2:

141

7.3d Solve Linear Equations – Special Cases

Sometimes the variable ____________________!

This means there is either __________________________ or _______________________________

Example 1:2 x+5=2 x−1

Example 2:3 x−9=3(x−3)

Example 3:6 x+2=3(2x+1)

Example 4:4 x+1=2(2 x+3)−5

Q 1: Q 2:

You have completed the videos for 7.3 Solve Linear Equations. On your own paper complete the homework assignment.

142

7.4a Formulas – Two Step Formulas

Solving formulas: Treat other variables like _______________

Final answer is an ________________

Example: 3 x=15 and wx= y

Example 1:Solve wx+b= y for x

Example 2:Solve ab+5 y=wx+ y for b

Q 1: Q 2:

143

7.4b Formulas – Multi-Step Formulas

Strategy:

IMPORTANT: Terms ________________ reduce

Example 1:Solve a (3 x+b )=by for x

Example 2:Solve 3 (a+2b )+5b=−2a+b for a

Q 1: Q 2:

144

7.4c Formulas – Fractions and Formulas

Clear fractions by _____________________________

Example 1:

Solve 5x+4 a=b

x for x

Example 2:

Solve A=12

hb1+12

hb2 for b1

Q 1: Q 2:

145

7.4d Formulas – Factoring the Variable

Distributive property in reverse (Factor): ab+ac=¿

Put all terms with the variable one ________________ and the other terms on the __________________

Factor out the _________________ and then ______________ to isolate

Example 1:Solve ax+b=cx+d for x

Example 2: Solve A=π r2+πrl for π

Q 1: Q 2:

You have completed the videos for 7.4 Formulas. On your own paper complete the homework assignment.

146

7.5a Absolute Value Equations – Two Solutions

|x|=5 so the x could be ____ or ____

What is inside the absolute value can be ___________________ or ____________________

This means we have _____________________________

Example 1:|2 x−5|=7

Example 2:|7−5 x|=17

Q 1: Q 2:

147

7.5b Absolute Value Equations – Isolate the Absolute Value

Before we look at our two equations, we must first ___________________________

Never _______________ through absolute value!

Never ___________ a term ________ an absolute value and a term ____________ an absolute value!

Example 1:5+2|3 x−4|=11

Example 2:−3−7∨2−4 x∨¿−31

Q 1: Q 2:

148

7.5c Absolute Value Equations – Dual Absolute Values

With two absolutes, we need _______________________

The first equation is ___________________________

The second equation is _______________________________

Example 1:|2 x−6|=¿4 x+8∨¿

Example 2:|3 x−5|=¿7 x−2∨¿

Q 1: Q 2:

You have completed the videos for 7.5 Absolute Value Equations. On your own paper complete the homework assignment.

149

7.6a Word Problems - Number

Translate:

Is/Were/Was/Will Be:

More than:

Subtracted from/Less than:

Example 1:Five less than three times a number is nineteen. What is the number?

Example 2:Seven more than twice a number is six less than three times the same number. What is the number?

Q 1: Q 2:

150

7.6b Word Problems – Consecutive Integers

Consecutive Numbers:

First:

Second:

Third:

Example 1:Find three consecutive numbers whose sum is 543.

Example 2:Find four consecutive integers whose sum is -222.

Q 1: Q 2:

151

7.6c Word Problems – Consecutive Even/Odd Integers

Consecutive Even:

First:

Second:

Third:

Consecutive Odd:

First:

Second:

Third:

Example 1:Find three consecutive even integers whose sum is 84

Example 2:Find four consecutive odd integers whose sum is 152

Q 1: Q 2:

152

7.6d Word Problems - Triangles

The angles of a triangle add to ___________

Example 1:Two angles of a triangle are the same measure. The third angle is 30 degrees less than the first. Find the three angles.

Example 2:The second angle of a triangle measures twice the first. The third angle is 30 degrees more than the second. Find the three angles.

Q 1: Q 2:

153

7.6e Word Problems - Perimeter

Formula for perimeter of a rectangle:

Width is the ____________ side

Example 1:A rectangle is three times as long as it is wide. If the perimeter is 112 cm, what is the length?

Example 2:The width of a rectangle is 6 cm less than the length. If the perimeter is 52 cm, what is the width?

Q 1: Q 2:

You have completed the videos for 7.6 Word Problems. On your own paper complete the homework assignment.

154

7.7a Age Problems – Unknown Now

Table:

Equation is always for the _______________________

Example 1:Alexis is five years younger than Brian. In seven years the sum of their ages will be 49 years. How old is each now?

Example 2:Maria is ten years older than Sonia. Eight years ago Maria was three times Sonia’s age. How old is each now?

Q 1: Q 2:

155

7.7b Age Problems – Given the Sum Now

Consider: Sum of 8…..

When we have the sum now, for the first box we use ____ and the second we use _________________

Example 1:The sum of the ages of a man and his son is 82 years. How old is each if 11 years ago, the man was twice his son’s age?

Example 2:The sum of the ages of a woman and her daughter is 38 years. How old is each if the woman will be triple her daughter’s age in 9 years?

Q 1: Q 2:

156

7.7c Age Problems – Unknown Time

If we don’t know the time:

Example 1:A man is 23 years old. His sister is 11 years old. How many years ago was the man triple his sister’s age?

Example 2:A woman is 11 years old. Her cousin is 32 years old. How many years until her cousin is double her age?

Q 1: Q 2:

You have completed the videos for 7.7 Age Problems. On your own paper complete the homework assignment.

157

7.8a Distance, Rate, Time Problems – Opposite Directions

The distance equation:

Opposite directions:

OR

Example 1:Brian and Jennifer both leave the convention at the same time, traveling in opposite directions. Brian drove 35 mph and Jennifer drove 50 mph. After how much time were they 340 miles apart?

Example 2:Maria and Tristan are 126 miles apart biking towards each other. If Maria bikes 6 mph faster than Tristan and they meet after 3 hours, how fast did each ride?

Q 1: Q 2:

158

7.8b Distance, Rate, Time Problems – Catch Up

_________ the head start to the ________________ of the person with the head start

Catch up:

Example 1:Raquel left the party traveling 5 mph. Four hours later Nick left to catch up with her, traveling 7 mph. How long will it take him to catch up with her?

Example 2:Trey left on a trip traveling 20 mph. Julian left 2 hours later, traveling in the same direction at 30 mph. After how many hours does Julian pass Trey?

Q 1: Q 2:

159

7.8c Distance, Rate, Time Problems – Total Time

Consider: Total time of 8…..

When we have the total time, we use ____ for the first time and _________________ for the second

Example 1:Lupe rode into the forest at 10 mph, turned around and returned by the same route traveling 15 mph. If her trip took 5 hours, how long did she travel at each rate?

Example 2:Ian went on a 230 mile trip. He started driving 45 mph. However, due to construction on the second leg of the trip, he had to slow down to 25 mph. If the trip took 6 hours, how long did he drive at each speed?

Q 1: Q 2:

You have completed the videos for 7.8 Distance, Rate, Time Problems. On your own paper complete the homework assignment.

160

7.9a Inequalities - Graphing

Inequalities:

Less than:

Less than or equal to:

Greater than:

Greater than or equal to:

Graphing on number line: Use for less/greater than and use when its “or equal to”

Example 1:Graph x≥−3

Example 2:Give the inequality

Q 1: Q 2:

161

7.9b Inequalities – Interval Notation

Interval notation: ( , )

Use for less/greater than and use when its “or equal to”

∞ and −∞ always use a

Example 1:Give interval notation

Example 2:Graph the interval (−∞,−1)

Q 1: Q 2:

162

7.9c Inequalities – Solving

Solving inequalities is very similar to solving _____________________________ (with one exception…)

Three steps with inequalities: ______________, then __________________, then _____________________

Example 1:7+5 x≤17

Example 2:3 ( x+8 )+2>5 x−20

Q 1: Q 2:

163

7.9d Inequalities – Multiply or Divide by a Negative

What happens to 5>−2 when we multiply both sides by −3? (−3 )5??−2(−3)

When __________________ or ______________________ by a ________________ you must

________________________________________


Example 1:7−3 x≤16

Example 2:4←2 x+16

Q 1: Q 2:

164

7.9e Inequalities - Tripartite

Tripartite inequalities:

When solving __________________________

When graphing _________________________


Example 1:2≤5 x+7<22

Example 2:5<5−4 x ≤13

Q 1: Q 2:

You have completed the videos for 7.9 Inequalities. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 7: Linear Equations and Applications. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

165

Unit 8:

Graphing Linear Equations



166

8.1a Slope – Graphing Points and Lines

The Coordinate Plane:

Give ________________________ to a point going _______________ then _________________ as ______

If we have an equation we can pick values for ______ and find values for ___

Example 1:Graph the points

(−2,3 ) , (4 ,−1 ) , (−2 ,−4 ) , (0,3 ) ,(−1,0) and (3,4)

Example 2:Graph the line y=2x−1

Q 1: Q 2:

167

8.1b Slope – Slope from Graph

Slope:

Negative Slope: Positive Slope: Big Slope: Small Slope:

Example 1:Find the Slope

Example 2:Find the Slope

Q 1: Q 2:

168

8.1c Slope – Slope from Points

Slope Equation:

Example 1:Find the slope between (7,2) and (11,4)

Example 2:Find the slope between (−2 ,−5) and (−17,4)

Q 1: Q 2:

You have completed the videos for 8.1 Slope. On your own paper complete the homework assignment.

