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"Uncle Dan’s Algebra"

"It sure helps to have a teacher!"

"Algebra Workbook"

There is a video and answer CD to this work book contact

E-mail: uncledan@homespun4homeschoolers.com

Part 1 - Basic Operations and Solving Equations

Introduction to Integers and the Number Line 1

Subtraction of Integers 2

QUIZ: (Adding and Subtracting Integers) 4

Packages - "Do Me First" 5

Multiplication of Integers 6

Division of Integers 6

Operations with Integers Using Packages 7

QUIZ: (Operations with Integers Using Packages) 8

TEST- Operations with Integers 9

Exponents and Raising a Number to a Power 10

Order of Operations 10

QUIZ: (Exponents and Powers) 13

Evaluating Variable Expressions 14

Evaluating Variable Expressions with Exponents 16

More Evaluating Expressions 18

TEST- Evaluating Variable Expressions 20

Solving Equations 21

Order of Operations When Solving Equations 25

Combining Like Terms 26

Combining Terms: First Step in Solving Equations 26

Transposing: Are You Ready to Skip a Step? 27

QUIZ: (Solving Equations) 27

TEST- Solving Equations 28

The Distributive Law 30

More Solving Equations 32

TEST- More Solving Equations 33

Part 2 - Solving Word Problems with Algebra

Writing Variable Expressions 34

Number Problems 36

Consecutive-Integer Problems 39

Age Problems 41

Geometry Problems: Angles and Degree Measure 43

Coin Problems: Penny, Nickel, Dime, Quarter 44

Uniform Motion Problems: Rate x Time = Distance 46

TEST- Solving Word Problems with Algebra 49

Part 3 - Coordinate Geometry and Systems of Equations

Graphing Linear Equations on a Coordinate Plane 51

Using the Slope-Intercept Form: y = mx + b 56

More on Slopes 58

Solving a System of Equations Graphically 62

TEST- Coordinate Geometry 65

Solving Systems of Equations in Two Variable 68

Problems Using Two Variables - Digit Problems 72

TEST- Systems of Equations 74

Part 4 - Polynomials

Introduction to Polynomials 75

Subtracting Polynomials 75

Multiplying and Powers: The Laws of Exponents for Multipliction 76

QUIZ: (Multiplying and Powers) 77

More Complicated Expressions Involving Multiplying and Powers 77

Multiplying a Monomial by a Polynomial 78

Multiplying a Polynomial by a Polynomial 79

Are You Ready to Skip a Step- The "FOIL" Method 80

The Laws of Exponents for Division 81

QUIZ: (The Laws of Exponents for Division) 84

Division of a Polynomial by a Monomial 85

Division of a Polynomial by a Binomial 85

TEST- Basic Operations with Polynomials 87

Special Products 88

Factoring Out a Common Monomial Factor 89

Factoring Special-Product Polynomials 89

Factoring Completely 91

TEST- Factoring 92

Part 5 - Fractions

Equivalent Fractions 93

Multiplying Fractions 96

Dividing Fractions 97

Adding and Subtracting Like Fractions 98

Adding and Subtracting Unlike Fractions 99

Simplifying Complex Fractions 103

Solving Equations Containing Fractions 106

Solving Quadratic Equations by Factoring 112

Fractional Equations that Transform into Quadratic Equations 113

TEST - Operations and Equations with Fractions 115

Solving Word Problems Involving Quadratic Terms, Factoring, Etc. 118

Solving Word Problems Involving Equations with Fractions 121

TEST - Equations involving Quadratic Equations and Fractions 124

Part 6 - Radicals and Roots

Introduction to Rational and Irrational Numbers 125

Square Roots 125

The Laws of Square Roots 126

Part 7 - More on Quadratic Equations

Solving Quadratic Equations by Completing the Square 144

The Derivation of the Quadratic Formula 147

The Discriminate and the Nature of the Roots 148

TEST - Completing the Square and The Quadratic Formula 150

Part 8 - Inequalities

Introduction to Inequalities 151

Solving Inequalities 154

Combining Inequalities 156

Absolute Values in Open sentences 161

TEST - Inequalities 164

Graphing Linear Inequalities in Two Variables 166

Solving a System of Inequalities Graphically 168

TEST - Graphing Linear Inequalities in Two Variables 169

Part 9 - Supplementary Topics

Calculating Square Roots 170

TEST - Calculating Square Roots 173

Changing Repeating Decimals to Fractions 174

TEST - Changing Repeating Decimals to Fractions 175

Scientific Notation 176

TEST - Scientific Notation 177

The Distance Formula 178

TEST - The Distance Formula 179

Workbook

Part 1 - Basic Operations and Solving Equations

Introduction to Integers and the Number Line

{1, 2, 3, 4, . . . } = the "counting numbers"

{0, 1, 2, 3, 4, . . . } = the "natural numbers"

{ . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . } = the "integers"

The Number Line:

l l l l l l l l l l l l l l l l l l l

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

= means "is equal to"

> means "is greater than" or "is to the right of"

< means "is less than" or "is to the left of"

5 = 5 3 + 2 = 5 5 > 3 3 < 5

-3 < 2 -4 < -2 -3 > -5 2 > -5

Practice: (Put in the correct sign: =, >, or <)

8 8 8 10 12 5

-4 -7 -6 1 -5 -2

3 -6 -3 0 0 4

5 + 2 7 3 + 1 5 8 - 7 4

Note: The value of a number not considering the sign is the "absolute value".

l 5 l = 5 l-5 l = 5

Practice:

l 8 l = l-8 l =

1

5 + 3 = (3) + (+5) =

(-5) + (-3) = (-3) + (-5) =

Note: Since numbers can be added in any order, we say that addition is

"commutative".

Rule: When adding two integers with different signs:

1) Find the difference between the absolute values of the two integers.

2) Use the sign of the integer with the greater absolute value.

5 + (-3) = (-5) + 3 =

Practice: (Show number line for each problem)

3 + 4 = 3 + (-4) =

-4 + 3 = -4 + (-3) =

37 + 45 = -33 + 78 =

53 + (-93) = -33 + (-47) =

Subtraction of Integers

Rule: To subtract one integer from another, add the "opposite" of the number to

be subtracted.

Note: You may use the number line to help you only after you have changed the

5 - (+3) = Note: 5 - (+3) is the same as : 5 - 3

When the double signs are different, it is the

same as if there were one "minus" sign.

-5 - (+3) =

2

5 - (-3) = Note: 5 - (-3) is the same as: 5 + 3

When the double signs are the same, it is the

same as if there were one "plus" sign.

-5 - (-3) = And: "minus a minus" is a "plus".

Practice:

7 - (+4) = 8 - 6 =

-6 - (+9) = -5 - 8 =

8 - (-10) = -4 - (-7) =

When there's more than two of them:

5 - 2 + (-4) - (+3) - (-6) + (+5) = 4 - 9 + (-3) - (-6) - 8 + 0 - (+4) - 0 =

Practice:

6 + 3 - (-7) + (-4) - (+5) + 8 + (+2) = -7 + (3) - (-9) - 8 + 4 - (+7) + (-5) =

- (-12) + (-18) + 8 - 0 -19 - (+36) = 7 + 0 - 14 + (-2) - (-6) - (+4) + 3 =

3

Part I - Show number line for each one

54 + (-36) =

28 - 42 =

-19 + (-48) =

-36 + 94 =

42 - (-14) =

39 + (-86) =

78 + (+17) =

-74 - (+37) =

21 + 65 =

-38 - (-26) =

Part II

-42 + 27 - (-17) - 38 + (-14) - 0 + (+31) - 25 =

4

Packages - "Do Me First"

( ) parentheses Note: "Packages" or "grouping symbols"

are usually arranged in this order:

[ ] brackets parentheses on the inside, then

brackets, then braces:

{ } braces { [ ( ) ] }

Rule: First perform the operations indicated within the innermost packages.

Work from the inside to the outside.

7 - (8 - 9) = [ 4 - (7 - 5) ] - [ -8 - (3 - 7) ] =

{ 12 - [ -18 + 7 - (3 - 4) ] } + { [ 5 - (-8) ] - 15 } =

Practice:

8 - (9 - 12) = [ (5 - 8) - (-4) ] + [ -9 + (6 - 9) ] =

{ [7 - (-8) ] - (3 - 4 ) } - { (-8 + 9) + [ 6 - (-9) ] } =

5

Multiplication of Integers

3 x 5 = 3 . 5 = 3 (5) = (3) 5 = (3) (5) =

Rule: When multiplying integers:

If the signs are the same, the product is positive.

If the signs are different, the product is negative.

(3) (5) = (-3) 5 = 3 (-5) = -3 . -5 =

Practice:

-4 (7) = (4) (8) = -9 . -7 = 8 (-9) =

Note: (8) (-9) is not the same as: (8) - (9)

Division of Integers

Note: The rule for division of integers is the same as the rule for multiplication.

10 ÷ 2 = -10 ÷

2 = 10 ÷

-2 = -10 ÷ -2 =

10 = -10 = 10 = -10 =

2 2 -2 -2

10 + -12 = -5 =

2 4 8

Practice:

-63 = 48 = -54 = 72 =

9 -6 -6 8

-8 - -16 = 3 =

2 8 -5

6

Operations with Integers Using Packages

-3 (7 - 9) = 4 [ 7 - (-3 + 8) ] =

2 ( 3 - 8) = 7 ( -12 + -20 )

-5 (12 - 9)

4

-5 =

5 ( 9 - 8 )

3

-2

Practice:

-8 ( 4 - 3 + 7 ) = 3 [ (4 - 9) - (-2) ] =

-6 [ 8 + (-3 + 10) ] -3 [ 14 + (6 - 5) ]

=

-2

5 [ -7 - (4 - 5) ]

=

2 [ 3 - ( -12 + 12 ) ]

-2 -3

7

QUIZ: (Operations with Integers Using Packages)

-5 [ (3 - 5) - (-5) ] 3 [ 4 - ( -12 + 3 ) ]

=

3

3 [ 3 - (2 - 4)] =

-2 { -4 + [ 4 - (3 - 7) ] }

-2

8

TEST - Operations with Integers

Part I - Show number line for each problem

8 + (-4) =

-7 - (-3) =

3 - (-9) =

-7 + (-8) =

Part II

5 . -4 = 4 (7) = (-3) (-8) = (-5) 9 =

Part III

-12 = -8 = 18 = -3 =

3 -2 -6 4

Part IV Part V

-4 + (-3) + 7 - (-5) + 0 - (+8) =

-3 [ 4 - (8 -10 ) ]

=

2 ( -16 + 15 )

-8 -3

9

Exponents and Raising a Number to a Power

52 = = 5 is the "base". 2 is the "exponent".

25 = = 2 is the "base". 5 is the "exponent".

25 is read: "2 to the fifth (power)".

22 = = 23 = =

34 = = 2 (32) = =

(2 . 3) 2 = =

[ 3 (4) ] 2 = =

Practice:

42 = = 33 = =

104 = = 5 (22) = =

(5 . 2) 2 = =

[ 4 (2) ] 2 = =

Order of Operations

Remember: Any quantities packaged by grouping symbols (parentheses,

brackets, braces, etc.) are saying, "Do me first!"

Rule: Whenever there are no grouping symbols, operations must be done

in the following order:

1) Quantities must be raised to the indicated power.

2) Do multiplications and divisions in order from left to right

3) Do additions and subtractions in order from left to right.

10

3 + 5 . 8 = 12 ÷ 2 + 4 = 12 ÷ 4 . 3 =

7 + 3 . 4 - 8 + 12 ÷ 2 = 8 . 4 - 6 ÷ 2 + 10 =

8 + 4 [ -7 + ( -3 + 8 ) ] = 9 - 3 { - 4 [ 2 - (-6) ] } =

32 . 4 = 2 . 32 = (4)2 . (2)3 = (22 + 42) ÷ 2 =

82 - 62 = [ ( 52 - 33 ) - 6 ] 2 = Note: (-8)2 = 64

8 - 6 - 82 = -64

2 [ 43 - 8 . 2 + (8 - 5) 3 ]

=

- 3 ( 36 - 4 . 22 ) 2

11

5 . 4 + 5 = 20 - 10 ÷ 5 = 8 . 6 ÷ 4 =

8 - 12 ÷ 3 + 4 - 2 . 3 = 7 . 3 + 9 ÷ 3 -10 =

- 8 + 3 [ - 4 - (8 - 5) ] = - 4 + 2 { - 5 [ -2 (6 - 7) ] } =

23 . 3 = 3 . 52 = (3)3 . (2)2 = (32 + 24) ÷ 5 =

42 - 52 = [ (34 - 52 ) - 50 ] 2 =

4 - 5

12

Practice:

= - 32 = -9

(24 ÷ 6 + 22 ) 2

QUIZ: (Exponents and Powers)

-2 { 3 [ 32 (23 - 42) ] }

=

4 + 3 ( 24 - 4 . 3 - 6)

13

[ 32 - 8 ÷ 4 + (5 - 8)2] 2 Note: (-3)2 = 9

Evaluating Variable Expressions

n + 3 = when n = 2 3 + n = when n = 2 n + 3 = when n = -2

n - 5 = when n = 2 5 - n = when n = 2 5 - n = when n = -2

5n = when n = 2 5n = when n = -2

-5n = when n = 2 -5n = when n = -2

n = when n = 4 n = when n = -4 2 = when n = -4

2 2 n

5n - 4 = when n = -3 -2n = when n = -6

3

14

Practice:

n + 4 = when n = 3 4 + n = when n = 3 n + 4 = when n = -3

n - 8 = when n = 3 8 - n = when n = 3 8 - n = when n = -3

4n = when n = 3 4n = when n = -3

-4n = when n = 3 -4n = when n = -3

n = when n = 6 n = when n = -6 3 = when n = -6

3 3 n

6n - 3 = when n = -4 -3n = when n = -4

2

15

Evaluating Variable Expressions with Exponents

n2 = when n = 2 n2 = when n = -2 - n2 = when n = 2

- n2 = when n = -2 (-n)2 = when n = 2 (-n)2 = when n = -2

- (-n)2 = when n = 2 - (-n)2 = when n = -2 3n2 = when n = 2

3n2 = when n = -2 - 3n2 = when n = 2 - 3n2 = when n = -2

(3n)2 = when n = 2 (-3n)2 = when n = 2 - (3n)2 = when n = 2

4n3 = when n = 2 4n3 = when n = -2

- (4n)3 = when n = -2 - (4n3) = when n = -2

16

n2 = when n = 3 n2 = when n = -3 - n2 = when n = 3

- n2 = when n = -3 (-n)2 = when n = 3 (-n)2 = when n = -3

- (-n)2 = when n = 3 - (-n)2 = when n = -3 4n2 = when n = 3

4n2 = when n = -3 - 4n2 = when n = 3 - 4n2 = when n = -3

(4n)2 = when n = 3 (-4n)2 = when n = 3 - (4n)2 = when n = 3

2n3 = when n = 3 2n3 = when n = -3

- (2n)3 = when n = -3 - (2n3) = when n = -3

17

Practice:

(7n - 5) + (-4n - 3) = when n = -3 - - 4n + 4 = when n = 2

3n - 8

2n2 - n = when n = 5 n2 - n - 6 = when n = -5

n + 3

4n3 - 3n2 = when n = 2 (2n)3 - 4n2 + 8 = when n = -2

4 - 2n2 - (2n)2 +3n - 2

18

More Evaluating Expressions

(-6n + 2) - (3n + 2) = when n = -2 - - 5n + 5 = when n = 3

3n - 8

3n2 - n = when n = 4 n2 - n - 8 = when n = -4

n + 2

2n3 - 2n2 = when n = 3 (3n)3 - 6n2 + 45 = when n = -3

9 - 3n2 - (3n)2 +2n + 5

19

Practice:

TEST - Evaluating Variable Expressions

Part 1

3n - 6 = when n = 4 10 + 5n = when n = -2

-5n - 8 = when n = -3 -7 + 8n = when n = -1

4n = when n = -6 n - 10 = when n = -6

-3 3n - 14

4n2 = when n = 3 - 3n2 = when n = -3

(-2n)3 = when n = 2 - (3n)3 = when n = -2

Part II

(4n3 - 5) - [ (2n2 + 8) + (8 - 3n) ] = when n = -3

- -2n3 - 3n2 + n = when n = -2

(2n)2 - 4n + 6

20

Solving Equations

x + 4 = 10 x + 7 = -4 x + 8 = 3 x + 5 = -3

Practice:

x + 5 = 20 x + 5 = -5 x + 9 = 6 x + 3 = -1

Using the opposite of subtraction:

x - 4 = 6 x - 6 = 2 x - 4 = -2 x - 3 = -8

Practice:

x - 8 = 5 x - 9 = 3 x - 7 = -5 x - 4 = -7

21

-x and the "minus-one maneuver":

4 - x = 3 4 - x = 3 -6 - x = 8 -6 - x = 8

8 - x = -9 8 - x = -9 -5 - x = -7 -5 - x = -7

Practice: (Complete the solution and do the check)

5 - x = 2 5 - x = 2 -5 - x = 10 -5 - x = 10

+ x + x -5 -5 + x + x +5 +5

6 - x = -9 6 - x = -9 -4 - x = -8 -4 - x = -8

+ x + x -6 -6 + x + x +4 +4

22

Using the opposite of multiplication:

5x = 25 3x = -12 -3x = 9 -7x = -35

Practice:

4x = 12 5x = -20 -6x = 30 -4x = -16

Using the opposite of division:

x = 8 x = -2 x = 4 x = -2

4 3 -8 -5

Practice:

x = 7 x = -3 x = 6 x = -7

5 4 -4 -4

23

When the "coefficient" of the "variable" is a fraction:

Remember: The quantity which "can be anything" and is represented by a letter

is called the "variable".

Note: When a variable is multiplied by a number, that number is called the

"coefficient" of that variable. In the expression 5x; 5 is the "coefficient of x".

Rule: In an equation where the coefficient of a variable is a fraction, the equation

can be solved by multiplying both sides of the equation by the "reciprocal" or "flipflop"

of the fraction.

Note: 2x is the same as 2 x

3

3

2x = 24 2 x = 24

3 3

4 x = 20 5 x = -30 - 7 x = 35 - 2 x = -30

5 6 8 5

Practice:

3 x = 36 3 x = -30 - 5 x = 20 - 2 x = -18

4 5 8 3

24

Note: x = 3 is the same as 1 x = 3

4 4

1 x = 3 Practice: 1 x = 5

4 3

Order of Operations When Solving Equations

Rule: The order of operations when solving equations is the reverse of the order

of operations when evaluating variable expressions:

1) Use additions and subtractions to eliminate terms

2) Use multiplications and divisions to eliminate terms

3x + 2 = 14 5x - 7 = 13 4x + 20 = 8 3x - 10 = -25

Practice:

7x + 8 = 43 4x - 3 = 13 4x + 2 = -10 3x - 5 = -20

25

Combining Like Terms

5x + 4x = 3x - 2x = -3x + 7x = -5x - 10x =

Practice:

3x + 8x = 7x - 10x = -2x + x = -3x - x =

More combining like terms:

5x - 2 - 8x + 4 = 3x - 2x + 4 - 5 = 3 - 9x - 4 + 10x - 1 - 2x =

-5x - 4 + 3x - 1 + 3 + 2x = 5x + 4 - 3x + 2x - 5 + 4 - 4x - 3 =

Practice:

7x - 1 - x + 4 = -8x + 2 - 3 - x = -4x +2 - 3x + 9x - 7 - x =

5 + 3x - 8 - 5x + 4 - 1 = -5 + 6x + 8 - 3x - 4 - 2x + 1 - x =

Combining Terms: First Step in Solving Equations

5x + 3 - 2x - 7 = -10 7x + 5 - 3x - 20 = -27

Practice:

8x + 3 - 10x + 4 = 9 -5x + 3 + 4x - 8 = 9 Hint: You might need

the "Minus-One

Maneuver".

26

Transposing: Are You Ready to Skip a Step?

5x - 3 = 2x + 9 5x + 4 - 2x + 3 = 7x - 5 + 4x - 12

Practice:

7x + 19 = 4x + 4 6x + 3 - 2x - 7 = 5x - 8 + 3x + 12

QUIZ: (Solving Equations)

5x - 9 - 3x + 2x + 8 - 11 = 7 - 4x - 5 + 11x - 6 - x

27

TEST - Solving Equations

(Please, no "skipping steps" until next page.)

Solve each equation and do the check:

x + 5 = 8 x + 7 = -4 x - 6 = 2 x - 4 = -2

(Complete) (Complete)

2 - x = 3 9 - x = -3 3x = -21 -6x = -18

+ x + x -9 -9

-4x = 16 x = -8 x = -4 x = 4

3

-6

-3

3 x = 21 1 x = -9 3x + 5 = -1 4x - 18 = 6

5 3

28

4x + 4 - 2x - 7 = 3 8x - 5 - 2x + 11 = -18

3x - 20 = -2x + 5 3x - 4 + 2x + 8 = 5x + 1 - x

29

The Distributive Law

4(3 + 2) = 4(3) + 4(2)

3(x + 2) = 4(x - 3) = 6(3x - 5) =

4(6 - 2x) = -3(-5x + 2) = -4(7 - 3x) =

Practice:

4(x +3) = 3(x - 2) = 2(2x - 6) =

3(5 - 3x) = -4(-2x + 3) = -2(8 - 4x) =

Using the distributive law and then combining terms:

5(x - 4) - 3(2x + 2) = -4(3 - 2x) + 3(6x - 4) =

Practice:

6(x + 2) - 4(-2x + 3) = -3(4 - 3x) + 2(5 - 2x) =

30

Distributing a "minus-one" to "subtract" a quantity in parentheses:

Note: A plus sign directly in front of a quantity in parentheses is the same as if

the parentheses were not even there.

4x + (2x - 1) = 4x + 2x - 1 = 6x - 1

But a minus sign directly in front of a quantity in parentheses requires

special attention.

4x - (2x - 1) = 4x - 1(2x - 1) = 4x - 2x + 1 = 2x + 1

Rule: Whether or not you think of it as distributing with a "minus-one",

to clear a quantity in parentheses which has only a minus sign in front of it,

change the sign of every term within the parentheses.

5x - (2x + 2) = 4x - (3 - 2x) =

(6x - 2) - (-3x + 1) = - (x - 2) - 3(-2x - 1) =

3(2x - 4) - 4(x - 2) - (6 - 3x) =

Practice:

6x - (5x + 4) = 3x - (5 + 6x) =

(3x - 2) - (-4x + 5) = - (3x + 1) - 4(5 - 2x) =

5(3x - 2) - 2(4x - 3) - (4 - 2x) =

31

More Solving Equations

Using the Distributive Law and Combining Terms as First Steps:

3(4x - 7) = 5(3x + 12) 5(x - 4) - (x - 3) = 3(2x - 1) + 2(2x - 4)

Practice:

4(-3x - 5) = -5(2x - 8) 7(3x - 7) - (3 + 2x) = 3(5x + 1) - (5 - 2x)

32

TEST - More Solving Equations

Solve and Check:

4(x - 5) - 3(x + 2) = 6(x - 3) - (2x - 4)

33

Workbook

Part 2 - Solving Word Problems with Algebra

Writing Variable Expressions

the sum of n and 6 _____ 4 more than n _____

n increased by 12 _____ 6 plus n _____

14 minus n _____ the difference when 12 is diminished by n _____

3 subtracted from n _____ the difference when n is subtracted from 15 _____

5 less than n _____ n decreased by 10 _____

5 times n _____ the product when n is multiplied by 8 _____

twice n _____ the product of 9 and n _____

n divided by 6 _____ the quotient when n is divided by 4 _____

18 divided by n _____ the quotient when 10 is divided by n _____

Practice:

the sum of n and 8 _____ 5 more than n _____

n increased by 10 _____ 4 plus n _____

18 minus n _____ the difference when 11 is diminished by n _____

2 subtracted from n _____ the difference when n is subtracted from 12 _____

3 less than n _____ n decreased by 9 _____

4 times n _____ the product when n is multiplied by 6 _____

twice n _____ the product of 3 and n _____

n divided by 2 _____ the quotient when n is divided by 7 _____

15 divided by n _____ the quotient when 8 is divided by n _____

More Writing Variable Expressions:

3 times n increased by 4 __________ 3 times the quantity n increased by 4 _________

the sum of 5 times n and 2 __________ 5 times the sum of n and 2 __________

34

5 more than n divided by 2 __________ 5 more than n all divided by 2 __________

the sum of 6 and n divided by 5 ________ the sum of 6 and n all divided by 5 _______

8 less than n divided by 4 ________ the quantity 8 less than n then divided by 4 _______

17 minus n divided by 3 __________ 17 minus n all divided by 3 __________

6 less than n divided by 5 _______ the quotient when 6 less than n is divided by 5 _______

4 times n divided by 3 __________

the sum of n divided by 3 and 4 __________

Practice:

4 times n increased by 5 __________ 4 times the quantity n increased by 5 __________

the sum of 6 times n and 3 __________ 6 times the sum of n and 3 __________

twice n diminished by 8 _________ twice the quantity n diminished by 8 __________

7 more than n divided by 3 __________ 7 more than n all divided by 3 __________

the sum of 9 and n divided by 4 ________ the sum of 9 and n all divided by 4 ________

6 less than n divided by 2 ________ the quantity 6 less than n then divided by 2 _______

16 minus n divided by 5 __________ 16 minus n all divided by 5 __________

4 less than n divided by 2 _______ the quotient when 4 less than n is divided by 2 _______

7 times n divided by 5 __________

the sum of n divided by 2 and 6 __________

A Useful Concept When Writing Variable Expressions:

The sum of two numbers is 12. If one of them is n, the other can be expressed as _________

Practice:

The sum of two numbers is 36. If one of them is n, the other can be expressed as _________

35

twice n diminished by 9 _________ twice the quantity n diminished by 9 __________

Number Problems

1) The sum of a number and twice that 2) The sum of a number and the same

number is 12. What is the number? number increased by 6 is 48. What is

the number?

3) Three times a number subtracted from 4) Ten more than twice a number is

8 times a number is 25. What is the 24. What is the number?

number?

5) 36 more than 5 times a number is -34. 6) When 8 more than twice a number is

What is the number? subtracted from 3 less than 4 times a

number, the result is 13. What is the

number?

7) Five times a number decreased by 18 is 8) 5 times a number exceeds 2 less than 2

equal to 3 times the number increased by 6. times a number by 14. What is the

What is the number? number?

36

number. Their sum is 31. What are number. Their sum is 17. What is the

the numbers? smaller number?

11) One number is 25 greater than a second 12) The sum of two numbers is 28. 5 times

number. If the lesser number is subtracted the saller diminished by 2 times the

from 3 times the greater number, the differ- larger is 14. What are the numbers?

ence is 195. What are the numbers?

Practice:

1) The sum of a number and twice that 2) The sum of a number and the same number

number is 63. What is the number? increased by 9 is 33. What is the number?

3) Four times a number subtracted from 6 times 4) Six more than twice a number is 56.

a number is 24. What is the number? What is the number?

37

9) One number is 5 more than another 10) One number is 3 less than another

5) Eight less than 3 times a number is -62. 6) Two more than a certain number is 15 less

What is the number? than twice the number. What is the

number?

7) Six times a certain number exceeds 8) The sum of 5 more than a certain number

4 times the number by 12. Find the and 10 more than 4 times the number is

number. equal to 6 times the number increased by 3.

Find the number.

9) One number is 8 more than another number. 10) One number is 7 less than another

Their sum is 54. What are the numbers? number. Their sum is -39. What

is the smaller number?

11) One number is 14 greater than a 12) The sum of two numbers is 25. 3 times

second number. If the lesser number the smaller is 5 more than twice the

is subtracted from 4 times the greater larger. Find the numbers.

number, the difference is 23. Find the

numbers.

38

Consecutive-Integer Problems

Set of consecutive integers: {-4, -3, -2, -1, 0, 1, 2, ....} {n, (n + 1), (n + 2), (n + 3), ....}

Set of consecutive odd integers: {-5, -3, -1, 1, 3, 5, ....} {n, (n + 2), (n + 4), (n + 6), ....}

Set of consecutive even integers: {-4, -2, 0, 2, 4, ....} {n, (n + 2), (n + 4), (n + 6), ....}

1) Find two consecutive integers 2) Find two consecutive even integers

whose sum is 375. whose sum is 446.

3) Find three consecutive odd integers 4) Find three consecutive integers if the

whose sum is -3. sum of the first and third is equal to 246.