169

8.2a Equations of Lines – Slope-Intercept Equation

Slope Intercept Equation:

Example 1:Give the equation of the line with

a slope of −34 and a y-intercept of 2

Example 2:Give the equation of the graph:

Q 1: Q 2:

170

8.2b Equations of Lines – Equation Through a Point

To find the y-intercept we use _________________ and solve for _____________

Example 1:Give the equation of the line that passes

through (6 ,−2) and has a slope of 4.

Example 2:Give the equation of the line that passes

through (−3,5 ) and has a slope of −23

Q 1: Q 2:

171

8.2c Equations of Lines – Put in Slope-Intercept Form

We may have to put the equation in _________________________

To do this we ________________________________

Example 1:Give the slope and y-intercept

5 x+8 y=16

Example 2:Give the slope and y-intercept

−3 x+2 y=8

Q 1: Q 2:

172

8.2d Equations of Lines – Graph a Linear Equation

We can graph an equation by identifying the ____________________ and _________________________

Start at the __________________ and use the __________________________ for changing to the next point

Remember slope is ________________ over _______________

Example 1:

Graph y=−34

x+2

Example 2:Graph 3 x−2 y=2

173

Q 1: Q 2:

8.2e Equations of Lines – Given Two Points

To find the equation of a line you must have the ________________________

Recall the slope formula:

To find the y-intercept we use _________________ and solve for _____________

Example 1:Find the equation of the linethrough (−3 ,−5) and (2,5)

Example 2:Find the equation of the line

through (1 ,−4) and (3,5)

Q 1: Q 2:

174

You have completed the videos for 8.2 Equations of Lines. On your own paper complete the homework assignment.

8.3a Parallel and Perpendicular Lines - Slopes

Parallel Lines:

Slope:

Perpendicular Lines:

Slope:

Neither:

Example 1:One line goes through (5,2) and (7,5). Another line goes through (−2 ,−6) and (0 ,−3). Are the lines

parallel, perpendicular, or neither?

Example 2:One line goes through (−4,1) and (−1,3). Another line goes through (2 ,−1) and (6 ,−7). Are the lines


Example 3:One line goes through (3,7) and (−6 ,−8). Another line goes through (5,2) and (−5 ,−4). Are the lines


Q 1:

175

Q 2: Q 3:

8.3b Parallel and Perpendicular Lines – Parallel Equations

Parallel lines have the _________ slope

Once we know the slope and a point we can use the formula:

Example 1:Find the equation of the line parallel to the line

y=−34

x+2 that goes through the point (−8,1)

Example 2:Find the equation of the line parallel to the line 2 x−5 y=3 that goes through the point (5,3)

Q 1: Q 2:

176

8.3c Parallel and Perpendicular Lines - Perpendicular Equations

Perpendicular lines have __________________________ slopes

Once we know the slope and a point we can use the formula:

Example 1:Find the equation of the line perpendicular to the line y=5 x+1 that goes through the point (−5,2)

Example 2:Find the equation of the line perpendicular to the line 3 x+2 y=5 that goes through the point (−3 ,−4)

Q 1: Q 2:

177

You have completed the videos for 8.2 Parallel and Perpendicular Lines. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 8: Graphing Linear Equations. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

178

Unit 9:

Polynomials



179

9.1a Exponents – Product Rule

a3 ∙ a2=¿

Product Rule: am ∙ an=¿

Example 1:(2 x3)(4 x2)(−3 x)

Example 2:(5a3b7 ) (2a9b2c4 )

Q 1: Q 2:

180

9.1b Exponents – Quotient Rule

a5

a3=¿

Quotient Rule: am

an =¿

Example 1:a7b2

a3b

Example 2:8m7n4

−6m5n

Q 1: Q 2:

181

9.1c Exponents – Power Rules

(ab )3=¿

Power of a Product: (ab )m=¿

( ab )

3

=¿

Power of a Quotient: ( ab )

m

=¿

(a2 )3=¿

Power of a Power: (am )n=¿

Example 1:(5a4b )3

Example 2:

(−5m3

9n4 )2

Q 1: Q 2:

182

9.1d Exponents – Zero Exponent

a3

a3=¿

Zero Power Rule: a0=¿

Example 1:(5 x3 y z5 )0

Example 2:(3 x2 y0 ) (5 x0 y 4 ) (x2 y3 )

Q 1: Q 2:

183

9.1e Exponents – Negative Exponents

a3

a5=¿

Negative Exponent Rules: a−m=¿ 1

a−m=¿ ( ab )

−m

=¿

Example 1:25a−4

Example 2:7 x−5

3−1 y z−4

Q 1: Q 2:

184

9.1f Exponents - Properties

am an=¿ am

an =¿ (ab )m=¿

( ab )

m

=¿ (am )n=¿ a0=¿

a−m=¿

To Simplify:

1a−m =¿ ( a

b )−m

=¿

Example 1:(4 x−5 y2 z )2 (2 x4 y−2 z3 )4

Example 2:(2x2 y−3 )−4 (x4 y−6 )−2

( x−6 y4 )2

Q 1: Q 2:

185

You have completed the videos for 9.1 Exponents. On your own paper complete the homework assignment.

9.2a Scientific Notation – Convert Scientific and Standard Notation

a×10b

a is

b is

b positive

b negative

Example 1:Convert to Standard Notation

5.23×105

Example 2:Convert to Standard Notation

4.25×10−4

Example 3:Convert to Scientific Notation

81,500,000

Example 4:Convert to Scientific Notation

0.0000245

Q 1: Q 2:

Q 3: Q 4:

186

9.2b Scientific Notation – Almost Scientific Notation

Put the number in front in ___________________________

Then use ______________________________ on the 10’s

Example 1:523.6×10−8

Example 2:0.0032×105

Q 1: Q 2:

187

9.2c Scientific Notation – Multiply or Divide

Multiply/Divide the _____________________________

Then use ______________________________ on the 10’s

Example 1:(3.4×105)(2.7×10−2)

Example 2:5.32×104

1.9×10−3

Q 1: Q 2:

188

9.2d Scientific Notation – Multiply or Divide where Answer is not in Scientific Notation

If our final answer is not in scientific notation we must _______________________

Example 1:(6.7×10−6)(5.2×10−3)

Example 2:2.352×10−6

8.4 ×10−2

Q 1: Q 2:

189

9.2e Scientific Notation – Multiply and Divide

Multiply/Divide the _____________________________

Then use ______________________________ on the 10’s

Example 1:(4.2×104 ) (8.1×10−6 )

1.4×105

Example 2:2.01×10−5

(1.5×10−3 ) (3.2×10−4 )

Q 1: Q 2:

190

You have completed the videos for 9.2 Scientific Notation. On your own paper complete the homework assignment.

9.3a Add, Subtract, Multiply Polynomials - Evaluate

Term:

Monomial:

Binomial:

Trinomial:

Polynomial:

Evaluate:

Example 1:5 x2−2 x+6 when x=−2

Example 2:−x2+2x−7 when x=4

Q 1: Q 2:

191

9.3b Add, Subtract, Multiply Polynomials – Add

To add polynomials:

To subtract polynomials:

Example 1:(5 x2−7 x+9 )+(2x2+5 x−14)

Example 2:(3 x3−4 x+7 )−(8 x3+9 x−2)

Q 1: Q 2:

192

9.3c Add, Subtract, Multiply Polynomials – Multiply Monomial by Polynomial

To multiply a monomial by a polynomial:

Example 1:5 x2(6 x2−2x+5)

Example 2:−3 x4(6 x3+2x−7)

Q 1: Q 2:

193

9.3d Add, Subtract, Multiply Polynomials – Multiply Binomials

To multiply a binomial by a binomial:

This is often called ___________ which stands for ______________________________________________

Example 1:(4 x−2)(5x+1)

Example 2:(3 x−7)(2 x−8)

Q 1: Q 2:

194

9.3e Add, Subtract, Multiply Polynomials – Multiply Trinomials

Multiplying trinomials is just like _____________________ we just have _____________________________

Example 1:(3 x−4)(9 x2+12 x+16)

Example 2:(2 x2−6 x+1)(4 x2−2x−6)

Q 1: Q 2:

195

9.3f Add, Subtract, Multiply Polynomials – Multiply Monomials and Binomials

Multiply _____________________ first, then _____________________ the ______________________

Example 1:4 (2 x−4 )(3 x+1)

Example 2:3 x (x−6)(2 x+5)

Q 1: Q 2:

196

9.3g Add, Subtract, Multiply Polynomials – Multiply Sum and Difference

(a+b ) (a−b )=¿

Sum and Difference Shortcut

Example 1:(x+5)(x−5)

Example 2:(5 x−2)(5 x+2)

Q 1: Q 2:

197

9.3h Add, Subtract, Multiply Polynomials – Perfect Squares

(a+b )2=¿

Notice that (a+b )2 is ___________________ a2+b2. That is to say, (a+b )2⎕ a2+b2

Perfect Square Shortcut:

Example 1:( x−4 )2

Example 2:(2 x+7 )2

Q 1: Q 2:

198

You have completed the videos for 9.3 Add, Subtract, Multiply Polynomials. On your own paper complete the homework assignment.

9.4a Polynomial Long Division – Division by Monomials

To divide a polynomial by a monomial we _____________ each ____________ by the __________________

Example 1:3x5+18 x4−9x3

3x2

Example 2:15a6−25a5+5a4

5a4

Q 1: Q 2:

199

9.4b Polynomial Long Division – Review Long Division

Long Division Review:

. 5)2632

Example 157376

Q 1:

200

9.4c Polynomial Long Division – Division by Binomial

Follow the same pattern as __________________________

On the division step focus only on the _________________________

Example 1:x3−2x2−15 x+30

x+4

Example 2:4 x3−6 x2+12x−5

2x−1

Q 1: Q 2:

201

9.4d Polynomial Long Division – Division with Missing Term

The exponents MUST ____________________________

If one is missing we will add ______________

Example 1:3x3−50 x+4

x−4

Example 2:2x3+4 x2+9

x+3

Q 1: Q 2:

You have completed the videos for 9.4 Polynomial Long Division. On your own paper complete the homework assignment.

202

Congratulations! You made it through the material for Unit 9: Polynomials. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

Unit 10:

Factoring



203

10.1a Factor Common Factors and Grouping – Find a GCF

Greatest Common Factor: _____________ factor that __________________ into each term

On variables we use the _________________ exponent

Example 1:Find the common factor:

15a4+10a2

Example 2:Find the common factor4 a4b7−12a2b6+20ab9

Q 1: Q 2:

204

10.1b Factor Common Factors and Grouping – Factor a GCF

Factor:

a (b+c )=¿

Put the ________ in front, and divide each ______. What is left goes into the _____________________

Example 1:9 x4−12x3+6 x2

Example 2:21a4b5−14a3b7+7a2b4

Q 1: Q 2:

205

10.1c Factor Common Factors and Grouping – Binomial GCF

The GCF can be a ____________________

Example 1:5 x (2 y−7 )+6 y (2 y−7)

Example 2:3 x (2 x+1 )−7(2 x+1)

Q 1: Q 2:

206

10.1d Factor Common Factors and Grouping - Grouping

Grouping: GCF of the _____________ and _________________

Then factor out the __________________________ (if it matches)

Example 1:15 xy+10 y−18 x−12

Example 2:6 x2+3xy+2 x+ y

Q 1: Q 2:

207

10.1e Factor Common Factors and Grouping – Grouping with Change of Order

If the binomials don’t match:

Example 1:12a2−7b+3ab−28a

Example 2:6 xy−20+8 x−15 y

Q 1: Q 2:

You have completed the videos for 10.1 Factor Common Factors and Grouping. On your own paper complete the homework assignment.

208

10.2a Factor Trinomials – Reverse FOIL

Recall FOIL: (a+b ) (c+d )=¿

________________ multiplies to _______________ and _________________ multiplies to _______________

The _________________ and __________________ must add to the ______________________

This may take some ___________________________

Example 1:3 x2+11 x+10

Example 2:12 x2+16 x−3

Q 1: Q 2:

209

10.2b Factor Trinomials – Two Variables

Be aware of _______________ variables when using reverse ____________

Example 1:12 x2−5xy−2 y2

Example 2:6 x2−17 xy+10 y2

Q 1: Q 2:

210

10.2c Factor Trinomials – With GCF

Always factor the _______ first!

Example 1:18 x4−21 x3−15 x2

Example 2:16 x3+28x2 y−30 x y2

Q 1: Q 2:

211

10.2d Factor Trinomials – Without a Leading Coefficient

If the leading coefficient (in front of x2) is a 1, then the two numbers will _________ to the ______________

Note: This only works if the leading coefficient is _____

Example 1:x2−2 x−8

Example 2:x2+7xy−8 y2

Q 1: Q 2:

You have completed the videos for 10.2 Factor Trinomials. On your own paper complete the homework assignment.

212

10.3a Factoring Tricks – Perfect Squares

(a+b )2=¿

If we can take the square root of the first and last term it _________ be a ______________________

Example 1:x2−10 x+25

Example 2:9 x2+30xy+25 y2

Q 1: Q 2:

213

10.3b Factoring Tricks – Difference of Squares

(a+b ) (a−b )=¿

Difference of Squares:

Example 1:a2−81

Example 2:49 x2−25 y2

Q 1: Q 2:

214

10.3c Factoring Tricks – Sum of Squares

Factor: a2+b2

Sum of squares is always _______________ (this means it ______________ be factored)

Example 1:x2+9

Example 2:32a2b+50b3

Q 1: Q 2:

215

10.3d Factoring Tricks – Sum and Difference of Cubes

Sum of Cubes: a3+b3=¿

Difference of Cubes: a3−b3=¿

Some cubes worth memorizing:

Example 1:m3+125

Example 2:8a3−27 y3

Q 1: Q 2:

216

10.3e Factoring Tricks – Difference of 4th Powers

The square root of x4 is ______

With fourth powers we can use ______________________ twice!

Example 1:a4−16

Example 2:81 x4−256

Q 1: Q 2:

217

10.3f Factoring Tricks – Difference of 6th Powers

The square root of x6 is _____ and the cubed root of x6 is _____

A difference of 6th powers may be a difference of ____________ or a difference of ________________

Use the ___________________ to decide which formula to use

Example 1:x6−49 y6

Example 2:8a6−27b6

Q 1: Q 2:

218

10.3g Factoring Tricks – With GCF


Example 1:9 x3−81 x

Example 2:2 x2 y−12 xy+18 y

Q 1: Q 2:

You have completed the videos for 10.3 Factoring Tricks. On your own paper complete the homework assignment.

219

10.4 Factoring Strategy


2 terms: 3 terms: 4 terms:

Example 1:Which method would you use?

25 x2−16


x2−x−20


xy+2 y+5x+10

Q 1:

Q 2: Q 3:

Q 4: Q 5:

You have completed the videos for 10.4 Factoring Strategy. On your own paper complete the homework assignment.

220

10.5a Solving Equations by Factoring – Zero Product Rule

Zero Product Rule: if ab=0 then ________________

To solve we set each ___________________ equal to _________________

Example 1:(5 x−1 ) (2x+5 )=0

Example 2:2 x ( x−6 ) (2 x+3 )=0

Q 1: Q 2:

221

10.5b Solving Equations by Factoring – Solve by Factoring

If there is an x2 and x in the equation, we need to __________________ before we ______________

Example 1:x2−4 x−12=0

Example 2:3 x2+x−4=0

Q 1: Q 2:

222

10.5c Solving Equations by Factoring – Must Equal Zero

Before we factor, the equation must equal _______________

To make factoring easier, we want the ______ term to be ________________________

Example 1:5 x2=2 x+16

Example 2:−2 x2=x−3

Q 1: Q 2:

223

10.5d Solving Equations by Factoring – Simplify First

Before we make the equation equal zero, we may have to __________________ first

Example 1:2 x ( x+4 )=3 x−3

Example 2:(2 x−3 ) (3x+1 )=−8 x−1

Q 1: Q 2:

224

10.5e Solving Equations by Factoring – GCF’s as Factors

When solving do not forget that the _______ is a _________________ also.

If there is no ___________________ in the GCF then we can ____________ it.

Example 1:4 x3−12x2=40x

Example 2:6 x2=36−15x

Q 1: Q 2:

You have completed the videos for 10.5 Solving Equations by Factoring. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 10: Factoring. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

225

Unit 11:

Rational Expressions



226

11.1a Reduce Rational Expressions – Evaluate Rational Expressions

Rational Expression: Quotient of two _____________________

Example 1:x2−2x−8

x−4 when x=−4

Example 2:x2−x−6x2+x−12

when x=2

Q 1: Q 2:

227

11.1b Reduce Rational Expressions – Review Reducing Fractions

To reduce fractions we ___________________ common ____________________

Example 1:2415

Example 2:48 x2 y18 x y3

Q 1: Q 2:

228

11.1c Reduce Rational Expressions – Reduce Rational Expressions

To reduce fractions we ___________________ common ____________________

This means we must first _____________________

Example 1:2x2+5 x−32x2−5 x+2

Example 2:9 x2−30x+259x2−25

Q 1: Q 2:

You have completed the videos for 11.1 Reduce Rational Expressions. On your own paper complete the homework assignment.