Practice:

1) Find three consecutive integers 2) Find three consecutive integers

whose sum is 213. whose sum is -153.

39

whose sum is 260. the second and fourth is 352.

5) Find two consecutive even integers if the 6) Find two consecutive integers if the lesser

larger is equal to 6 less than twice is equal to one more than twice the greater.

the smaller.

7) Find four consecutive even integers if 8) Find four consecutive odd integers if twice

twice the third is equal to the sum of the the sum of the first and the third is equal to

fourth and 3 times the second. 38 more than three times the sum of the

second and fourth.

40

3) Find four consecutive even integers 4) Find four consecutive integers if the sum of

Age Problems

1) Jack is 20 years older than Jill. Five 2) Jack is now 8 years old. Jill is now 2 years

years ago Jack was 5 times as old as Jill old. In how many years will Jack be twice

was then. How old is each now? as old as Jill will be then?

Now 5 years ago Now In x years .

Jack Jack .

Jill Jill .

Let x = Let x =

3) The sum of Jack's age and Jill's age 4) The sum of Jack's age and Jill's age

is 50. Eight years from now Jack is 32. Jack's age one year from now

will be twice as old as Jill will be then. will be 7 times Jill's age one year ago.

How old is each now? How old is each now?

Now 8 years from now Now 1 year from now 1 year ago

Jack Jack

Jill Jill .

Let x = Let x =

Practice:

1) Jack is 3 times as old as Jill. Ten years 2) Jack is 32 years old and Jill is 14. How

from now Jack will be twice as old as Jill many years ago was Jack 4 times as

will be then. How old is each now? old as Jill was then?

now 10 years from now now x years ago

Jack Jack .

Jill Jill .

Let x = Let x =

41

3) Jack is 4 times as old as Jill. Eight years 4) Jack's age exceeds Jill's by 16 years.

from now, Jack's age will exceed 3 times Four years ago Jack was twice as old

Jill's age by 2 years. How old is each of as Jill was then. How old is each of

them now? them now?

now 8 years from now now 4 years ago

Jack Jack .

Jill Jill .

Let x = Let x =

5) Five years from now, Jack will be 3 times as old as Jill will be then.

Four years ago Jack was 30 years older than Jill was then.

How old is each of them now

4 years ago now 5 years from now

Jack .

Jill .

Hint: Let x = Jill's age 5 years from now

42

Geometry Problems: Angles and Degree Measure

Remember:

1) The sum of the measures of the angles in any triangle is 180o.

2) Two angles are supplementary if their sum is equal to 180o. If the measure of one of

the angles is xo, the measure of the other can be expressed as (180 - x)o

3) Two angles are complementary if their sum is equal to 90o. If the measure of one of

the angles is xo, the measure of the other can be expressed as (90 - x)o

1) Find the measures of two complementary 2) Find the measure of an angle for which the

angles if the second measures 12o more sum of the measures of its complement and

than twice the first. its supplement is 194o .

3) The first angle of a triangle measures 3 times the second angle. The third angle

measures 4o less than the sum of the measures of the other two angles. Find the

measure of each angle.

Practice:

1) Find the measures of two supplementary 2) Find the measure of an angle whose

angles if the second measures 100o less supplement measures 8o more than

than three times the first. twice its complement.

43

3) The first angle of a triangle measures 20o more than the second angle. The third angle

measures 30o less than twice the sum of the first two angles. Find the measure of each

angle in the triangle.

Coin Problems: Penny, Nickel Dime, Quarter

Note:

In the USA: a penny = 1 cent, a nickel = 5 cents, a dime = 10 cents, a quarter = 25 cents

1) A jar contains 45 coins, all dimes and nickels, with a total value of \$3.50. How many are

there of each kind of coin?

number of coins x value of each = total value .

dimes .

nickels . .

2) A jar contains 3 more nickels than dimes, with a total value of \$1.95. How many are there

of each kind of coin?

number of coins x value of each = total value .

dimes .

nickels .

44

1) A jar contains 50 coins, all dimes and nickels, with a total value of \$4.00. How many are

there of each kind of coin?

number of coins x value of each = total value .

dimes .

nickels .

2) A jar has has twice as many dimes as nickels, and 2 more quarters than nickels. The total

value is \$4.00. How many are there of each kind of coin?

number of coins x value of each = total value .

dimes .

nickels .

quarters .

3) A jar has 50 coins; all pennies, nickels and dimes; with a total value of \$3.10. There are

twice as many nickels as pennies. How many are there of each kind of coin?

number of coins x value of each = total value .

pennies .

nickels .

dimes .

Hint: Whatever is neither pennies nor nickels, must be dimes.

45

Practice:

Uniform Motion Problems: Rate x Time = Distance

1) In a car race, one car went 12 miles/hr. 2) A helicopter flew at 150 miles/hr. to rescue

faster than the other and got to the finish the boat people who were 330 miles away.

line in 6 hours. The slower car took an The boat continued toward the helicopter at a

hour longer. How fast was each car going? speed of 15 miles/hr. How long did it take

How many miles was the race? the helicopter to reach the boat people?

r x t = d r x t = d .

slow car helicopter .

fast car boat .

3) The slow car leaves town 3 hours before 4) John rides his motorcycle from city "A" to city

the fast car. The slow car travels 40 "B" at 30 miles/hr. He makes the return trip

miles/hr, and the fast car 50 miles/hr. at 24 miles/hr. If he makes the entire trip in

How long before the fast car overtakes 9 hours, how far apart are the cities?

the slow car?

r x t = d r x t = d .

slow car going .

fast car returning

46

different route which was 8 miles longer, but he increased his speed by 12 miles/hr and got

home in one hour less time. Find the rate that he traveled from home to the hotel, the rate

he traveled on the way home, and the total number of miles traveled round trip.

r x t = d .

going .

returning .

Practice:

1) A freight train and a passenger train left 2) Two planes fly toward each other from cities

city "A" at 11 A.M.. The freight train 2050 miles apart. One leaves at 9 A.M. and

got to city "B" at 4 P.M. The passenger and flies at 150 miles/hr. The other leaves

train got there at 2 P.M. If the passenger at 11 A.M. and flies at 200 miles/hr. At

train went 40 miles/hr faster than the freight what time will they pass each other?

train, how fast was each train going?

r x t = d r x t = d .

freight slow plane _

passenger fast plane .

47

5) A man drove his car from his home to the hotel in 4 hours. On the return trip he followed a

3) Jack and Jill start at the same place on their 4) Joe swam the length of the pond using the

motorbikes. If Jack leaves at 12 noon and butterfly stroke at a rate of 1.5 m/sec. He

travels at 22 miles/hr and Jill leaves 2 swam back using the breast stroke at

hours later and travels at 30 miles/hr, at what 1.2 m/sec. The entire swim took 3 minutes

time will Jill be only 4 miles behind Jack? How long is the pond? (3 min. = 180 sec.)

r x t = d r x t = d .

Jack butterfly .

Jill breast .

5) A man drove from his home to the beach at 25 miles/hr and returned home at 30 miles/hr

by a different route 5 miles longer. The return trip home took 1 hour less than the trip to

the beach. How long did each trip take and how many miles were traveled in all?

r x t = d .

going .

returning .

48

TEST - Solving Word Problems with Algebra

1) The VCR costs \$90 less than twice the 2) Find three consecutive odd integers such

cost of the CD player. The TV costs that the sum of the second and the third is 9

\$10 more than the VCR and the CD less than the first.

player put together. All three together

cost \$550. How much does each cost?

3) Today Billy is 3 years older than Bobby. 4) Find the angle whose supplement is 30o

Four years ago Billy's age was 11 years more than twice its complement.

less than twice the age that Bobby was then.

How old is each now? (Make your own chart.)

49

5) A passenger train left Northvalley at 11 A.M. heading towards Southberg at 90 miles per

hour. Two hours later a freight train left Southberg heading towards Northvalley at 80

miles per hour. If the two cities are 1030 miles apart, at what time do the two trains pass

by each other? At that time, how many miles had the freight train traveled? (Make your

own chart.)

6) A jar has 47 coins: pennies, nickels, dimes, and quarters. The total value of the coins is

\$3.48. There are 2 more dimes than quarters, and there are 4 more nickels than dimes.

The rest of the coins are pennies. How many are there of each kind of coin? (Make your

own chart.)

50

Workbook

Part 3 - Coordinate Geometry

and

Systems of Equations

Graphing Linear Equations on a Coordinate Plane

Vocabulary: coordinate geometry, coordinate plane, Rene Descartes, Cartesian

coordinates, x-axis, y-axis, origin, ordered pair, coordinates, x-coordinate,

absissa, y-coordinate, ordinate, plotting points

y

Plot these points:

A (3, 5)

B (5, 3)

C (-6, -4)

D (-4, 7)

E (4, -7) x

F (0, 5)

G (6, 0)

H (0, 0)

Practice: y

Plot these points:

A (2, 7)

B (7, 2)

C (-5, -3)

D (-3, 4)

x

E (3, -4)

F (0, 4)

G (5, 0)

H (0, 0)

51

Vocabulary: table of values, graph, linear equations, dependent variable, independent

variable, constant

1) Graph the equation: y = 2x - 4 y

x 2x - 4 y

x

2) Graph the equation: y = -1/2x + 3 y

x 1/2 x + 3 y

x

52

x y

x

4) Graph the equation: -2x + 3y = 3 y

x y

x

53

3) Graph the equation: 4x - y = 3 y

Practice:

1) Graph the equation: y = 3x - 5 y

x 3x - 5 y

x

2) Graph the equation: y = -1/3x + 1 y

x - 1/3 x + 1 y

x

54

3) Graph the equation: 3x - y = 4 y

y x

x

4) Graph the equation: -3x + 2y = 6 y

x y

x

55

Using the Slope-Intercept Form: y = mx + b

Vocabulary: slope, y-intercept, y = mx + b, positive slope, negative slope

1) Graph the equation: y = 2x - 4 2) Graph the equation: y = -1/2 x + 3

y y

x x

3) Graph the equation: 4x - y = 3 4) Graph the equation: -2x + 3y = 3

y y

x x

56

Practice:

1) Graph the equation: y = 3x - 5 2) Graph the equation: y = -1/3x + 1

y y

x x

3) Graph the equation: 3x - y = 4 4) Graph the equation: -3x + 2y = 6

y y

x x

57

More on Slopes

Rule: The slope of a line through two given y2 - y1

points is equal to the difference in "y" divided m =

by the difference in "x": x2 - x1

Find the slopes of the lines that go through

the given pairs of points:

(0, 0) (2, 6) (3, 2) (6, 4) (2, 3) (6, 1) (-5, 4) (4, -5)

Practice:

(2, 1) (4, 5) (-6, -1) (2, 3) (-6, 0) (3, -3) (-1, 2) (4, -3)

Note: It is possible to find the equation of a line once you know the slope of the

line and the co-ordinates of just one point on the line. Since we have already

figured out the slopes of the lines that go through the pairs of points given above,

now let's find the equations of those lines by substituting the co-ordinates of one

of the given points, and also the slope, into the equation: y = mx + b

y = mx + b y = mx + b

y y

x x

58

y = mx + b y = mx + b

y y

x x

Practice:

y = mx + b y = mx + b

y y

x x

59

y = mx + b y = mx + b

y y

x x

The slope of a line parallel to A line parallel to the "y axis" is said to

the "x-axis" is zero. have no slope. This is because for

( Example: y = 2 ) any difference in y, the difference in x

is zero, and it is "impossible" to divide

by zero. Example: ( x = 2 )

y y

x x

60

If two different lines have the same If two lines are perpendicular to each

slope, then those lines are parallel. other, the slope of one is the "negative

Example: y = 2x + 2 and y = 2x - 3 reciprocal" of the other. (The product

of the two slopes is -1) Example:

y = 2x + 2 and y = -1/2 x + 1

y y

x x

Show that these two lines are perpendicular: y

2x + 3y = 3 and -3x + 2y = -8

x

61

Practice: Show that these two lines are perpendicular: y

2x + y = 2 and -x + 2y = -2

x

Solving a System of Linear Equations Graphically: y

2x + y = 6 and y - x = 3

x

Practice: Solve the system of equations graphically: y

3x - 4y = 4 and y = 2x + 4

x

62

Now let's try these: y

x + y = 3 and y = 5 - x

x

2x + y = 1 and 3y = 3 - 6x y

x

Note: Two equations are said to be "consistent" if their slopes are unequal

and thus their graphs intersect in a common point.

Two equations are said to be "inconsistent" if they have the same

slope but different y-intercepts; their graphs will be parallel and will not intersect

in a common point.

Two equations are said to be "dependent" if they are actually

different forms of the same equation and represent the same line.

63

Tell whether the following pairs of equations are consistent (intersecting lines),

inconsistent (parallel lines), or dependent (the same line)

x + y = 5 and x + y = 10 x + y = 12 and x - y = 2

2x = 5y + 5 and 4x - 10y = 10 y = x and 3y - 3x = 4

Practice:

2x + 3y = 6 and 3x + 2y = 10 x - 4y = 7 and x - 9 = 4y

x + 2y = 6 and x = 2 - 2y 3x - 2y = 6 and 4y = 6x - 12

64

TEST - Coordinate Geometry

Complete the table of values and graph the equation:

3x + 2y = 8 y

x y

x

Convert to y = mx + b form and graph the equations:

-2x + y = - 4 x - 2y = 4

y y

x x

65

Find the slope of the line going through the Show that the graphs of these two equations

given pair of points. Then using the slope are perpendicular by showing that the product

and one of the given points, substitute into of the slopes is -1. Then graph the lines:

y = mx + b to find the y-intercept. Finally,

write the equation of the line and do the graph. x + 3y = 6 3x - y = 1

(-2, 5) (3, -5) y = mx + b

y y

x x

Solve this system of linear equations graphically. In the spaces provided, write the ordered

pair of the point of intersection of the graphs, and the solutions. Also do the checks by

substituting the solutions into the original equations: y

( , )

x + y = 3 and x - 2y = 0 x =

y =

x

66

Tell whether the following pairs of equations are consistent (intersecting lines), inconsistent

(parallel lines), or dependent (the same line).

2x - y = 5 and 2x - y = 10 2x + y = 8 and x - y = 7

3x = 2y + 7 and 6x - 4y = 14 x = y and 2y - 2x = 6

67

Solving Systems of Equations in Two Variables

Solving by Substitution:

x + y = 12 4x + 3y = 12

3x - y = 4 x - 4y = 3

5x - 3y = -1 3x - 4y = 5

x + y = 3 x + 7y = 10

Practice:

x - y = 1 2x - y = 2

x + y = -5 3x - 2y = 3

68

3x - 5y = 8 4x - 3y = 15

x + 2y = -1 x - 2y = 0

x + y = 7 x - y = 4

x - y = 3 4x - y = -2

3y - 5x = -19 (watch the order) 3x - 5y = -36

2x + 3y = -5 3x + 5y = 12

69

x - y = -9 3x - y = 10

x + y = 17 2x - y = 7

2y = 21 - 5x (watch the order) -8 = 2y - 3x (watch the order)

7x = 2y + 39 16 = 6y - 3x

x + 2y = 1 3x + 5y = 9

3x + y = 8 9x + 2y = -12

70

Practice:

2x + 3y = 0 -5x + 3y = 25

5x - 2y = -19 4x + 2 y = 2

Practice:

x + 11y = -6 5 x + 12 y = 24

2x + y = 9 9x + 4y = 8

5x + 6y = 16 4 x + 20 y = 8

6x - 5y = 7 5x + 8y = 27

71

Problems Using Two Variables - Digit Problems

Note: If u represents the units digit, and t represents the tens digit,

then the value of a two-digit number can be expressed as: 10t + u

For example: if u = 7 and t = 6, then 10t + u = 10(6) + 7 = 67

The value of the number with the digits reversed can be expressed as: 10u + t

In the case of this example: 10u + t = 10(7) + 6 = 76

1) The sum of the digits of a two-digit 2) The sum of the digits of a two-digit

number is 11. The value of the number number is 9. If the order of the digits

is 13 times the units digit. Find the is reversed, the result is a number

number. exceeding the original number by 9.

Find the original number.

3) The units digit of a two-digit number 4) --Extra for Experts-- Find a three-

is 3 times the tens digit. The sum digit number whose units digit is 3 times

of the digits is 12. Find the number. its hundreds digit and 2 times its tens

digit, and the sum of whose digits is 11.

72

1) The sum of the digits of a two-digit 2) The sum of the digits of a two-digit

number is 9. The number with the number is 14. If 18 is subtracted from

digits reversed is 9 times the original the number, the result is the number

tens digit. Find the original number. with the digits reversed. Find the

original number.

3) The units digit of a two-digit number 4) --Extra for Experts-- A three-digit

exceeds 2 times the tens digit by 2. The number is 198 more than itself with the

sum of the digits is 11. Find the number. digits reversed. The sum of the digits is

19. The hundreds digit is 3 times the

tens digit. Find the original number.

73

Practice:

TEST - Solving Systems of Equations

1) Solve by Substitution: 2) Solve by addition: (Change order as necessary.)

3x + 4y = 6 2x - 3y = 9

x - 5y = -17 3y = 9 - 4x

3) Solve by multiplication and then addition: 4) When the digits of a two-digit number

(One of the answers is a fraction) are reversed, the new number is 10

more than twice the original number.

8x - 2y = 10 The units digit of the original number

20x - 3y = 21 is 3 times its tens digit. Find the

original number.

74

Workbook

Part 4 - Polynomials

Introduction to Polynomials

Expressions with only one term are called "monomials": -2, 3x, 2x2, (2x2)3, -[4x3y2]2

Expressions with more than one term are called "polynomials": 5x3 - 2x2 + 3x - 4

If a polynomial has two terms it is called a "binomial": 5x + 2, 3x + 2y, x2 + 3y2

If a polynomial has three terms it is called a "trinomial": 3x2 - 2x + 4, x2 + 2xy + y2

(5x + 3) + (2x - 4) = 5x + 3 (3x2 - 4x + 1) + (2x2 - 5) = 3x2 - 4x + 1

2x - 4 2x2 - 5

+ +

Practice:

(2x - 3) + (-5x + 5) = 2x - 3 (5x2 - 2x + 5) + (3x2 + 5x) = 5x2 - 2x + 5

-5x + 5 3x2 + 5x

+ +

Subtracting Polynomials

(5x - 2) - (-3x + 4) = 5x - 2 (4x2 - 3x + 1) - (2x2 - 3x + 5) = 4x2 - 3x + 1

-3x + 4 2x2 - 3x + 5

- -

Practice:

(-3x - 4) - (7x - 3) = -3x - 4 (3x2 + 2x - 5) - (3x2 + 5x - 7) = 3x2 + 2x - 5

7x - 3 3x2 + 5x - 7

- -

75

Multiplying and Powers: The Laws of Exponents for Multiplication

The Multiplication Law:

22 . 23 = Note: 25 = 2 . 2 . 2 . 2 . 2 = 32

24 = 2 . 2 . 2 . 2 = 16

23 = 2 . 2 . 2 = 8

x2 . x3 = (x2)(x) = x2 . x2 . x . x4 = 22 = 2 . 2 = 4

21 = 2

xa . xb = xa+b Rule: The Multiplication Law - When multiplying like

quantities raised to powers, add the exponents.

(2x2)(3x) = (3x)(x4) =

(x3)(4x) = -2(-x2)(4x3) = -(3x3)(-x2)= (-2x2)(x4)(-x) =

Practice:

x4 . x = (x2)(x5 ) = x . x2 . x3 . x4 = -x2 . x4 =

(x2)(2x) = (-3x3)(x2) = -(-2x2)(x3) = (-5x4)(-4x5) =

The-Power-of-a-Power Law:

(22)3 = (x2)3 =

(xa)b = xab Rule: The Power-of-a-Power Law - When raising a power of a

quantity to a power, multiply the exponents

(x3)2 = (x3)3 = (x3)4 = (x4)3 =

Practice:

(x5)6 = (x4)4 = (x2)5 = (x5)2 =

The Power-of-a-Product Law:

(2 . 3)2 = (xy)2 = (3x)2 =

(xy)a = xaya Rule: The Power of a Product Law: - When the product of two

quantities is raised to a power, the result is equal to the

product of each of those quantities raised to that power.

76

(4x)2 = (-2x)3 = (5xy)3 =

(2x2)2 = (-2x2)3 = - (-3x2y)2=

Practice:

(5x)2 = (-3x)3= (4xy)4 =

(3x3)2 = (-3x3)3 = - (2x3y2)2 =

Remember: The Laws of Exponents for Multiplication

xa . xb = xa+b The Multiplication Law

(xa)b = xab The Power of a Power Law

(xy)a = xaya The Power of a Product Law

QUIZ: (Multiplying and Powers)

x(x3) = (x2)(x6) = (x3)4 =

(xy)2 = (xy2)2 = (4x)2 =

(4x4)2 = (-3x2)3 = - (4x3)3=

(5x3y)3 = (-3x5)4 =

More Complicated Expressions Involving Multiplying and Powers

Simplify:

5x3 - x(-3x2) = (3x2y)(2x2y3) + (-xy)4 =

3(xy)2 + 2xy(xy) - 4x(2xy2) = (6x2yz3)(-3xy2z) + (-2z2)(-9x3y2)(yz2) =

77

(-3x2y)3 + (2x2y)3 = (2xy2)4 (-3x2y4)2 - (-4x4y8)2 =

Practice:

(x2)(2x) + (x)(x2) = (2x3)(2x)4 - (4x)2(2x5) =

3xy(xy) + 8x(xy2) - 2(xy)2 = (7y2)(2x2) - (-4y)(-3y)(-5x2) + (5y2)(-2x)(3x) =

x(-2x3) + 3x2(x2) - (2x2)2 = (-2x3y3)2 - (3x2y2)3 =

(2xy)(-3x2y3)3 + (xy6)(-2x3y2)2 =

Multiplying a Monomial by a Polynomial

Rule: To multiply a monomial by a polynomial, use the distributive law and

multiply the monomial by each term of the polynomial.

4(x - 3) = 3x(2x + 5) =

4x2(x - 2) = -2x3(3x2 - 2x + 5) =

Practice:

-3(2x + 5) = -5x(3x - 2) =

-3x2(4x - 8) = 3x3(x2 - 2x + 1) =

78

Multiplying a Polynomial by a Polynomial

Rule: To multiply two polynomials, multiply each term of one by each term of the

other.

(x + 2)(x - 3) = x + 2 (5x + 2)(3x - 1) = 5x + 2

x - 3 3x - 1

(2x - 3y)(-5x - 2y) = 2x - 3y (x - 3)(x2 + 3x + 9) = x2 + 3x + 9

-5x -2y x - 3

(x + 3)(x2 - 4) = x2 - 4 Note: A space is left for the

x + 3 missing term, 0x

.

Practice:

(x + 1)(x - 4) = x + 1 (3x - 2)(2x - 3) = 3x - 2

x - 4 2x - 3

(5x - 4y)(2x + 3y) = 5x - 4y (x - y)(x2 + xy + y2) = x2 + xy + y2

2x + 3y x - y

79

(x - 5)(2x2 + 3) = 2x2 + 3

x - 5

Are You Ready to Skip a Step? The "FOIL" Method

Note: You can multiply two binomials by "inspection", or by doing it "mentally" or

(2x + 1) (3x + 2) =

First Outside-Inside Last

(x + 3)(x + 2) = (x + 3)(x - 2) = (x - 3)(x - 2) =

(2x + 3)(3x - 4) = (4x + y)(x - 2y) =

Practice:

(x + 1)(x + 4) = (x + 1)(x - 4) = (x - 1)(x - 4) =

(3x - 2)(5x + 3) = (7x - 3y)(2x - 4y) =

80

The Laws of Exponents for Division

The Division Law:

25

= (2)(2)(2)(2)(2) = Note: 23 = (2)(2)(2) = 8

23

(2)(2)(2) 22 = (2)(2) = 4

21 = 2

23

= (2)(2)(2) = 20 = 1

25 (2)(2)(2)(2)(2) 2-1

= 1 = 1

21 2

x4 = x2 = x2 = 2-2

= 1 = 1

x2 x2 x4 22 4

xa

= xa - b Rule: The Division Law - When dividing like quantities raised to

xb powers, subtract the exponent of the denominator from the

exponent of the numerator.

Simplify:

x3

= x6

= 4x4

= 8x2

= 6x2

=

x x9 2x3 2x2 8x2

Practice:

x5

= x7 = 9x3

= 10x2

= 6x2

=

x2 x11 3x 5x2 10x2

Simplify:

-12x2y = 18x6y4

= 10x3y4

=

4x2 -3x3y 5x4y3

-6x2y3 = - 42x3y3 = 14x6y10

=

-12xy4 6x4y3 28x7y11

Practice:

15x5y3

= -8xy2

= 8x2y =

5x2y 2x -24x4

-21x5y2 = - 27x7y4 = -12x4y =

-7x4y4 54x6y 48xy

81

Simplify:

(-4xy2)3 = (-3xy2)2(-2x2y)3 =

(2xy2)4 (9x2y)(-4x2y)2

Practice:

(4xy2)2 = (4x2y3)2(-xy2)3 =

-8x2y4 - (2x2y2)2(x3y)2

The Power-of-a-Quotient Law:

.2. 2

= (2)2

=

.x.a

= (x)a Rule: The Power-of-a-Quotient Law - The power of

.3ÿ (3)2 .yÿ (y)a a quotient (a division expressed as a fraction)

is equal to the power of the numerator over

the power of the denominator.

Simplify:

.x.3

= .2x.2

= .-3x.3

= .2x

2y. 3

=

.yÿ . y2ÿ . x3 ÿ . 3y

3 ÿ

Practice:

.x.2

= . x .3

= .-2x.3

= .3x

3y

2. 2

=

.yÿ .3yÿ . x2 ÿ . -2y

3 ÿ

Negative Exponents:

2

-2 = 1 = 1 x

-a = 1 Rule: Negative exponents - The negative

22 4 xa power of any number is equal to the

reciprocal of its positive power.

Note: By changing the sign of its exponent, any factor of the numerator may be

made a factor of the denominator, or any factor of the denominator may be

made a factor of the numerator.

82

2

-3 = x

-2 = 1 = 1 =

2x

-2 (2x)

-3

Practice:

3-2

= x-3

= 1 = 1 =

2x

-3 (3x)-2

Simplify: (Express the result using only positive exponents.)

x

-2 = x2y

-3 = 5x

-1y 2 = -2x

-2y =

y

-3

5 = 2 = 2x

-2 = (2x)

-2 =

x

-2 3x

-3

(x2y)

-3 = (xy

-2)3 = .1 . -1

= .x.-2 =

.x ÿ .yÿ

(3x

-2)4 = x3 = 2x

-2 =

x

-4 x

-4

Practice:

x

-3 = x3y

-4 = 4x

-2y

3 = -4x

-3y

2 =

y

-2

3 = 3 = 3x

-3 = (3x)

-3 =

y

-4 4x

-2

(x3y)

-2 = (xy

-3)2 = .1.-2

= .x.-1 =

.xÿ .yÿ

83

(2x

-3)3 = x2 = 3x

-3 =

x

-3 x

-5

Simplify: (Express the results without any denominators.)

x

2 = x2 = 5xy

3 = 8x

-3y

3 =

y

3 y

-1 z4 (2x

-4)2

Practice:

x

3 = x3 = 3x2y = 12x

2y

-

3 =

y

2 y

-2 z3 (2y

-3)2

QUIZ: (Laws of Exponents for Division)

Simplify:

x4 = 2x3 = 14x4y2 = -15x5y2 =

x3 x4 -7x2y3 -3x6y2

5x4y3 = (3x2y)3 =

15x5y5 (-2xy3)3

(-xy3)3(2x2y2)2 =

(3x2y)2(-2x3y2)2

.2x2. 3 = . xy . 3 =

. -x ÿ .-2x2yÿ

Simplify: (Express results using only positive exponents.)