229

11.2a Multiply and Divide Rational Expressions – Review Multiply and Divide Fractions

To multiply we ___________________ common ____________________ then multiply _________________

Division is the same, with one extra step at the start: _________________ by the ___________________

Example 1:635

∙ 2110

Example 2:58

÷ 103

Q 1: Q 2:

230

11.2b Multiply and Divide Rational Expressions – Multiply or Divide Rational Expressions



Division is the same, with one extra step at the start: _________________ by the ___________________

Example 1:x2+3 x+24 x−12

∙ x2−5 x+6x2−4

Example 2:3x2+5 x−2x2+3 x+2

÷ 6 x2+x−12x3−6 x2−8 x

Q 1: Q 2:

231

11.2c Multiply and Divide Rational Expressions – Multiply and Divide Rational Expressions

To divide:



Example 1:x2+3 x−10x2+6 x+5

∙ 2x2−x−32x2+x−6

÷ 8 x+206 x+15

Example 2:x2−1

x2−x−6∙ 2x2−x−153 x2−x−4

÷ 2 x2+3 x−53 x2+2 x−8

Q 1: Q 2:

You have completed the videos for 11.2 Multiply and Divide Rational Expressions. On your own paper complete the homework assignment.

232

11.3a Add and Subtract Rational Expressions – Review LCD of Numbers with Prime Factorization

Prime Factorization:

To find the LCD use _________ factors with ____________ exponents

Example 1:Find the LCD:20 and 36

Example 2:Find the LCD:18 ,54 and 81

Q 1: Q 2:

233

11.3b Add and Subtract Rational Expressions – LCD of Monomials

To find the LCD with variables use _________ factors with ____________ exponents

Example 1:Find the LCD:

5 x3 y2 and 4 x2 y5

Example 2:Find the LCD:7ab2c and 3a4b

Q 1: Q 2:

234

11.3c Add and Subtract Rational Expressions – LCD of Polynomials

To find the LCD with polynomials use _________ factors with ____________ exponents



x2+3x−18 and x2+4 x−21


x2−10 x+25 and x2−x−20

Q 1: Q 2:

235

11.3d Add and Subtract Rational Expressions – Review Adding and Subtracting Fractions

To add or subtract we ___________ the denominators by _____________ by the missing ________________

Example 1:521

+ 715

Example 2:814

− 310

Q 1: Q 2:

236

11.3e Add and Subtract Rational Expressions – Add and Subtract with Common Denominator

Add the __________________ and keep the __________________

When subtracting we will first _________________ the negative

Don’t forget to ______________

Example 1:x2+4 x

x2−2x−15+ x+6

x2−2x−15

Example 2:x2+2 x

2x2−9 x−5− 6 x+52 x2−9 x−5

Q 1: Q 2:

237

11.3f Add and Subtract Rational Expressions – Add and Subtract with Different Denominators

To add or subtract we ___________ the denominators by _____________ by the missing ________________

This means we must first _____________________ the denominators

Example 1:2 x

x2−9+ 5

x2+x−6

Example 2:2 x+7

x2−2x−3− 3 x−2

x2+6 x+5

Q 1: Q 2:

You have completed the videos for 11.3 Add and Subtract Rational Expressions. On your own paper complete the homework assignment.

238

239

11.4a Dimensional Analysis – One Step Conversions

Consider: ¿

Divide out units by placing them in the __________________ part of the fraction

Conversion factor: Same _________ in numerator and denominator, but different _________

Dimensional analysis: Multiply by a _____________________ to convert units

Example 1:Convert 3.8 inches to centimeters

Example 2:Convert 48 kilograms to pounds

Q 1: Q 2:

11.4b Dimensional Analysis – Multi-Step Conversions

240

If we do not have the correct conversion factor we can convert using _______________ conversion factors

Example 1:Convert 5 feet to meters

Example 2:Convert 3 miles to yards

Q 1: Q 2:

11.4c Dimensional Analysis – Dual Unit Conversions

241

Dual Unit:

“Per” is the _________________________

With dual units we convert ___________________________

Example 1:Convert 100 ft per sec to mi per hr

Example 2:Convert 25 mi per hr to km per min

Q 1: Q 2:

You have completed the videos for 11.4 Dimensional Analysis. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 11: Rational Expressions. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

242

Unit 12:

Proficiency Exam #2


1. Complete the practice tests on your own paper.2. Take the (two part) unit exam.

243

Unit 13:

Compound Inequalities



244

13.1a Review Inequalities – Graph and Interval Notation

Graphing Inequalities: Use ______ for greater/less than and use ______ for “or equal to”

Interval Notation: Use _____ for greater/less than and use ______ for “or equal to”

Example 1:Graph and give interval notation:

x≥3

Example 2:Graph and give interval notation:

x←2

Q 1: Q 2:

245

13.1b Review Inequalities – Solve

Inequalities solve just like equations except if we _________________ or ________________ by a

_________________, then we must ___________________________

Example 1:2 x−7>4 x+1

Example 2:5−4 x ≥13

Q 1: Q 2:

You have completed the videos for 13.1 Review Inequalities. On your own paper complete the homework assignment.

246

13.2a Compound Inequalities – OR (two directions)

OR:

First we will _____________ each part above the number line, then we will _____________ the union (OR)

Symbol for Union:

Example 1:4 x+7←5 OR −4 x−8≤−20

Example 2:8 x+9<4 x−19 OR 2 (4 x−8 )−2≤12 x−50

Q 1: Q 2:

247

13.2b Compound Inequalities – OR (one direction)

With an OR if both graphs go the same direction than we use the _______________________

Example 1:4 x−6>10 OR 5−2x ≤7

Example 2:3 x+5<2 x−9 OR 7 x+3≤5(x−1)

Q 1: Q 2:

248

13.2c Compound Inequalities – AND (between)

AND:

First we will _____________ each part above the number line, then we will use the _____________ (AND)

Example 1:6 x+5<11 AND −7 x+2≤44

Example 2:11 x−10>3 x−2 AND 2 (5x−3 )+2≥18 x−52

Q 1: Q 2:

249

13.2d Compound Inequalities – AND (one direction)

With an AND if both graphs go the same direction than we use the _______________________

Example 1:5 x−6≥26 AND 3 x+1> x−9

Example 2:2 (4 x+4 )>6 x+2 AND 7−x≤3+x

Q 1: Q 2:

250

13.2e Compound Inequalities – Special Cases

OR can give us ____________ of number line or ________________________, in interval notation _________

AND can give us ___________ of the number line or _______________________, in interval notation ______

Example 1:2 x+1<x−3 OR 3 ( x+1 )≥ x−15

Example 2:−3 (4 x−1 )≤15 AND 2 x−3≤−9

Q 1: Q 2:

You have completed the videos for 13.2 Compound Inequalities. On your own paper complete the homework assignment.

251

13.3a Absolute Value Inequalities – GreatOR Than

|x|>2 means the __________ from zero is ____________ than 2.

This is a graph of a compound ______ inequality. It can be written as _______________________

If the absolute value is greatOR than a number we set up an _____

Example 1:|2 x−1|≥7

Example 2:|7 x+4|>32

Q 1: Q 2:

252

13.3b Absolute Value Inequalities – Less Than

|x|<2 means the __________ from zero is ____________ than 2.

This is a graph of a compound ______ inequality. It can be written as _______________________

If the absolute value is less than a number we set up an _____

Example 1:|3 x+7|<6

Example 2:|4 x+1|≤2

Q 1: Q 2:

253

13.3c Absolute Value Inequalities – Isolate Absolute Value

Before setting up a compound inequality, we must first ________________ the absolute value!

Beware: with absolute value we cannot _____________ or _______________________________

Example 1:2−7|3 x+4|←19

Example 2:5+2|4 x−1|≤17

Q 1: Q 2:

You have completed the videos for 13.3 Absolute Value Inequalities. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 13: Compound Inequalities. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

254

Unit 14:

Systems of Equations



255

14.1a Systems by Graphing - Solutions

The points on a line are the ___________________ to the equation

The intersection of two lines is the _______________ to both equations!

Other options: ______________ lines have ____ solutions. ______________ lines have __________ solutions

Example 1:What is the solution for both lines?



Q 1:

256

14.1b Systems by Graphing – Solve with Intercept Form

To graph lines remember the equation _________________________

Start with the ____________________ or ____ and use the _____________ or _____ to find the next point

Example 1:

y=−23

x+3 y=2x−5

Example 2:2 x− y=−4x+ y=1

Q 1: Q 2:

You have completed the videos for 14.1 Systems by Graphing. On your own paper complete the homework assignment.