2 = 3 = (x-2y2)3 =

4x

-3 (2x)-3

3x = 8x2y

-

4 =

x

-2 (2y

-2)3

84

Division of a Polynomial by a Monomial

25x - 15 = 12x2 + 4x = 16x3 + 12x2 - 40x =

5 2x 4x

9x2y3 + 27xy2 - 72xy = 18x2y + 30xy2 - 12x2y2 =

9x2y2 -6x2y2

Practice:

24x + 16 = -48x2 + 36x = 24x4 -16x3 + 32x2 =

8 6x -8x2

21x2y + 63xy - 7xy2 = 40xy2z2 - 24x2yz2 + 36x2y2z =

7xy2 8x2y2z2

Division of a Polynomial by a Binomial

x2 - 3x -10 = 3x2 + 12 + 13x = 2x2 + 11x - 18 =

x + 2 3x + 4 2x - 3

85

3x3 - 4x - 1 = 4x3 - 2x2 + 18 =

x +1 2x + 3

Practice:

x2 - 5x + 6 = 2x2 + 4 - 6x = 12x2 + 4x - 18 =

x - 2 2x - 2 2x + 3

5x3 + 2x2 - 48 = 6x3 + 4x - 56 =

x - 2 2x - 4

86

TEST - Basic Operations with Polynomials

Perform the indicated operations:

(3x2 - 2x + 5) + (-7x2 + 4x - 5) = (7x2 - 2x - 3) - (-3x2 + 2x - 5) =

2x2(x4) = (-3xy2)2 = (-2x2)3 = (4x)2(-x2)3 =

2x2(-x)2 - (3x2)2 = -4x2(-2x2 + 3x - 2) = (3x + 2)(-2x + 1) =

(3x - 2)(2x2 + 3x + 2) = (x - 4)(x2 + 3) =

3x3 = -8x5y = (2xy4)2 = . x2y .3 =

x5 2x4y3 (-4xy3)3 .-2xy2 ÿ

Express results using only positive exponents:

3x2 = (4x3y

-2)2 =

6x

-3 (2y

-3)2

Divide and check:

4x2y - 8xy2 = 6x2 - 6 - 5x = 6x3 - x2 - 18 =

-2xy 2x - 3 2x - 3

87

Special Products

Squaring a Binomial:

(x + 1)2 = (x - 1)2 = (2x + 3)2 = (2x - 3)2 =

(x + y)2 = (x - y)2 = (2x + 3y)2 = (2x - 3y)2 =

(a + b)2 = Rule: To square a binomial:

(a - b)2 = Square the first term of the binomial

Square the second term of the binomial

Double the product of the two terms

The result is called a "trinomial square",

or a "perfect-square trinomial".

Practice:

(Use the "FOIL method".)

(x + 2)2 = (x - 2)2 = (3x + 2)2 = (3x - 2)2 =

(Use the "shortcut".)

(y + x)2 = (y - x)2 = (3x + 2y)2 = (3x - 2y)2 =

The Difference of Two Squares:

(x + 1)(x - 1) = (2x + 3)(2x - 3) = (x + y)(x - y) = (2x + 3y)(2x - 3y) =

(a + b)(a - b) = Rule: When the first terms of the binomials are the

same, but the second terms are the opposites

of each other, the "middle term" drops out, and

the result is called "The Difference of Two Squares".

Practice:

(x + 2)(x - 2) = (3x + 2)(3x - 2) = (y + x)(y - x) = (3x + 2y)(3x - 2y) =

88

Factoring Out a Common Monomial Factor

4x - 8 = 21x2 + 14x = 15x2y - 10xy + 20xy2 =

4x3 - 8x2 + 12x = 9x3y2 + 12x2y +12xy2 + 9x2y3 =

Practice:

9x - 12 = 20x2 + 15x = 16x2y - 8xy + 12xy2 =

5x3 - 10x2 + 15x = 14x3y2 + 21x2y + 21xy2 + 14x2y3 =

Factoring Special-Product Polynomials

Factoring Trinomial Squares:

x2 + 2x + 1 = x2 - 6x + 9 =

9x2 - 24x + 16 = 4x2 + 4xy + y2 =

Practice

x2 + 4x + 4 = x2 - 8x + 16 =

25x2 - 60 x + 36 = 9x2 + 6xy + y2 =

Factoring the Difference of Two Squares:

x2 - 4 = 4x2 - 9 =

x2 - y2 = 9x2 - 16y2 =

89

x2 - 9 = 9x2 - 16 =

y2 - x2 = 25x2 - 36y2 =

x2 + 5x + 6 = x2 - 9x + 8 = x2 - 2x - 8 =

x2 + 2x - 35 = -x2 + 3x + 10 = -x2 - x + 6 =

Practice:

x2 + 6x + 8 = x2 - 10x + 16 = x2 - 4x - 21 =

x2 + 3x - 4 = -x2 + 10x - 24 = -x2 + 5x + 24 =

When the coefficient of the quadratic term is other than 1 or -1:

3x2 + 4x + 1 = 2x2 - 3x + 1 = 4x2 + 3x - 1 =

6x2 + 5x + 1 = 3x2 - 2x - 5 = 4x2 + 25x - 21 =

16x2 - 50 x + 25 = 12x2 - 7x - 12 = 6x2 - 25x - 9 =

Practice:

2x2 + 3x + 1 = 3x2 - 4x + 1 = 3x2 - 2x - 1 =

8x2 + 6x + 1 = 3x2 + 4x - 7 = 4x2 + 23x - 6 =

8x2 + 3x - 5 = 18x2 - 9x - 14 = 6x2 - 13x + 6 =

90

Practice:

Note: To factor a polynomial completely means to keep factoring as long

as is possible. Sometimes you can factor out a common monomial factor

and then continue to factor what remains.

5x2 - 45 = x3 - 25x = 2x2 - 8x + 8 =

3x2 - 6x - 24 = 3x3 - 12x2 + 9x = 2x3 + 16x2 - 40x =

-3x2 - 15x + 18 = 36x3 - 21x2 - 30x =

Practice:

5x2 - 5 = 9x3 - x = 12x2 - 36x + 27 =

6x2 - 24x -72 = x3 + 7x2 + 12x = 3x2 - 21x - 24 =

-18x2 + 9x + 9 = 24x3 + 22x2 - 10x =

91

Factoring Completely

TEST - Factoring

Factor as completely as possible:

4x2 + 12x + 9 = 4x2 - 9 =

x2 - 2x - 3 = 6x2 - 5x - 4 =

-6x2 + 13x + 5 = 16x3 - x =

6x2 + 9x - 6 = -8x3 + 46x2 + 12x =

92

Workbook

Part 5 - Fractions

Equivalent Fractions

Note: Fractions which have the same value but different forms are called

"equivalent fractions".

Rule: The value of a fraction does not change when both the numerator and

denominator are multiplied by the same quantity. The result is equivalent

to the original, and it is said that the fraction has been raised to "higher terms".

(2) (x – 2)

4 = 4 =

5 5

(x + 2)

(x + 2) =

(x - 2)

Practice: Raise each fraction to higher terms by multiplying both numerator

and denominator by the given factor:

(x + 2) (x - 2) (2x - 3)

3 = x - 2 = 2x - 3 =

5 x + 2 2x + 3

Rule: The value of a fraction does not change when both the numerator and

denominator are divided by the same quantity. The result is equivalent to the

original, and it is said that the fraction has been "reduced to lower terms".

8 =

10

Note: To reduce a fraction to lowest terms:

8 = Factor completely both numerator and denominator.

10 Cancel out the common factors.

4x - 8 = x2 + 4x + 4 =

5x - 10 x2 - 4

Simplify: Reduce to lowest terms:

4 - 8x = x2 - 1 =

1 - 4x2 2x + 2

Practice:

2x - 2y = x2 - 3x =

4x2 - 4xy x2 - 4x + 3

93

1) The sign before the fraction

2) The sign of the numerator

3) The sign of the denominator

Rule: The value of a fraction is not changed when any two of the three signs

associated with it are changed.

Note: Often it is useful to change the sign of either the numerator or the

denominator of a fraction, which you may do if you change the sign in front of

the fraction as well.

4 = - -4 Changing the sign before the fraction and the sign of the numerator.

2 2

4 = - 4 Changing the sign before the fraction and the sign of the denominator.

2 -2

Remember: The most useful application of the above -2 = 2 = - 2

rule is that if only one of the three signs is negative, it 3 -3 3

may be placed in any one of the three positions.

Rule: If the numerator or the denominator of a fraction is a polynomial, when you

change the sign of that polynomial, you must change the sign of each of its terms.

Change the sign of each expression from positive to negative:

(x - y) 1 - x

2

Simplify: Reduce to lowest terms.

2x - 3 = - y - x =

9 - 4x2 x y

(x - 5)2 = 4(x - y)2 =

5 - x 8(y - x)

94

Note: There are three signs which are associated with a fraction:

x2 - 2x + 1 x2 - 9

3x2 - 10x + 3 =

12 - x - x2

Practice:

2 - 5x = 9x2 - 1 =

4 - 25x2 3x - 1

(x - 6)2 = 8x3 - 2x =

6 - x 2x2 - 5x + 2

x2 - y2 = 4 - x =

x2 - 3xy + 2y2 2x2 - 9x + 4

3 - 5x - 2x2 = 4y - 8x =

4x2 - 1 2x2 + 3xy - 2y2

Rule: When the numerator or the denominator is expressed in factored form,

the value of the fraction is not changed when the signs of an even number of

factors in the numerator or denominator are changed.

(2 - x)(5 - x) =

(x - 2)

Practice:

(x - 1)(x - 2) = 5(y - x)2

=

(3 - x)(2 - x)(1 - x) 10(x - y)

95

3 - 3x = 6 - x - x2 =

Multiplying Fractions

Rule: To multiply fractions:

Factor completely numerator and denominator.

Cancel out any factors common to any numerator and any denominator.

Multiply across the remaining factors.

9x2 - 25 . 3x - 3 = x2 - 5x + 6 . x + 2 =

x2 - 1 9x - 15 2x + 4 2x - 6

2x + 1 . x2 - 4x = 3x - 6 . 3x2 - 12 =

x2 - 16 4x2 - 1 6x + 12 x2 - 5x + 6

Practice:

3x - 6 . x2 - x - 6 = 4x - 4y . xy =

5x x2 - 4 xy 2x - 2y

5x . 6x + 18 = 4x + 8 . 6x + 15 =

x2 - 9 15x3 4x2 - 25 2x2 + 4x

3x2 + 2x - 1 . 10x2 - 13x - 3 = x2 + 9x + 14 . x2 + 2x - 35 =

5x2 - 9x - 2 2x2 - x - 3 x2 - 3x - 10 x2 + 4x - 21

96

Dividing Fractions

Rule: To divide by a fraction, multiply by the reciprocal of the fraction.

5x + 15 ÷ 10x2 + 10x = x2 - 9y2 ÷ 3x - 9y =

x2 - 9 4x - 12 8x + 4y 12x + 6y

x2 - x - 20 ÷ x2 - 7x + 10 = 2x - 8 ÷ 4 - x =

x2 + 7x + 12 x2 + 9x + 18 x2 - 4 x2 + 3x + 2

Practice:

x - 5 ÷ x2 - 25 = 12 + 6x ÷ 4 - x2 =

x x2 15 - 3x 25 - x2

x - 7 ÷ 3x - 21 = x2 - 3x ÷ x2 - x - 6 =

3x2 - 8x + 4 6x2 - 24 x2 + 3x - 10 x2 - 4

x2 - 5x + 6 ÷ x2 - 2x - 3 = 4x2 - 1 ÷ 2x2 + x - 1 =

x2 - 4 x2 + 3x + 2 x2 + 2x - 3 1 - x2

97

Rule: To Add or Subtract Like Fractions:

Add or subtract the numerators as indicated by their signs.

Write the result over their common denominator.

Reduce to lowest terms.

2 + 3 = 2 + 3 = 5 7 - 2 = 7 - 2 = 5 7 - 3 + 4 = 7 - 3 + 4 = 8 = 4

8 8 8 8 9 9 9 9 10 10 10 10 10 5

3 + 2 - 4 = 5x - 2x + 3x = 2x - x - 3y =

5x 5x 5x 8 8 8 x - y x - y

2x + 2y = 3x + y + x - 4y + 2x - 3y = x - y =

x + y x + y 4(x - y) 4(x - y) 4(x - y) x2 - y2 x2 - y2

x2 + 3x - 2x + 12 = 3 + 1 = Note: Sometimes you can make

x2 + 2x - 15 x2 + 2x - 15 x - 4 4 - x denominators alike by changing

the sign of one of them.

Practice:

5 - 2 + 4 = 3x - 7x + 9x = 2y - 3x - 5y =

9x 9x 9x 10 10 10 x + y x + y

98

3x - 3y = 6x + 3x - 10y - x - 6y = 2x - 3y =

x - y x - y 2x - y 2x - y 2x - y 4x2 - 9y2 4x2 - 9y2

-x + 6 - x = 7 - x + 2x - 3 = 5 + 1 =

2(3 - x) 2(3 - x) x2 + 3x - 4 x2 + 3x - 4 2x - 2 2 - 2x

x + 1 = x + 3 = 3x - y - x + y =

x + 1 x + 1 x - 3 3 - x 2x2 - 3xy + y2 2x2 - 3xy + y2

Note: The LCD, Least Common Denominator, of two or more given fractions,

is the product of all the factors in the denominators, each being taken the

greatest number of times it occurs in any one of the given denominators.

LCD (20, 50) = LCD (x2 + 2x + 1, x2 - 1) =

Practice:

LCD (18, 24) = LCD (x2 - 2x + 1, x2 + x - 2) =

99

Rule: To add and subtract unlike fractions,

Factor each denominator and find the LCD.

Replace each fraction with an equivalent one having the LCD as denominator

Find the sum or difference and simplify.

Simplify:

5x + 2x = x + 3 + x + 4 =

6 3 x 3x

1 - x = 3 - 2x - 4 - 3x =

x - y 3x - 3y 6 8

x + 3 + x + 1 = 3 + 4 =

4x 3x x - 4 x + 4

6 - 3 = 2 + 3 =

x - 2 x + 2 x2 - 4 x2 + 5x + 6

100

3x - 3 - 3 = 2 + 3 =

5x - 15 2x - 6 x2 - 2x + 1 x2 + x - 2

3 + 5 - 6 = 9 + -7 - 3x - 1 =

x - 2 x2 - 4 x + 2 2 + x x - 2 4 - x2

Practice:

3x - 5x = x + 1 + x + 3 =

4 2 3x x

3 + x - 2x = 9 - 4x - 3 - 5x =

x - y 2x - 2y 12 9

101

x + 1 + x - 3 = 4 - -3 =

2x 3x x + 5 x - 5

4 - 6 = -7 + 8 =

x - 1 x + 1 x2 - 9 x2 + x - 12

3 - 4 = 3 - 2 =

x2 - 4 2 - x x2 - x x2 + x - 2

5 - 6 - 1 = 3 - 2 =

x - 3 x2 - 3x x x2 - 3x + 2 x2 - 1

102

Simplifying Complex Fractions

Note: A complex fraction contains a fraction within the numerator or

denominator or both.

Rule: To simplify a complex fraction:

Combine the terms in the numerator and/or denominator into single

fractions.

Set the fraction up as a division problem dividing the numerator by the

denominator.

Invert the divisor and multiply.

Simplify and reduce the result.

x - 2 x - 4

2

=

x

=

x - 2 1 + 2

x

x

Note: If the denominators of both the "top" and "bottom" of the main fraction

are the same, they can be eliminated by multiplying both the "top" and "bottom"

of the main fraction by that factor.

x2 - y2 1 - 1

x + 2y

=

x y

=

x - y 1 + 1

x + 2y x y

Simplify:

x + 1 x .

.

y

=

x + y

=

x - 1

y

.

y

x + y

103

x2 - 4y2 3 - 1

x2 =

x

=

x + 2y 3 + 1

x x

Practice:

x - 4 x - 9

4

=

x

=

x - 4 1 + 3

x

x

x2 - y2 2 + 1

x - 2y

=

x y

=

x + y

2 - 1

x - 2y x y

1 - 3 x + y

x

=

y x

=

1 - 9 1 .

x2

xy

104

1 + 1 x .

x2 y2 =

x + 3

=

2 1 - x .

xy x + 3

x - y

y x

=

y

- 1

x

2 - 2 .

x + y x - y

=

4 .

x2 - y2

105

Solving Equations Containing Fractions

Rule: To solve an equation containing fractions:

Clear the equation of fractions by multiplying both members of the

equation by the LCD of the denominators.

Solve the resulting equation.

Check to see that the "root", or answer, "satisifies" the given equation

Solve and Check:

x + 2 = x x + x = 3

3 2 2 4

3x + 4x + 1 = 4 - 2x 3x + 5 + 3x - 4 = 3x + 2

4 6 -3 3 6

106

x - 4 + x + 3 = 5 2 (4x - 1) - 3 (x + 1) = 7

3 2 6 3 5

Practice:

5x + 2 = 7 5 - 31x = 2 - 7x 5x - 29 = 5 - 3x - 4x

6 6 9 3 2 5 2 15

x + 5 - x - 3 = 7 4x + 1 - 2x - 1 = 3 4 + x - 3 = 3x - 1

2 5 10 7 6 2 3 4 6

107

x + 2 - x - 3 = 1 7x + 5 - 3x + 15 = 2

4 3 2 8 10

x - 3 + x + 2 = - 7 3 (2x + 1) - 1 (4x - 1) = - 19

2 4 16 2 3 6

108

Rule: When a variable appears in the denominator of one or more terms of an

equation, you must eliminate as a "root" or "solution" to the equation, any value for the

variable that would result in a division by zero. Any solution that causes a division by zero

in the original equation is called an "extraneous root". If the only solution to the equation

is an "extraneous root", the equation is said to have "no solution".

Solve and Check:

5 = 6 x - 1 = 2 3 - 2 = x .

x 18 x + 3 3 x + 2 3 x + 2

5 + 3 = 6 x + 3 = x + 2

x + 1 x - 1 x2 - 1 x + 1 x + 4

5 + 2 = 3 6x - x - 6 = 5x2 - 12

x - 2 2 - x 2 x + 3 x - 3 x2 - 9

109

Practice:

x = 3 1 = 6 - 5 2 = 5 .

x + 2 5 2 x 2x x x - 3

11 = 2 + 5 3 + 6 = 14 x + 3 + x - 7 = 5

.

x x 3x - 6 x - 2 3 x x 3

2 + x = 3 + 1 3 - 1 = 7 - 9

.

6x 5x 30 x 5x 5

110

4 - 6 = 24 4x + x - 3 = 5x2 - 33

.

x - 3 x + 3 x2 - 9 x + 2 x - 2 x2 - 4

Note: If the equation consists of only one fraction on each side, and is thus in

the form of a "proportion", then you can apply the rule "the product of the means

equals the product of the extremes":

If a = c , then ad = bc

b d

5 = 6 5 = 6 x - 1 = 2 x - 1 = 2

x 18 x 18 x + 3 3 x + 3 3

Practice: (Just do them as proportions)

x = 3 2 = 5 .

x + 2 5 x x - 3

111

Rule: If the product of two factors is zero, then either one or both of the factors is

zero. So; if ab = 0, then a = 0 or b = 0 or both.

Note: An equation in one unknown in which the highest power of the variable

is the second power, is called a "quadratic equation".

Rule: To solve a quadratic equation:

Write all the terms on one side, so that the other side equals zero.

Factor the non-zero side of the equation.

Set each factor equal to zero and solve.

Check each root by substituting into the original equation:

x2 - 2x - 15 = 0 x2 + 10 = 7x 2x2 + x = 15

Practice:

x2 + 6x + 8 = 0 x2 + 8x = -15 10 = -13x + 3x2

23x - 6 = -4x2 x + 4 = 3x2 2x2 - 3x = -1

112

Fractional Equations that Transform into Quadratic Equations

11x - 12 = 2x 4 - 3 = 5 .

x x 2x + 3

3 + 10 = 5 .

x2 - 1 x - 1

113

Practice:

-8x + 3 = x 4 - 7 = 2 .

3x x - 2 x - 3 15

5 + 12 = 3 .

x2 - 4 x - 2

114

TEST - Operations and Equations with Fractions

Simplify:

x2 - 4x = 3 - x = 3x - 3y =

x2 - x - 12 3x2 - 5x - 12 6y - 6x

2x + 8 . 3x2 - 12x =

2x2 - 11x + 12 2x2 + 5x - 12

x2 - 9 ÷ x2 - 3x =

3x - 6 x2 + 2x - 8

2x + 3x - 4 = 1 + x - 1 =

x - 2 2 - x x - 1 x + 1 x2 - 1

115

x + 3 2 - 2 .

2x x y

= =

4x2 - 9 x2 - y2 .

2x y

Solve and Check: (Show any values of x that cause division by zero

and declare any "extraneous roots")

x - 2 + x + 5 = 29 5 + 4 = 11 .

3 5 15 x - 2 x + 2 x2 - 4

116

10x2 = -4x + 6 2 + 3x = 18x .

x + 3 x - 3 x2 - 9

117

Solving Word Problems Involving Quadratic Terms, Factoring, Etc.

1) The side of one square is 2 cm longer than the 2) The difference of the squares of two

side of another. The area of the smaller square consecutive positive integers is 55.

is 36 cm2 less than the area of the larger square. Find both integers.

Find the length of the side of each square.

3) Find two consecutive positive integers 4) The square of a positive integer exceeds

whose product is 42. 3 times the integer by 54. Find the

integer.

5) The dimensions of a rectangular garden are 6) The sum of two numbers is 17 and their

12 m by 8 m. The garden is surrounded by product is 42. Find the numbers.

a walkway of uniform width. The combined

area of the garden and the walkway is 192 m2.

Find the width of the walkway.

118

perimeter is 60 cm and whose area is 161 cm2. is 4 cm less than twice the length of the

other leg. The area of the triangle is 24

cm2. Find the length of each leg. (Area

of right triangle is equal to one half the

product of the legs.)

Practice:

1) A rectangular picture is 3 cm longer than it 2) The difference of the squares of two

is wide. It is surrounded by a frame that is consecutive odd integers is 72. Find

2 cm wide The area of the frame and picture both integers.

together is 100 cm2 greater than that of the picture

alone. Find the dimensions of the picture.

3) Find two consecutive positive integers 4) Find two consecutive negative integers

whose product is 72. such that the sum of their squares is 61.

119

7) Find the dimensions of a rectangle whose 8) The length of one leg of a right triangle

5) The length of a rectangle is 3 cm less than 6) The sum of two integers is 4. The sum

twice its width. The area of the rectangle is of the squares of the two integers is 16

77 m2. Find the dimensions of the rectangle. greater than the product of the two

integers. Find the integers.

7) Find the dimensions of a rectangle whose 8) The length of one leg of a right triangle

perimeter 48 cm and whose area is 95 cm2. is 2 cm more than twice the length of the

other leg. The area of the triangle is 30

cm2. Find the length of each leg. (Area

of right triangle is equal to one half the

product of the legs.)

120

Solving Word Problems Involving Equations with Fractions

1) Five sixths of a number is 6 more than 2) What number added to both the numerator

half the number. What is the number? and the denominator of the fraction 4/7

results in a fraction equivalent to 4/5?

3) The sum of the reciprocals of two positive 4) Find two consecutive even integers such

consecutive integers is 11/30. Find the that 5 times the reciprocal of the lesser is

integers. equal to 6 times the reciprocal of the

greater.

5) One number is 16 more than another 6) The denominator of a fraction is 4 less

number. Four ninths of the greater number than 5 times the numerator. The fraction

is equal to four fifths of the lesser number. is equivalent to 1/4. Find the fraction.

Find the numbers.

121

7) When a number is added to the numerator 8) Find two consecutive integers such that

of the fraction 4/7, and twice the same number the reciprocal of the lesser increased by

is subtracted from the denominator, the result 3 times the reciprocal of the greater is

is equal to 7. Find the number. equal to 9 times the reciprocal of the

product of those integers.

Practice:

1) Three fourths of a number is 1 more than 2) What number added to both the numerator

two thirds of the number. What is the number? and the denominator of the fraction 5/9

results in a fraction equivalent to 3/4?

3) The sum of the reciprocals of two positive 4) Find two consecutive even integers such

consecutive integers is 9/20. Find the that 9 times the reciprocal of the lesser is

integers. equal to 10 times the reciprocal of the

greater.

122

5) One number is 4 less than another 6) The numerator of a fraction is 10 less

number. Three fifths of the greater number than 2 times the denominator. The fraction

is equal to two thirds of the lesser number. is equivalent to 3/4. Find the fraction.

Find the numbers.

7) When a number is subtracted from the 8) Find two consecutive integers such that

denominator of the fraction 5/9, and one the reciprocal of the lesser increased by

more than twice the same number is added 2 times the reciprocal of the greater is

to the numerator, the result is equal to 2. equal to 25 times the reciprocal of the

Find the number. product of those integers.

123

TEST - Problems Involving Quadratic Equations and Fractions

1) Find the dimensions of a rectangle whose perimeter is 20 cm and whose area is 21 cm2.

2) Find two consecutive odd integers such that 3/5 the reciprocal of the lesser plus 5/8 the

reciprocal of the greater is equal to 13/40.

124

Workbook

Part 6 - Radicals and Roots

Introduction to Rational and Irrational Numbers

Remember:

{1, 2, 3, 4, . . . } = the "counting numbers"

{0, 1, 2, 3, 4, . . . } = the "natural numbers"

{ . . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . } = the "integers"

Note: The counting numbers are a subset of the natural numbers.

The natural numbers are a subset of the integers.

The integers are a subset of the rational numbers.

A "rational number" is a number that can be expressed in the form of a quotient

or ratio or as one integer divided by another. All rational numbers can also

be expressed as either terminating or repeating decimals.

Examples of rational numbers:

1 3 5 1 5 - 2 - 4 6 -7

2 4 8 3 3 5 3 1

An "irrational number" is a number that cannot be expressed as a ratio or as one

integer divided by another.

___

Example: \/ 2

Neither the rational nor the irrational numbers are subsets of each other.

Together they make up the set called the "real numbers".

Square Roots

Note: Just as the inverse operation of addition is subtraction, and the inverse

operation of multiplication is division, so the inverse operation of squaring a

number is called finding a "square root".

___

Example: 22 = 4 \/ 4 = 2

The square root of a number is one of its two equal factors.

A positive real number has two square roots: a positive or "principal" square root,

and a negative square root.

____ ____

\/ 4 = 2 because (2)(2) = 4, but (-2)(-2) = 4 also, so \/ 4 = 2 or -2

__ __

The principal square root of a number is represented by: \/ x , so: \/ 4 = 2

____ ____

The negative square root of a number is represented by: - \/ x , so: - \/ 4 = -2

125

Unless otherwise stated, when we speak of the square root of a number as \/ x ,

we will be talking only about the positive or principal square root.

____

\/ -4 is not a real number. It is called an "imaginary" number. The square root of

any negative quantity is "imaginary".

__ __ __ __ __

\/ 22 = \/ 4 = 2 \/ 32 = \/ 42 = \/ x2 =

Simplify:

___ ___ __ ____

\/ 49 = - \/ 16 = \/ 82 = \/ (-3)2 =

____ ___ ___ ___

- \/ (-5)2 = \/ -36 = \/ y2 = - \/ x2 =

Practice:

___ ___ ___ ____

\/ 64 = - \/ 25 = \/ 72 = \/ (-4)2 =

____ ___ ___ ___

- \/ (-9)2 = \/ -9 = \/ x2 = - \/ y2 =

____

Note: The symbol \/ is called the "radical sign". The quantiy placed beneath

the radical sign is called the "radicand", and the whole quantity is called called

Rule: The first requirement for a radical to be in "simplified form" is that the

radicand has no factor that is a perfect square.

Note: When we simplify a radical we speak of removing a quantity that is a

perfect square and "taking it out from under the radical."

The Laws of Square Roots

___

\/ 36 = Rule: The Product Property of Square Roots -

The square root of a product is equal to the

___ __ __ the product of the square roots of its factors.

\/ ab = \/ a \/ b

126

Simplify:

__ ___

\/ 8 = \/ 48 =

___ ___

3 \/ 75 = \/ 40 =

2

Practice:

___ ___

\/ 12 = \/ 45 =

___ ____

\/ 72 = \/ 250 =

___ ___

2 \/ 28 = 2 \/ 18 =

___ ___

\/ 54 = \/ 32 =

3 6

Simplify:

___ ___ ___ ___

\/ 4x2 = \/ 3x2 = \/ 16x = \/ x2y =

_____ ____ ____ ____

\/ 9x2 y2 = \/ 4xy2 = \/ 7x2y = 3 \/ 8xy2 =

127

___ ___ ____ ___

\/ 9x2 = \/ 5x2 = \/ 25x = \/ xy2 =

______ _____ _____ _____

\/ 16x2y2 = \/ 36xy2 = \/ 10x2y = 2 \/ 12x2y =

Simplify:

___ ___ ___

\/ x4 = \/ x3 = \/ x

8 =

___ ___ ____

\/ x7 = \/ 9x8 = \/ 16x5 =

___ ___

\/ 5x3 = \/ 8x9 =

Practice:

___ ___ ___

\/ x6 = \/ x5 = \/ x10 =

___ ___ ____

\/ x9 = \/ 4x6 = \/ 25x7 =

____ ____

\/ 11x3 = \/ 20x9 =

Simplify:

________ _______ ___________

\/ (x2 + y2)2 = \/ 3(x + y)3 = \/ x2 + 2xy + y2 =

128

Practice:

_______ _______ ____________

\/ (x2 - y2)2 = \/ 5(x + y)5 = \/ 4x2 + 4xy + y2 =

Rule: The Quotient Property of Square Roots -

The square root of a fraction or quotient is equal to the quotient of the

square roots of the numerator and the denominator.