14.2a Systems – Introduction to Substitution

257

Substitution: Replace the _______________ with what it ____________________

Example 1:x=−3

2 x−3 y=12

Example 2:4 x−7 y=11

y=−1

Q 1: Q 2:

14.2b Systems – Substitute an Expression

258

Just as we can replace a variable with a number, we can also replace it with an ___________________

Whenever we substitute it is important to remember __________________________

Example 1:y=5 x−3

−x−5 y=−11

Example 2:2 x−6 y=−24

x=5 y−22

Q 1: Q 2:

259

14.2c Systems – Solve for a Variable

260

To use substitution we may have to ___________________ a lone variable

If there are several lone variables _________________________

Example 1:6 x+4 y=−14x−2 y=−13

Example 2:−5 x+ y=−177 x+8 y=5

Q 1: Q 2:

261

14.2d Systems – Substitution Special Cases

262

If the variables subtract out to zero then it means either there is _____________________________

or _____________________________

Example 1:x+4 y=−7

21+3 x=−12 y

Example 2:5 x+ y=38−3 y=15 x

Q 1: Q 2:

263

14.2e Systems – Addition/Elimination

264

If there is no lone variable, it may be better to use ______________________

This method works by adding the _______________ and ______________ sides of the equations together

Example 1:−8 x−3 y=−122 x+3 y=−6

Example 2:−5 x+9 y=295 x−6 y=−11

Q 1: Q 2:

14.2f Systems – Addition/Elimination and Multiplying an Equation

265

Addition only works if one of the variables have _______________________________________

To get opposites we can multiply ___________________________ of an equation to get the value we want

Be sure when multiplying to have a ________________ in front of either the ___ or the ____

Example 1:2 x−4 y=−44 x+5 y=−21

Example 2:−5 x+3 y=−3−7 x+12 y=14

Q 1: Q 2:

266

14.2g Systems – Addition/Elimination and Multiplying Both Equations

267

Sometimes we may have to multiply ________________________ by something to get opposites

The opposite we look for is the ________ of both coefficients

Example 1:−6 x+4 y=264 x−7 y=−13

Example 2:3 x+7 y=2

10 x+5 y=−30

Q 1: Q 2:

268

14.2h Systems – Addition/Elimination Special Cases

269

If the variables subtract out to zero than it means either there is _____________________________

or _____________________________

Example 1:2 x−4 y=163 x−6 y=20

Example 2:−10 x+4 y=−625 x−10 y=15

Q 1: Q 2:

270

You have completed the videos for 14.2 Systems. On your own paper complete the homework assignment.

14.3a Systems with Three Variables – Simple

271

To solve systems with three variables we must _______________ the ____________ variable _____________

This will give us ______ equations with _______ variables we can then solve for!

Example 1:3 x−3 y+5 z=162 x−6 y−5 z=35

−5 x−12 y+5 z=28

Example 2:−x+2 y+4 z=−20−2 x−2 y−3 z=54 x−2 y−2 z=26

Q 1: Q 2:

272

14.3b Systems with Three Variables – Multiply to Eliminate

273

To eliminate a variable we may have to __________________ one or more equations to get ______________

Example 1:−2 x−2 y+3 z=−63 x−3 y−2 z=−175 x−4 y+5 z=11

Q 1:

274

You have completed the videos for 14.3 Systems with Three Variables. On your own paper complete the homework assignment.

14.4a Applications of Systems – Value Comparison

275

Define the ______________________

Make an equation for the ___________________


Example 1:Brian has twice as many dimes as quarters. If the value of the coins is $4.95, how many of each does he have?

Example 2:A child has three more nickels than dimes in her piggy-bank. If she has $1.95 in her bank, how many of each does she have?

Q 1: Q 2:

276

14.4b Applications of Systems – Value with Total

277

Define the ______________________



Example 1:Scott has $2.25 in his pocket made up of quarters and dimes. If there are 12 coins, how many of each coin does he have?

Example 2:If 105 people attended a concert and tickets for adults cost $2.50 while tickets for children cost $1.75 and total receipts for the concert were $228, how many children and how many adults went to the concert?

Q 1: Q 2:

278

14.4c Applications of Systems – Interest Comparison

279

Define the ______________________



Beware: When using a percent we must ___________________

Example 1:Sophia invested $1900 in one account and $1500 in another account that paid 3% higher interest rate. After one year she had earned $113 in interest. At what rates did she invest?

Example 2:Carlos invested $2500 in one account and $1000 in another which paid 4% lower interest. At the end of a year he had earned $345 in interest. At what rates did he invest?

Q 1: Q 2:

280

14.4d Applications of Systems – Interest with Total Principle

281

Define the ______________________



Beware: When using a percent we must ___________________

Example 1:A woman invests $4600 in two different accounts. The first paid 13%, the second paid 12% interest. At the end of the first year she had earned $586 in interest. How much was in each account?

Example 2:A bank loaned out $4900 to two different companies. The first loan had a 4% interest rate; the second had a 13% interest rate. At the end of the first year the loan had accrued $421 in interest. How much was loaned at each rate?

Q 1: Q 2:

282

14.4e Applications of Systems – Mixture with Starting Amount

283

Define the ______________________



Example 1:A store owner wants to mix chocolate and nuts to make a new candy. How many pounds of chocolate which costs $1.50 per pound should be mixed with 40 pounds of nuts that cost $3.00 per pound to make a mixture worth $2.50 per pound?

Example 2:You need a 55% alcohol solution. On hand, you have 600 mL of 10% alcohol mixture. You also have a 95% alcohol mixture. How much of the 95% mixture should you add to obtain your desired solution?

Q 1: Q 2:

284

14.4f Applications of Systems – Mixture with Final Amount

285

Define the ______________________



Example 1:A chemist needs to create 100 mL of a 38% acid solution. On hand she has a 20% acid solution and a 50% acid solution. How many mL of each should she use?

Example 2:A coffee distributor needs to mix a coffee blend that normally sells for $8.90 per pound with another coffee blend that normally sells for $11.16 per pound, how many pounds of each kind of coffee should they mix if the distributer needs 50 pounds of the new mix to sell for $9.85?

Q 1: Q 2:

286

You have completed the videos for 14.4 Applications of Systems. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 14: Systems of Equations. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

287

Unit 15:

Radicals



288

15.1a Simplify Radicals - Variables

Radical: n√a=b where ______________. The n is called the ________________.

Square Root: √a=b where ______________. The index on a square root is always ____

Radicals divide the ______________ by the __________________

The whole number is how many “things” __________ and the remainder is how many “things” ___________

Example 1:√a3

Example 2:4√b19

Q 1: Q 2:

289

15.1b Simplify Radicals – Several Variables

Work with ___________ variable at a time

Example 1:√a5b8 c15

Example 2:4√a13 b23 c10 d3 e36

Q 1: Q 2:

290

15.1c Simplify Radicals – Using Prime Factorization

Prime Factorization:

To find a prime factorization we _______________ by _______________

A few prime numbers:

Roots of numbers are difficult, find the ____________________ so that we can divide the ______________

by the __________________

Example 1:3√750

Example 2:9√250 x4 y z5

Q 1: Q 2:

291

15.1d Simplify Radicals - Binomials

We can only pull ___________________ (separated by _________________________) out of a radical

If we have ______________ (separated by ____ or _____) we must ______________ first!

Example 1:√100 x2−16 x4

Example 2:3√216 x6−27 x9

Q 1: Q 2:

You have completed the videos for 15.1 Simplify Radicals. On your own paper complete the homework assignment.

292

15.2a Add, Subtract, and Multiply Radicals – Add Like Radicals

Simplify: 2 x−5 y+4 x+2 y

Simplify: 2√3−5√7+4 √3+2√7

When adding and subtracting radicals we can ____________________________________

Example 1:−4√6+2√11+√11−5√6

Example 2:3√5+3√5−8 3√5+2√5

Q 1: Q 2:

293

15.2b Add, Subtract, and Multiply Radicals – Add with Simplifying

Before adding radicals together _______________________

Example 1:5√50x+5√27−3√2 x−2√108

Example 2:3√81x3 y−3 y 3√32 x2+ x 3√24 y− 3√500 x2 y3

Q 1: Q 2:

294

15.2c Add, Subtract, and Multiply Radicals – Multiply Monomial Radical Expressions

Product Rule: a n√b ∙ c n√d=¿

Always be sure your final answer is ______________________

Example 1:4 √6∙2√15

Example 2:−3 4√8 ∙7 4√10

Q 1: Q 2:

295

15.2d Add, Subtract, and Multiply Radicals – Multiply Monomial by Binomial Radical Expressions

Recall: a (b+c )=¿


Example 1:5√10 (2√6−3√15 )

Example 2:7√3 (√6+9 )

Q 1: Q 2:

296

15.2e Add, Subtract, and Multiply Radicals – Multiply Binomial Radical Expressions

Recall: (a+b ) (c+d )=¿


Example 1:(3√7−2√5 ) (√7+6√5 )

Example 2:(2 3√9+5)(4 3√3−1)

Q 1: Q 2:

297

15.2f Add, Subtract, and Multiply Radicals – Square Binomial Radical Expression

Recall: (a+b )2=¿


Example 1:(√6−√2 )2

Example 2:(2+3√7 )2

Q 1: Q 2:

298

15.2g Add, Subtract, and Multiply Radicals – Multiply Conjugates

Recall: (a+b ) (a−b )=¿


Example 1:(4+2√7 ) (4−2√7 )

Example 2:(2√3−√6 ) (2√3+√6 )

Q 1: Q 2:

You have completed the videos for 15.2 Add, Subtract, and Multiply Radicals. On your own paper complete the homework assignment.