__ __ __ __

4 = \/ 4 = 2

a = \/ a .

\ 9 \/ 9 3 \ b \/ b

Rule: There are three requirements for a radical to be in "simplified form":

1) The radicand has no factor that is a perfect square.

2) The radicand does not contain a fraction.

3) No radical appears in a denominator.

Note: In order to fulfill the requirement that no radical appears in a denominator

we employ a process called "rationalizing the denominator".

__ __ __ __ __ __

\/ 2 \/ 2 = \/ 3 \/ 3 = \/ 5 \/ 5 =

__

4 =

\ 5

2 =

_

\/ 8

Simplify:

___ ___

3 = 1 =

\ 4 \ 2

1 = 3 =

\/ 3 \/ 2

129

Practice:

9 = 5 =

\ 32 \ 12

12 = 4 =

_

\/ 18 \/12

Practice:

___ ___

7 =

2 =

\ 16 \ 5

3 = 5 =

\/ 5 \/ 8

___ ___

16 =

5 =

\ 27 \ 50

8 = 15 =

\/ 8 \/20

130

Simplify:

___

1 =

1 =

\ x \/ x3

x = 8 =

\/ y \/ x

___ ___

3x = x =

\ y \ y2

__ __

x = 2 =

\ 6 \ 8x

___ ___

2 \/ 3x = x3 =

\/ x3 \ xy

Practice:

___

2 =

1 =

\ x \/ x5

__

y \/ x = 8 =

_

\/ y \/ 2x

131

_____ ___

2x = x2 =

\ y \

y

__ __

12 = 9 =

\

3x \

5x

___ ___

3\/ 5x = x5 =

_

\/ x3 \ 2xy

Rule: To add or subtract like radicals, apply the rules for combining terms.

To add or subtract unlike radicals, it must be possible to transform

Simplify:

__ __ __ __ __ ___

6 \/ 3 + 9 \/ 3 = 8 \/ 2 - 5 \/ 2 = \/ 3 + \/ 12 =

___ ___ ___ __ __ ___

\/ 45 + \/ 20 = \/ 50 - \/ 8 = 3 \/ 5 - \/ 20 =

132

__ __ ___ ___ ___ __ ___

3 \/ 2 + 2 \/ 8 - 3 \/ 32 = 7 \/ 28 - 4 \/ 63 = 4 \/ 8 - \/ 98 =

____ ____ ___ ____ ___ __ _____

\/ 16x + \/ 25x = \/ 5x2 + \/ 20x2 = \/ xy2 - y \/ x + \/ 16xy2 =

Practice:

__ __ __ __ __ ___

5 \/ 2 + 6 \/ 2 = 9 \/ 5 - 7 \/ 5 = \/ 5 + \/ 45 =

___ ___ ___ ___ __ ___

\/ 32 + \/ 50 = \/ 75 - \/ 27 = 3 \/ 2 + 2 \/ 18 =

__ ___ ___ ___ ___ ___ __ ___

3 \/ 5 - \/ 20 + \/ 45 = 4 \/ 72 - 2 \/ 50 = \/ 32 - 5 \/ 8 + 2 \/ 50 =

___ ____ ___ ___ ___ __ _____

\/ 9x - \/ 64x = \/ 9x3 - x \/ 4x = y \/ x2y - 2x \/ y3 + \/ 9x2y3 =

133

Simplify:

__ __ ___ __ __ __

\/ 8 \/ 2 = 2 \/ 50 3 \/ 2 = \/ 6 \/ 3 =

__ __ ___ __ __

2 \/ 6 \/ 2 = 4 \/ 18 5 \/ 3 = ( 5 \/ 3 )2 =

__ __ __ __ __

\/ 2 ( 4 + \/ 3 ) = \/ 5 ( \/ 2 - \/ 3 ) =

__ __ __ __

( 4 + \/ 2 ) ( 3 + \/ 2 ) = ( 3 + \/ 2 )( 1 - \/ 2 ) =

Note: Two binomial factors of the form: __ __

(a + \/ b ) and (a - \/ b )

are called "conjugates" of each other. Their product is a "difference of

two squares", and the "middle term" drops out leaving a rational number

Simplify:

__ __ __ __

( 5 + \/ 2 )( 5 - \/ 2 ) = ( 5 + 6 \/ 3 ) ( 5 - 6 \/ 3 ) =

__ __ __ __

( 4 \/ 3 + \/ 5 ) ( 4 \/ 3 - \/ 5 ) =

134

Practice:

___ __ ___ __ ___ __

\/ 27 \/ 3 = 3 \/ 12 4 \/ 3 = \/ 10 \/ 5 =

__ __ ___ __ __

3 \/ 6 \/ 8 = 4 \/ 10 5 \/ 6 = ( 4 \/ 2 )2 =

__ __ __ __ __

\/ 5 ( 3 + \/ 2 ) = \/ 2 ( \/ 5 - \/ 3 ) =

__ __ __ __

( 3 + \/ 3 ) ( 2 + \/ 3 ) = ( 2 + \/ 5 ) ( 8 - \/ 5 ) =

__ __ __ __

( 4 + \/ 3 )( 4 - \/ 3 ) = ( 7 + 2 \/ 5 ) ( 7 - 2 \/ 5 ) =

__ __ __ __

( 5 \/ 2 + \/ 3 ) (5 \/ 2 - \/ 3 ) =

Simplify:

__ __ ___ ___ ____ ____

\/ x \/ x3 = \/ 3x \/ 2x4 = \/ 5x2y \/ 10y =

135

Practice:

___ ___ ___ ___ ___ ____

\/ 2x \/ 8x = \/ 4x \/xy = \/ 3x3y \/ 8xy2 =

Simplify:

___ ___

\/ 50 = 6 \/ 45 =

\/ 2 3 \/ 5

___ ___

\/ 24 = 50 \/ 24 =

\/ 3 5 \/ 2

___ ___

6 \/ 27 = 2 \/ 24 =

12 \/ 3 4 \/ 3

___ ___ ___ ___

\/ 32 + \/ 50 = 10 \/ 72 - 15 \/ 18 =

\/ 2 5 \/ 3

136

Practice:

___ ___

\/ 48 = 6 \/ 96 =

\/ 3 2 \/ 6

___ ___

\/ 54 = 8 \/ 60 =

\/ 2 4 \/ 5

___ ___

4 \/ 32 = 4 \/ 72 =

12 \/ 2 8 \/ 6

___ ___ ___ ___

\/ 18 + \/ 8 = 8 \/ 60 - 6 \/ 15 =

\/ 2 2 \/ 3

Simplify:

___ _____ ______

\/ x3 = \/ 12x5 = \/ 75x3y =

\/ x \/ 3x \/ 3xy

137

___ _____ _______

\/ x7 = \/ 18x3 = \/ 125x3y4 =

\/ x3 \/ 2x \/ 5xy2

Simplify: (Rationalize the denominators.)

6 =

2 - \/ 2

__

\/ 2 =

\/ 7 + \/ 2

__

\/ 3 + 1 =

\/ 3 - 1

Practice:

12 =

3 - \/ 5

__

\/ 3 =

\/ 5 + \/ 3

__

\/ 2 + 4 =

\/ 2 - 1

138

Practice:

Note: A "radical equation" is one in which the unknown appears under a radical sign.

Usually the equation is transformed into one without the radical (the square root)

by squaring both sides. After solving the resulting equation, you must check to

see if the answer satisfies the original equation. If not, then the answer is

considered to be an extraneous root and the equation may have no solution.

When doing the check, you may consider only the principal square root, not the

negative square root.

__ ___ ___ __

\/ x = 7 \/ 5x = 5 \/ 3x = 2 4 \/ x = 8

___ ____ ______

5 \/ 5x = 20 \/ x - 5 = 3 \/ 2x + 2 = 6

_____ __ _______

\/ x2 + 3 = x + 1 2x + 6 = 3 \/ x \/ x2 + 15 = 5 - x

\/ x

139

____ ____ ______

2 \/ x - 1 = -4 3 + \/ 5 - x = 2 \/ 12 + x = x

Practice:

__ ___ ___ __

\/ x = 4 \/ 6x = 6 \/ 5x = 3 2 \/ 3x = 6

___ _____ _____

6 \/ 6x = 12 \/ x + 3 = 5 \/ 5x - 1 = 3

140

______ ____ _____

\/ x2 + 27 = x + 3 3 \/ x - 7 = 6 \/ x2 - 7 = x - 1

\/ x - 7

_____ ___ ______

\/ 3x - 2 = -1 2 - \/ 3x = 5 \/ 2x + 3 = x

141

Simplify:

___ ___

2 \/ 24 = \/ 63 =

3

___

3 = 4 =

\ 5 \/ 8

___ ___ __ __ ___

4 \/ 27 - \/ 48 = \/ 3 (\/ 5 + \/ 20 ) =

___ __ __ __

3 \/ 50 3 \/ 2 = ( 3 + \/ 3 ) ( 3 - \/ 3 ) =

___ ___

\/ 48 = 3 \/ 72 =

\/ 3 9 \/ 6

___ ___ __

\/ 40 - 2 \/ 90 = \/ 3 =

\/ 2 \/ 3 + 1

142

______ ____

2 \/ 12x2y5 = x7 =

\ x2y

___ ____ ____ ____ ______

\/ x2y + \/ 9x2y = \/ 3xy2 \/ 8x3y = \/ 12x2y3 =

\/ 3y

Solve and Check:

____ _____

2 \/ x - 2 = 6 \/ 5 + x = 2x

143

Workbook

Part 7 - More on Quadratic Equations

Solving Quadratic Equations by Completing the Square

Note: We can solve some quadratic equations by factoring, those that are factorable

and have roots that are rational numbers. By another process called

"completing the square" we can solve any quadratic equation, even those that

are not factorable and have roots that are irrational numbers.

Remember: When we square a binomial, the result is called a "trinomial

square" or a "perfect-square trinomial".

Rule: In every perfect square trinomial of the form: ax2 + bx + c the constant

term is always equal to the square of one-half the coefficient of x.

Examples: x2 + 8x + 16 = One-half of 8 = 4; then 42 = 16

(x + 4)(x + 4) =

(x + 4)2

x2 - 10x + 25 = One-half of -10 = -5; then (-5)2 = 25

(x - 5)(x - 5) =

(x - 5)2

Make each binomial into a perfect square trinomial by completing the square.

Then show each perfect square trinomial as the square of a binomial:

x2 + 6x x2 - 2x x2 + x

x2 - 3x x2

- 1 x x2 + 3 x

2

4

Practice:

x2 + 10x x2 - 12x x2 + 7x

x2 - 5x x2

- x x2 - 2 x

3

144

Rule: To solve a quadratic equation by completing the square:

Put all terms that contain the unknown on the left side and the constant on the right side.

If the coefficient of x2 is not 1, divide all terms by the coefficient of x2.

Find one-half the coefficient of x. Square it, and add the result to both sides.

Express the left side as the square of a binomial and simplify the right side.

Take the square root of both sides. Write + before the square root of the right side.

Solve the two separate equations, one for the positive value and one for the negative value.

x2 - 6x + 8 = 0 x2 + 2x - 7 = 0

2x2 - 3x - 2 = 0 x2 - x - 3 = 0

145

Practice:

x2 + 10x + 16 = 0 x2 + 4x - 14 = 0

3x2 + 2x - 1 = 0 x2 + 7x + 2 = 0

146

The Derivation of the Quadratic Formula

ax2 + bx + c = 0

Solve and check:

2x2 - 5x + 2 = 0 3x2 - 3x - 2 = 0

147

Practice: (Solve and check)

2x2 - x - 3 = 0 3x2 + 5x + 1 = 0

The Discriminate and the Nature of the Roots

Note: When solving equations of the form: ax2 + bx + c

The expression b2 - 4ac is called the "discriminant" because it can

tell us about the "nature of the roots".

Rule: If the value of the discriminate is a negative number, the roots of the

equation are imaginary.

If the value of the discriminate is zero, the equation will have just

one real root. (Actually there are two roots, but they are equal, so the root

is sometimes called a "double root".)

If the value of the discriminant is a positive number, there are two

unequal real roots. (They will be irrational numbers unless the value of the

discriminate is a "perfect square", in which case the roots will be rational.

148

Find the value of the discriminate for each equation and tell the nature of the roots:

x2 + 3x + 4 = 0 3x2 - 6x + 3 = 0

2x2 - 5x - 4 = 0 5x2 - 11x + 2 = 0

Practice:

4x2 - 3x + 5 = 0 4x2 + 4x + 1 = 0

x2 - 2x - 10 = 0 x2 - 7x + 10 = 0

149

TEST - Completing the Square and The Quadratic Formula

Solve each equation, first by Completing the Square and then again by the

Quadratic Formula. Do the check for each equation just once. Finally, show

how the discriminant confirms the nature of the roots:

2x2 + 5x + 3 = 0 x2 - 2x - 2 = 0

150

Workbook

Part 8 - Inequalities

Introduction to Inequalities

Remember: Example:

= means "is equal to" 4 = 4

> means "is greater than" or "is to the right of" 4 > 3

< means "is less than" or "is to the left of" 3 < 4

Note:

> means "is greater than or equal to" 4 > 3, 4 > 4

< means "is less than or equal to" 3 < 4, 3 < 3

A mathematical sentence containing the = symbol is called an equation.

A mathematical sentence containing any of the symbols >, <, >, or < is called an inequality.