299

15.3a Rationalize Denominators – Simplifying with Radicals

Expression with radicals: Always ______________ the _________________ first

Before _____________________ with fractions, be sure to _____________ first

Example 1:15+√17510

Example 2:8−√486

Q 1: Q 2:

300

15.3b Rationalize Denominators – Quotient Rule

Quotient Rule: √ ab=¿

It may be helpful to reduce the _________________ first and the ______________ second

Example 1:√48√150

Example 2:

√ 225 x7

20 x3

Q 1: Q 2:

301

15.3c Rationalize Denominators – Rationalize Monomial Roots in the Denominator

Rationalize Denominators: Never leave a _________________ in the ______________________

To clear radicals: ________________ by extra needed factors in denominator (same in numerator!)

It may be helpful to _________________ first

Hint: _____________ numbers!

Example 1:57√b2

Example 2:3√ 79a2b

Q 1: Q 2:

302

15.3d Rationalize Denominators – Rationalize Binomial Denominators

What does not work: 1

2+√3=¿

Recall: (2+√3 )( )=¿

Multiply by the __________________________

Example 1:6

5−√3

Example 2:3−5√24+2√2

Q 1: Q 2:

You have completed the videos for 15.3 Rationalize Denominators. On your own paper complete the homework assignment.

303

15.4a Rational Exponents – Convert

If we divide the exponent by the index, then n√am=¿

The index is the _______________________

Example 1:Write as an exponent: 7√m5

Example 2:Write as a radical: (ab )2/3

Example 3:Write as a radical: x−4 /5

Example 4:

Write as an exponent: 1

( 3√5x )2

Q 1: Q 2:

Q 3: Q 4:

304

15.4b Rational Exponents – Evaluate

To evaluate a rational exponent ______________ to a ________________________

Example 1:Evaluate: 322 /5

Example 2:Evaluate: 27−4 /3

Q 1: Q 2:

305

15.4c Rational Exponents – Simplify

Recall Exponent Properties:

am an=¿ am

an =¿ (ab )m=¿

( ab )

m

=¿ (am )n=¿ a0=¿

a−m=¿

To Simplify:

1a−m =¿ ( a

b )−m

=¿

Example 1:x4/3 y2/7 x5/4 y3 /7

x1/2 y6 /7

Example 2:

( 256 x3/ 2 y−1 /3

x1 /4 y3 /2 x−5/2 )−1/ 8

306

Q 1: Q 2:

You have completed the videos for 15.4 Rational Exponents. On your own paper complete the homework assignment.

307

15.5a Radicals of Mixed Index – Reduce Index

Using rational exponents: 8√ x6 y2=¿

To reduce the index ____________ the ___________ and the ________________ by the _______

Without using rational exponents: 8√ x6 y2=¿

Hint: ______________ any numbers

Example 1:15√x3 y9 z6

Example 2:25√32a10 b5 c20

Q 1: Q 2:

15.5b Radicals of Mixed Index – Multiply Mixed Index 308

Using rational exponents: 3√a2b ∙ 4√ab2=¿

Get a ____________________ by ________________ the _______________ and ________________

Without using rational exponents: 3√a2b ∙ 4√ab2=¿

Hint: __________________ any numbers

Always be sure your final answer is ___________________

Example 1:4√m3n2 p ∙ 6√m n2 p3

Example 2:3√4 x2 y ∙ 5√8 x4 y2

Q 1: Q 2:

15.5d Radicals of Mixed Index – Divide Mixed Index

309

Division with mixed index – get a __________________________

Hint: __________________ any numbers

May have to _____________ the denominator (cannot be under a ___________ and under a ____________)

Example 1:√ab33√ab2

Example 2:4√2 x3 y26√32 y 4

Q 1: Q 2:

You have completed the videos for 15.5 Radicals of Mixed Index. On your own paper complete the homework assignment.

15.6a Complex Numbers – Square Roots of Negatives

310

Define: √−1=¿ and therefore i2=¿

Now we can calculate √−25=¿

Expressions with radicals: Always ______________ the _________________ first

Example 1:√−45

Example 2:√−6 ∙√−10

Q 1: Q 2:

15.6b Complex Numbers – Simplify Square Roots of Negatives

311

Before _____________________ with fractions, be sure to _____________ first

Example 1:15+√−300

5

Example 2:20+√−80

8

Q 1: Q 2:

15.6c Complex Numbers – Add and Subtract

312

i works just like ____________________

This means we can _________________________________

Example 1:(5−3i )+(6+i)

Example 2:(−5−2i )−(3−6 i)

Q 1: Q 2:

15.6d Complex Numbers – Multiply

313

i works just like ____________________

Remember i2=¿

Example 1:(−3 i ) (6 i )

Example 2:2 i(5−2 i)

Example 3:(4−3i)(2−5 i)

Example 4:(3+2 i )2

Q 1: Q 2:

Q 3: Q 4:

15.6e Complex Numbers – Rationalize Monomial Denominators

314

If i=¿ then we can rationalize it by just multiplying by _____

Example 1:5+3 i4 i

Example 2:2−i−3 i

Q 1: Q 2:

15.6f Complex Numbers – Rationalize Binomial Denominators

315

Similar to other radicals we can rationalize a binomial by multiplying by the _____________________

(a+bi ) (a−bi )=¿

Example 1:4 i2−5 i

Example 2:4−2 i3+5 i

Q 1: Q 2:

You have completed the videos for 15.6 Complex Numbers. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 15: Radicals. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

316

Unit 16:

Quadratics, Rational Equations,

and Applications



317

16.1a Complete the Square – Find c

a2+2ab+b2 is easily factored to ______________________

To make x2+bx+c a perfect square, c=¿

Example 1:Find c and factor the perfect square:

x2+10x+c

Example 2:Find c and factor the perfect square

x2−7 x+c


x2−37

x+c


x2+ 65

x+c

Q 1: Q 2:

Q 3: Q 4:

318

16.1b Complete the Square – Rational Solutions

If x2=9 then there are ____ solutions for x, ______ and ________. We can write this as ___________

To complete the square on a x2+bx+c=0

1. Separate ________________ and __________________

2. Divide by _______ (everything)

3. Find the _________ and _________ to ________________________

Example 1:x2−x−6=0

Example 2:3 x2=15 x−18

319

Q 1: Q 2:

320

16.1c Complete the Square – Irrational and Complex Solutions

If we can’t simplify the _______________ we _____________________ what we can.

Example 1:5 x2−3 x+2=0

Example 2:8 x+28=4 x2

321

Q 1: Q 2:

You have completed the videos for 16.1 Complete the Square. On your own paper complete the homework assignment.

322

16.2a Quadratic Formula – Finding the Formula

Solve by Completing the Square:

a x2+bx+c=0

(How we found the formula is useful to know for the test!)

323

16.2b Quadratic Formula – Using the Formula

If a x2+bx+c=0 the x=¿

Example 1:6 x2+7 x−3=0

Example 2:5 x2−x+2=0

Q 1: Q 2:

324

16.2c Quadratic Formula – Make Equation Equal Zero

Before using the quadratic formula, the equation must equal ______ and be in _____________

That is the equation should look like:

Example 1:2 x2=15−7 x

Example 2:3 x2+5 x+2=7

Q 1: Q 2:

325

16.2d Quadratic Formula – Missing Terms

If a term is missing, we use _____ in the quadratic formula

Example 1:3 x2+54=0

Example 2:5 x2=2 x

Q 1: Q 2:

You have completed the videos for 16.2 Quadratic Formula. On your own paper complete the homework assignment.