A mathematical sentence containing a variable is called an open sentence.

Examples of open sentences:

x = 4 is read, "x is equal to 4" (an equation).

x < 5 is read, "x is less than 5 (an inequality).

10 > x is read, "10 is greater than or equal to x" (an inequality).

The set of numbers that a variable could possibly stand for is called the

replacement set or the domain of that variable.

The set of numbers that satisfies an equation or an inequality within the

specified domain is called the solution set.

Example: For the inequality x > 5 to be solved within the domain of the

integers, the solution set is: {6, 7, 8, 9, ...}

Find the solution set given that the domain is the integers:

x > 3 x < 2

x > -2 x < -1

x > 4 x < 7

x > -1 x < 0

151

Practice:

x > 4 x < 6

x > -1 x < -2

x > -2 x < 6

x > 0 x < -1

When the domain of an inequality is the set of real numbers, rather than the

set of just the integers, the solution set includes rational numbers and

irrational numbers as well as integers:

Example: For the inequality x > 2 the solution set would include, among others,

the following members: 2.1, 2 1/2, \/ 5, 3, etc.

Thus, when the domain is the set of real numbers, it is best to show the solution

set as a graph on a number line.

O .

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > 2 x < 5

o means that 2 is not a member of the solution set. . means that 5 is a member of the solution set.

The arrowhead on the graph means "and so on" in the indicated direction.

Use graphs to show the solution set of each inequality:(the domain is the real

numbers).

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > 3 x < 2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > -2 x < - 1

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > 4 x < 7

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > -2 x < 0

152

Practice:

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > 4 x < 6

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > -1 x < - 2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > -5 x < 6

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x > 0 x < -1

Given the graph, write the inequality underneath the number line:

.

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

o .

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

o o

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Practice:

o o

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

. .

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

. o

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

153

o

Solving Inequalities

The procedures for solving an inequality are the same as for solving an equation

with one important exception:

Rule: If you multiply or divide each side of an inequality by the same negative

number, you must also reverse the direcion of the inequality symbol.

Solve each inequality and show the solution set as a graph on the number line.

The domain is always the real numbers:

x + 5 > 7 -2 > x - 4

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

4x < -12 7 - 4x > 15

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x + 1 < 4 x + 3 < -2

2 -4

.

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

154

Note: We will do this problem two ways.

3x + 5 < 4x + 3 3x + 5 < 4x + 3 2(x - 4) - 9 > 3 (x - 6)

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Practice:

x - 4 > 2 3 < x - 2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

3x > -15 8 - 3x < 2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-1 < 1 + x x - 2 > 2

5 -3

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

155

Note: Please do this problem two ways.

3x + 8 < 5x + 4 3x + 8 < 5x + 4 3(x + 5) - 4 < 2(2x + 4)

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Combining Inequalities

We can combine two inequalities by connecting them with either the word "and" or the word "or".

If we put them together with the word "and", the result is called a "conjunction". The solution set of the

conjunction, contains only those elements that are common to both of the solution sets of the separate

parts. This is also known as the "intersection" of those two solution sets.

If we put them together with the word "or", the result is called a "disjunction". The solution set of the

disjunction, contains those elements that are in either one (or both) of the solution sets of the separate

parts. This is also known as the "union" of those two solution sets.

Examples: "x > -3 and x < 4" First, x > -3 can be turned around to be -3 < x

Then -3 < x and x < 4 can be put together into -3 < x < 4

Notice that this can be done because the inequality symbols are in the same direction and

because the last part of the first inequality is identical to the first part of the second inequality.

However, be careful: You can't connect 3 < x and x < 2 to make 3 < x < 2

because 3 is not less than 2.

Now, look at the graphs of each part and of the conjunction: -3 < x < 4 o

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > - 3

.

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x < 4

o .

| | l l l l l l l l l l l l l l l l l l l -3 < x < 4

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

In words, the solution set for the conjunction, -3 < x < 4

being the intersection of the two parts, is:

"4 and all the real numbers between -3 and 4".

156

o

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x > - 3

.

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 x < 4

| | l l l l l l l l l l l l l l l l l l l x > -3 or x < 4

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

In words, the solution set for for the disjunction, x > -3 or x < 4

being the union of the two parts, is:

"all the real numbers".

Solve each inequality and graph the solution sets:

-2 < x - 1 < 5 3 < 2x + 1 < 13

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

5 > 2 - 3x > -7 10 - 2x > 12 and 7x < 4x + 9

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

157

Now, look at the graphs of each part and of the disjunction: x > -3 or x < 4

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

8 - 2x > -6 or 8 - 2x < 6 x - 1 < 2x + 3 < x + 4

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

2x - 1 < 3(x - 1) < 2(x - 2)

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

158

-3 < -2x - 1 and 0 > -1 - x x + 2 > 6 or x + 2 < -6

-1 < x + 3 < 5 -14 < 4x - 2 < -6

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

8 > 4 - 2x > -6 6x - 12 > 12x and 13 + 4x < 1

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-2 < x + 1 and -3 < 5 - 2x 4 + x < -3 or 3 + x > 4

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

159

Practice:

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

8 - 3x < x - 4 < -2x + 5

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

3x + 8 < 2 or x + 5 > 1 7 - 3x > 6 - 4x > 4 - 3x

160

Absolute Values in Open Sentences

Remember: The value of a number not considering the sign is the "absolute value".

l5l = 5 l-5l = 5

Rule: On a number line, the absolute value of a number is its distance from the

origin not considering the sign. Thus you travel in both the positive and

negative directions to graph the absolute value of a number.

Example: lxl = 3 . .

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The graph consists of the two points that are three units from the origin, 3 and -3.

This is true because: If x = -3, then l-3l is 3. If x = 3, then l3l is 3.

The value of x could be either one, -3 or 3, to produce an absolute value of 3.

Thus, lxl = 3 is equivalent to the disjunction: x = -3 or x = 3.

Example: lxl < 3 o o

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The graph consists of all the points less than three units from the origin.

This is equivalent to the conjunction: -3 < x < 3.

Example: lxl < 3 . .

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The graph consists of all the points at most three units from the origin.

This is equivalent to the conjunction: -3 < x < 3.

Example: lxl > 3 o o

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The graph consists of all the points more than three units from the origin.

This is equivalent to the disjunction: x < -3 or x > 3.

Example: lxl > 3 . .

| | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

The graph consists of all the points at least three units from the origin.

This is equivalent to the disjunction: x < -3 or x > 3.

161

Solve the open sentence and graph the solution set:

lx - 4l = 6 l1 - 4xl = 3

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

lxl < 1 l2x - 9l < 1

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

l3 - xl > 5 l3 - 3xl > 7

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

162

lx + 3l = 5 l1 - 3xl = 10

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

lx - 3l < 6 l3x - 7l < 2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

l4 - xl > 3 l5 - 2xl > 3

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

163

Practice:

TEST - Inequalities

Solve each Inequality and graph the solution set:

3 > x - 4 4 - 3x < 13

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

x + 4 < 6 3(x - 1) - 1 < 4(x + 1)

2

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

4x - 5 > 7 or 4x - 5 < - 9 7 - 4x > 6 - 5x > 4(1 - x)

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

164

l2x + 3l < 11 l4 - 2xl > 4

| | l l l l l l l l l l l l l l l l l l l | | l l l l l l l l l l l l l l l l l l l

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

165

Graphing Linear Inequalities in Two Variables

1) Graph the inequality: y > 2x - 4 2) Graph the inequality: y > -1/2 x + 3

y y

x x

3) Graph the inequality: 4x - y < 3 4) Graph the inequality: -2x + 3y < 3

y y

x x

166

Practice:

1) Graph the inequality: y > 3x - 5 2) Graph the inequality: y > - 1/3x + 1

y y

x x

3) Graph the inequality: 3x - y < 4 4) Graph the inequality: -3x + 2y < 6

y y

x x

167

Solving a System of Inequalities Graphically: y

2x + y > 6 and y - x < 3

x

Practice: Solve the systems of inequalities graphically: y

3x - 4y > 4 and y > 2x + 4

x

y

x + y > 4 and x - y < 2

x

168

TEST - Graphing Linear Inequalities in Two Variables

Solve the system of inequalities graphically:

y

2x + 3y < 6 and 2x - y > 6

x

169

Workbook

Part 9 - Supplementary Topics

Calculating Square Roots

Calculate the square roots:

________ _______ _______

\/ 1 2 1 \/ 2 2 5 \/ 7 8 4

________ _________ ___________

\/ 1 3 6 9 \/ 9 2 1 6 \/ 1 3 4 5 6

_____________ _____________

\/ 1 1 9 0 2 5 \/ 2 5 7 0 4 9

170

Practice:

_______ _______ _______

\/ 1 6 9 \/ 3 6 1 \/ 4 4 1

________ _________ ___________

\/ 3 3 6 4 \/ 7 5 6 9 \/ 5 0 6 2 5

______________ _____________

\/ 1 2 0 4 0 9 \/ 1 6 4 8 3 6

171

Rule: When asked to round off a square root to a certain decimal place, always carry

out the operation one place beyond the place you are asked to round the answer to.

Calculate the square roots. Round off the answers to the nearest tenth:

______________ _____________

\/ 2. 0 0 0 0 \/ 7 7. 0 0 0 0

Practice: (Calculate the square roots. Round off the answers to the nearest tenth)

______________ _____________

\/ 3. 0 0 0 0 \/ 9 1. 0 0 0 0

172

TEST - Calculating Square Roots

Calculate the square root:

___________

\/ 7 5 6 2 5

Calculate the square root. Round off the answer to the nearest tenth:

_____________

\/ 4 3. 0 0 0 0

173

Changing Repeating Decimals to Fractions

Remember: All rational numbers can also be expressed as either terminating

or repeating decimals

Examples: 1 = .5 1 = .125 1 = .3 1 = .16 1 = .09 2 1 = 2.1 2 8 3 6 11 9

Rule: To change a repeating decimal into an equivalent fraction:

1) Set up an equation of the form: Let n = "the repeating decimal".

2) Multiply both sides of the equation by a power of 10 whose

exponent equals the number of digits in the repeating block.

3) Subtract the first equation from the second.

4) Solve the resulting equation to obtain the equivalent fraction.

Change these repeating decimals into equivalent fractions:

.3 .16 .09

.123 2.1

174

Practice:

.6 .83 .18

.321 4.2

TEST - Changing Repeating Decimals to Fractions

Change these repeating decimals into equivalent fractions:

.416 3.27

175

Scientific Notation

In "scientific notation", a number is expressed as the product of two numbers,

one of them being a number between 1 and 10, and the other being an integral

power of 10.

Scientific notation is used for two reasons:

1) To provide a more concise notation for very large or very small numbers.

2) To simplify computations by applying the laws of exponents to numbers

expressed in scientific notation.

Examples: 700 = 7 X 102 .75 = 7.5 X 10-1

750 = 7.5 X 102 .075 = 7.5 X 10-2

7,500 = 7.5 X 103 .0075 = 7.5 X 10-3

75,000 = 7.5 X 104 .00075 = 7.5 X 10-4

Change from decimal notation to scientific notation:

842 = 735,000 =

The approximate distance from our sun to Alpha Centauri, the nearest star

is 25,000,000,000,000 miles.

25,000,000,000,000 =

.01 = .000068 =

The mass of a molecule of water is approximately .00000000000000000000003

gram.

.00000000000000000000003 =

Practice:

7,634 = 9,380,000 =

A light year is the distance light travels in one year, approximately

5,900,000,000,000 miles.

5,900,000,000,000 =

.009 = .0000000056 =

The radius of the orbit of an electron of a hydrogen atom is approximately

.0000000000053 meter.

.0000000000053 =

176

Change from scientific notation to decimal notation:

8 X 103 = 7.4 X 106 =

5 X 10-4 = 5.12 X 10-5 =

Practice:

7 X 104 = 2.8 X 108 =

3 X 10-3 = 6.52 X 10-7 =

Use scientific notation and the laws of exponents to perform these computations:

(327,000)(200) =

(82,000)(3,000) =

252,000 =

.00126

Practice:

(2,570,000)(3,000) =

(38,000)(400) =

.0096 =

800

TEST - Scientific Notation

Use scientific notation and the laws of exponents to perform these computations:

(432,000,000)(.000007) =

.000072 =

300

177

The Distance Formula

Remember: The Pythagorean Theorm: In any right triangle the square of the

hypotenuse is equal to sum of the squares of other two sides.

y c2 = a2 + b2

It is possible to determine the distance

between any two points that lie in the

coordinate plane.

Example:

x Plot these two points: A (-2,2) B (6, -4)

Now plot a third point: C (-2,-4)

Connecting these three points forms

a right triangle. Label the sides as follows:

Side a is opposite vertex A.

Side b is opposite vertex B.

Side c is opposite vertex C.

It is easy to determine the lengths of the two legs as one is parrallel to the x-axis and the other

to the y-axis.

The length of the leg parallel to the x-axis is found by subtracting one of the x-coordinates from

the other. BC = [-2 - 6] = -8. Taking the absolute value, the number not considering the sign,

the length of the leg is 8 units. You would get the same result if you subtracted the

x-coordinates in the opposite order. [6 -(-2)] = 8 (This is the side we labeled a ).

The length of the leg parallel to the y-axis is found by subtracting one of the y-coordinates from

the other. AC = [-4 - 2] = -6. Taking the absolute value, the number not considering the sign,

the length of the leg is 6 units. You would get the same result if you subtracted the

y-coordinates in the opposite order. [2 - (-4)] = 6 (This is the side we labeled b ).

Now, using the Pythagorean Theorem, you can determine the length of the hypotenuse, which is

also the distance between points A and B.

c2 = a2 + b2

= 82 + 62

= 64 + 36

= 100

c = \/ 100

= 10

This process leads to a general formula for the distance between any two points in the

coordinate plane:

d = \/ (x2 - x1)2 + (y2 - y1)2

178

(-2, -1) and (2, 2) (4, 2) and (6, -1) (-3,0) and (0, -3)

d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2

Practice:

(-6, -4) and (6, 1) (-3, 2) and (5, 1) (-2,-3) and (2, 5)

d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2

TEST - The Distance Formula:

(1, 2) and (5, 5) (2, 3) and (-3, 4) (-4,-6) and (2, -4)

d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2 d = \/ (x2 - x1)2 + (y2 - y1)2

179

Using the distance formula, find the distance between the given points:

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