326

16.3a Equations with Radicals – Odd Roots

The opposite of taking a root is to do an ________________

3√ x=4 then x=¿ (Note: This only works for an _________ index)

Example 1:3√2x−5=6

Example 2:5√4 x−7=2

Q 1: Q 2:

327

16.3b Equations with Radicals – Even Roots

The opposite of taking a root is to do an ________________

With even roots we must ___________ the answer in the original equation! (called ____________________)

Recall: (a+b )2=¿

Example 1:x=√5 x+24

Example 2:√40−3 x=2x−5

Q 1: Q 2:

328

16.3c Equations with Radicals – Isolate Radical

IMPORTANT: Before we can clear a radical it must first be _________________

Example 1:4+2√2 x−1=2x

Example 2:2√5 x+1−2=2x

329

Q 1: Q 2:

You have completed the videos for 16.3 Equations with Radicals. On your own paper complete the homework assignment.

330

16.4a Equations with Exponents – Odd Exponents

The opposite of taking an exponent is to do a ________________

If x3=8, then x=¿ (Note: This only works for an _________ exponent)

Example 1:(3 x+5 )5=32

Example 2:(2 x−1 )3=64

Q 1: Q 2:

331

16.4b Equations with Exponents – Even Exponents

Consider: (5 )2=¿ and (−5 )2=¿

When we clear an even exponent we have _____________________________

Example 1:(5 x−1 )2=49

Example 2:(3 x+2 )4=81

Q 1: Q 2:

332

16.4c Equations with Exponents – Isolate Exponent

IMPORTANT: Before we can clear an exponent it must first be _________________

Example 1:4−2 (2 x+1 )2=−46

Example 2:5 (3 x−2 )2+6=46

Q 1: Q 2:

333

16.4d Equations with Exponents – Rational Exponents

To multiply to one: ab ∙( )=1We clear a rational exponent by using a _______________________

Recall am/n=¿

Recall: Check if original rational exponent has _______________________

Recall: Two solutions if original rational exponent has ___________________

Example 1:(3 x−6 )3 /2=64

Example 2:(5 x+1 )4 /5=16

Q 1: Q 2:

You have completed the videos for 16.4 Equations with Exponents. On your own paper complete the homework assignment.

334

16.5a Rectangle Problems – Area Problems

Area of a rectangle:

To help visualize the rectangle, __________________________________________

There are three ways to solve any quadratic equation

1.

2.

3.

Example 1:The length of a rectangle is 2 ft longer than the width. The area of the rectangle is 48 ft2. What are the dimensions of the rectangle?

Example 2:The area of a rectangle is 72 cm2. If the width is 6 cm less than the length, what are the dimensions of the rectangle?

Q 1: Q 2:

335

16.5b Rectangle Problems – Perimeter Problems

Perimeter of a rectangle:

Tip: Solve the ________________ equation for a variable and ________________ in the _________ equation.

Example 1:The area of a rectangle is 54m2. If the perimeter is 30 meters, what are the dimensions of the rectangle?

Example 2:The perimeter of a rectangle is 22 inches. If the area of the same rectangle is 24 in2, what are the dimensions?

Q 1: Q 2:

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16.5c Rectangle Problems – Bigger

We may have to draw _____________ rectangles

Multiply/Add to the _________ to make it equal the ____________ rectangle

Example 1:Each side of a square is decreased 6 inches. When this happens, the area of the larger square is 16 times the area of the smaller square. How many inches is the side of the original square?

Example 2:The length of a rectangle is 9 feet longer than it is wide. If each side is increased 9 feet, then the area is multiplied by 3. What are the dimensions of the original rectangle?

Q 1: Q 2:

337

16.5d Rectangle Problems – Frames

To help visualize the frame __________________________

Remember the frame is on the __________ and ______________, also the ___________ and _____________

Example 1:A frame measures 13 inches by 10 inches and is of uniform width. If the area of the picture inside is 54 square inches, what is the width of the frame?

Example 2:An 8 inch by 12 inch drawing has a frame of uniform width around it. The area of the frame is equal to the area of the picture. What is the width of the frame?

Q 1: Q 2:

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16.5e Rectangle Problems – Percent of a Field

Clearly identify the area of the ___________________ and ______________________ rectangles!

Be careful with ___________________, is it talking about the ___________, ______________, or _________?

Example 1:A man mows his 40 ft by 50 ft rectangular lawn in a spiral pattern starting from the outside edge. By noon he is 90% done. How wide of a strip has he cut around the outside edge?

Example 2:A woman has a 50 ft by 25 ft rectangular field that he wants to increase by 68% by cultivating a strip of uniform width around the current field. How wide of a strip should she cultivate?

Q 1: Q 2:

You have completed the videos for 16.5 Rectangle Problems. On your own paper complete the homework assignment.

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16.6a Rational Equations – Clear Denominator

Recall: 34

x−12=56

Clear fractions by multiplying each _________________ by the ________________

Example 1:5x= 37 x

−4

Example 2:4

x+5+x= −2

x+5

Q 1: Q 2:

340

16.6b Rational Equations – Factoring Denominator

To identify all the factors in the ________ we may have to ______________ the ____________________

Example 1:x

x−6+ 1

x−7= −3 x−8

x2−13 x+42

Example 2:2

x+3− 9x

x2−9= 1

x−3

Q 1: Q 2:

341

16.6c Rational Equations – Extraneous Solutions

Because we are working with fractions, the _____________ cannot be _____________

Example 1:x

x−8− 2

x−4= −3 x+56

x2−12 x+32

Example 2:x

x−2+ 2

x−4= 4 x−12

x2−6 x+8

Q 1: Q 2:

You have completed the videos for 16.6 Rational Equations. On your own paper complete the homework assignment.

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16.7a Work Problems – One Unknown Time

Adam does a job in 4 hours. Each hour he does ______ of the job.

Betty does a job in 12 hours. Each hour she does ______ of the job.

Together, each hour they do _____________________ of the job

This means together it would take them _________ hours to do the entire job.

Work equation:

Example 1:Catherine can paint a house in 15 hours. Dan can paint it in 30 hours. How long will it take them working together?

Example 2:Even can clean a room in 3 hours. If his sister Faith

helps, it takes them 225 hours. How long will it take

Faith working alone?

Q 1: Q 2:

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16.7b Work Problems – Two Unknown Times

Be sure to clearly identify who is the _________________

Example 1:Tony does a job in 16 hours less time than Marissa, and they can do it together in 15 hours. How long will it take each to do the job alone?

Example 2:Alex can complete his project in 21 hours less than Hillary. If they work together it can get done in 10 hours. How long does it take each working alone?

Q 1: Q 2:

You have completed the videos for 16.7 Work Problems. On your own paper complete the homework assignment.

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16.8a Distance and Revenue Problems – Simultaneous Products

Simultaneous product: ________ equations with _________ variables that are __________________

To solve: ________________ both by the same ______________________. Then ____________________.

Example 1:xy=72

( x−5 ) ( y+2 )=56

Q 1:

345

16.8b Distance and Revenue Problems – Revenue

Revenue Equation:

Beware: Profit =

To solve: Divide by what we _______________

Example 1:A group of college students bought a couch for $80. However, five of them failed to pay their share so the others had to each pay $8 more. How many students were in the original group?

Example 2:A merchant bought several pieces of silk for $70. He sold all but two of them at a profit of $4 per piece. His total profit was $18. How many pieces did he originally purchase?

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Q 1: Q 2:

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16.8c Distance and Revenue Problems – Distance

Distance Equation:

To solve: Divide by what we _______________

Example 1:A man rode his bike to a park 60 miles away. On the return trip he went 2 mph slower which made the trip take 1 hour longer. How fast did he ride to the park?

Example 2:After driving through a construction zone for 45 miles, a woman realized that if she had just driven 6 mph faster she would have arrived 2 hours sooner. How fast did she drive?

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Q 1: Q 2:

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16.8d Distance and Revenue Problems – Streams and Wind

Downwind/stream:

Upwind/stream:

Example 1:Zoe rows a boat downstream for 80 miles. The return trip upstream took 12 hours longer. If the current flows at 3 mph, how fast does Zoe row in still water?

Example 2:Darius flies a plane against a headwind for 5084 miles. The return trip with the wind took 20 hours less time. If the wind speed is 10 mph, how fast does Darius fly the plane when there is no wind?

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Q 1: Q 2:

You have completed the videos for 16.8 Distance and Revenue Problems. On your own paper complete the homework assignment.

351

16.9a Compound Fractions – Numbers

Compound/Complex Fractions:

Clear ___________________ by multiplying each _________ by the _______________ of everything

Example 1:34+ 56

12−43

Example 2:12+2

1+ 94

Q 1: Q 2:

352

16.9b Compound Fractions – Monomials

Recall: To find the LCD with variables, use the __________________ exponents

Be sure to check for _______________________ by __________________ the numerator and denominator

Example 1:

1− 9x2

1x+ 3

x2

Example 2:1y3

− 1x3

1x2 y3

− 1x3 y2

Q 1: Q 2:

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16.9c Compound Fractions – Binomials

The LCD could have one or more _______________ in it!

Example 1:5

x−2

3+ 2x−2

Example 2:x

x−9+ 5

x+9x

x+9− 5

x−9

Q 1: Q 2:

354

16.9d Compound Fractions – Negative Exponents

Recall: 5 x−3=¿

If there is any ______ or __________ we can’t just _________ terms. Instead make ___________________

Example 1:1+10x−1+25 x−2

1−25 x−2

Example 2:8b−3+27a−3

4 a−1b−3−6a−2b−2+9a−3b−1

Q 1: Q 2:

You have completed the videos for 16.9 Compound Fractions. On your own paper complete the homework assignment.

355

Congratulations! You made it through the material for Unit 16: Quadratics, Rational Equations, and Applications. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

Unit 17:

Functions



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17.1a Evaluate Functions – Functions

Function:

If it is a function we often write ____ which is read __________________________

A graph is a function if it passes the ______________________________, or each ____ has at most one ____

Example 1:Is the graph a function?

Example 2:Is the graph a function?

Q 1: Q 2:

357

17.1b Evaluate Functions – Function Notation

Function notation:

What is inside of the function ___________________ the ________________________

Example 1:f ( x )=−x2+2 x−5

Find f (3)

Example 2:g ( x )=√2x+5

Find g(20)

Q 1: Q 2:

358

17.1c Evaluate Functions – Evaluate Function at an Expression

When replacing a variable we always use _________________________

What is inside of the function ___________________ the ________________________

Example 1:f ( x )=√2 x+3 x

Find f (8x2 )

Example 2:p (n )=n2−2n+5

Find p(n−3)

Q 1: Q 2:

359

17.1d Evaluate Functions – Domain

Domain:

Fractions:

Even Radicals:

Example 1:Find the domain:f ( x )=3 4√2 x−6+4

Example 2:Find the domain:

g(x )=3|2 x+7|2−4

Example 3:Find the domain:

h ( x )= x−1x2−x−2

Q 1:

Q 2: Q 3:

You have completed the videos for 17.1 Evaluate Functions. On your own paper complete the homework assignment.

360

17.2a Operations on Functions – Add Functions

Add Functions: ( f +g ) (x )=¿

With a number we will ________________ both, then ______________ the results

With a variable we will ____________ the two functions __________________. Use ___________________!

Example 1:f ( x )=x−4

g ( x )=x2−6 x+8Find ( f +g)(−2)

Example 2:f ( x )=x2−5 xg ( x )=x−5

Find ( f +g)(x)

Q 1: Q 2:

361

17.2b Operations on Functions – Subtract Functions

Subtract Functions: ( f −g ) ( x )=¿




g ( x )=x2−6 x+8Find ( f −g)(−2)


Find ( f−g)(x)

Q 1: Q 2:

362

17.2c Operations on Functions – Multiply Functions

Multiply Functions: ( f ∙ g ) ( x )=¿




g ( x )=x2−6 x+8Find ( f ∙ g)(−2)


Find ( f ∙ g)(x)

Q 1: Q 2:

363

17.2d Operations on Functions – Divide Functions

Divide Functions: ( fg ) ( x )=¿



Beware of ______________ of fractions, the ________________ cannot be ____________


g ( x )=x2−6 x+8

Find ( fg )(−2)


Find ( fg )(x )

Q 1: Q 2:

364

17.2e Operations on Functions – Composition of Functions

Composition of Functions:

( f ∘ g ) (x )=¿

With numbers, _________________ the __________________ and put ______________ in ______________

With a variable, put the _________________ in for the ___________________ in the __________________

Example 1:f ( x )=√x+6g ( x )=x+3

( f ∘ g ) (7 )=¿

g[ f (7 )]=¿

Example 2:p ( x )=x2+2 xr ( x )=x+3

( p∘r ) ( x )=¿

r [ p (n )]=¿

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Q 1: Q 2:

366

17.2f Operations on Functions – Compose a Function with Itself

A function can be composed with _____________

Example 1:f ( x )=2 x−4

Find ( f ∘ f )(−2)

Example 2:g ( x )=x2−3 xFind g[ g ( x )]

Q 1: Q 2:

367

17.2g Operations on Functions – Composition of Several Functions

If we are composing several function, start in the _______________ and work __________

Example 1:f ( x )=x+2g ( x )=x2−5h ( x )=√3 x

Find ( f ∘ g∘h)(2)

Example 2:f ( x )=x+2g ( x )=x2−5h ( x )=√3 x

Find ( f ∘ g∘h)(a)

Q 1: Q 2:

You have completed the videos for 17.2 Operations on Functions. On your own paper complete the homework assignment.

368

17.3a Inverse Functions – Show Functions are Inverses

Inverse Function:

To test if functions are inverses, calculate __________ and ___________, the answer to both should be ___

Example 1:Are they inverses?

f ( x )=3 x−8

g ( x )= x3+8

Example 2:Are they inverses?

f ( x )= 5x−3

+6

g ( x )= 5x−6

+3

Q 1: Q 2:

369

17.3b Inverse Functions – Finding an Inverse Function

To find an inverse function _______________ the ____ and ____ , then solve for ____.

(the ____ is the y!)

Example 1:Find the inverse:

h ( x )= −3x−1

−2

Example 2:Find the inverse:g ( x )=5 3√x−6+4

370

Q 1: Q 2:

371

17.3c Inverse Functions – Inverse of Rational Functions

Clear fractions by ________________________________

Put the terms with ____ on one side and ____________________ on the other side

Factor out the _____ and _________________ to get it alone


f ( x )=2 x−5x+3


g ( x )= 5 x+12x−5

372

Q 1: Q 2:

You have completed the videos for 17.3 Inverse Functions. On your own paper complete the homework assignment.

373

17.4 Graphs of Quadratic Functions – Key Points

Quadratic Equations are of the form:

Quadratic Graph: Key points:

Example 1:Graph the function:

f ( x )=x2−2x−3

Example 2:Graph the function

f ( x )=−3x2+12 x−9

374

Q 1: Q 2:

You have completed the videos for 17.4 Graphs of Quadratic Functions. On your own paper complete the homework assignment.

375

17.5a Exponential Equations – With Common Base

Exponential functions:

Solving exponential functions: If the ______________ are equal, then the ___________________ are equal

Example 1:73 x−6=75 x+2

Example 2:45− x=43x

Q 1: Q 2:

376

17.5b Exponential Equations – Find a Common Base

If we don’t have a common base, then we find the _____________________ of the base

Recall exponent property: (am )n=¿

When using the above property we may have to ________________________

Example 1:272 x=9

Example 2:82 x−4=16x+3

Q 1: Q 2:

377

17.5c Exponential Equations – With Negative Exponents

Fractions are created by _____________________________

Example 1:

( 13 )x

=814 x

Example 2:

( 125 )3 x−1

=1254 x+2

Q 1: Q 2:

You have completed the videos for 17.5 Exponential Equations. On your own paper complete the homework assignment.

378

17.6a Compound Interest – N Compound a Year

Compound interest:

n compounds per year: A=P(1+ rn )

nt

A=¿

P=¿

r=¿

n=¿

t=¿

Example 1:Suppose you invest $13,000 in an account that pays 8% interest compounded monthly. How much would be in the account after 9 years?

Example 2:A bank loans out $800 at 3% interest compounded quarterly. If the loan is paid in full after five years, what is the balance owed?

Q 1: Q 2:

379

17.6b Compound Interest – Continuous Interest

Continuous interest:

A=P ert

A=¿

P=¿

e=¿

r=¿

t=¿

Example 1:An investment of $25,000 is at an interest rate of 11.5% compounded continuously. What is the balance after 20 years?

Example 2:What is the balance at the end of 10 years on an investment of $13,000 at 4% compounded continuously?

Q 1: Q 2:

You have completed the videos for 17.6 Compound Interest. On your own paper complete the homework assignment.

380

17.7a Logarithms – Convert Between Logs and Exponents

Logarithm:

bx=a can be written as __________________

Example 1:Write as a log:

m2=25

Example 2:Write as an exponent:

log x64=2

Q 1: Q 2:

381

17.7b Logarithms – Evaluate Logs

To evaluate a log: make the equation ______________________ and convert to an ________________

Example 1:log 464

Example 2:

log3( 181 )

Q 1: Q 2:

382

17.7c Logarithms – Solve Log Equations

To solve a log equation: convert to an ________________

Example 1:log x8=3

Example 2:log5(2x−6)=2

Q 1: Q 2:

You have completed the videos for 17.7 Logarithms. On your own paper complete the homework assignment.

Congratulations! You made it through the material for Unit 17: Functions. It is time to prepare for your exam. On a separate sheet of paper complete the practice test. Once you have completed the practice test, ask your instructor to take the test. Good luck!

383

Unit 18:

Proficiency Exam #3


1. Complete the practice tests on your own paper.2. Take the (two part) unit exam.

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Documents

· Web viewThis work is a combination of original work and a derivative of Prealgebra Workbook, Beginning Algebra Workbook, and Intermediate Algebra Workbook by Tyler Wallace which