Developmental Algebra Beginning and Intermediate ... · c. x + 2 = 15 Conditional, the value of x is not known. Example A Determine whether the equation is true, false, or conditional

Developmental AlgebraBeginning and Intermediate - Preparing for College Mathematics

By Paul Pierce

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Beginning and Intermediate

ALGEBRAPreparing for College Mathematics

By Paul Pierce

Texas Tech UniversityLubbock, Texas

Copyright © 2011 by Paul Pierce. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of University Readers, Inc.

First published in the United States of America in 2011 by Cognella, a division of University Readers, Inc.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explana-tion without intent to infringe.

15 14 13 12 11 1 2 3 4 5

Printed in the United States of America

ISBN: 978-1-60927-926-4

Table of Contents

Chapter 1: One-Variable Linear Equations and Inequalities ..........................................................................................1

1.1 Solving Linear Equations: The Addition Principle of Equality ....................................................................................................................2

1.2 Solving Linear Equations: The Multiplication Principle of Equality ...........................................................................................................6

1.3 Solving Linear Equations: Combining Like Terms ......................................................................................................................................9

1.4 Solving Linear Equations: The Distributive Property.................................................................................................................................13

1.5 Solving Linear Equations: Fractions and Decimals ....................................................................................................................................16

1.6 Solving Linear Equations: A General Strategy...........................................................................................................................................19

1.7 Solving Linear Inequalities .........................................................................................................................................................................23

1.8 Solving Absolute Value Equations and Inequalities...................................................................................................................................30

1.9 Applications Involving One-Variable Linear Equations.............................................................................................................................33

Chapter 2: Two-Variable Linear Equations and Inequalities .......................................................................................39

2.1 The Rectangular Coordinate System...........................................................................................................................................................40

2.2 Intercepts and Graphing Lines ....................................................................................................................................................................47

2.3 Slopes and Graphing Lines .........................................................................................................................................................................54

2.4 Equations of Lines ......................................................................................................................................................................................61

2.5 Parallel and Perpendicular Lines.................................................................................................................................................................69

2.6 Graphing Two-Variable Linear Inequalities ...............................................................................................................................................74

Chapter 3: Systems of Two-Variable Linear Equations and Inequalities.....................................................................81

3.1 Solving Systems of Linear Equations: Graphing Method.......................................................................................................................... 82

3.2 Solving Systems of Linear Equations: Substitution Method...................................................................................................................... 87

3.3 Solving Systems of Linear Equations: Elimination Method ...................................................................................................................... 93

3.4 Graphing Systems of Linear Inequalities ................................................................................................................................................... 99

3.5 Applications Involving Systems of Linear Equations .............................................................................................................................. 104 Chapter 4: Polynomials ................................................................................................................................................... 113

4.1 Basic Rules of Exponents......................................................................................................................................................................... 114

4.2 Introduction to Polynomials ..................................................................................................................................................................... 121

4.3 Adding and Subtracting Polynomials ....................................................................................................................................................... 126

4.4 Multiplying Polynomials .......................................................................................................................................................................... 132

4.5 Dividing Polynomials............................................................................................................................................................................... 142

4.6 Operations on Polynomials with Multiple Variables ............................................................................................................................... 147 Chapter 5: Factoring Polynomials.................................................................................................................................. 151

5.1 Greatest Common Factors (GCF)............................................................................................................................................................. 152

5.2 Factoring by Grouping ............................................................................................................................................................................. 157

5.3 Factoring Trinomials of the Form x2 + bx + c .......................................................................................................................................... 159

5.4 Factoring Trinomials of the Form ax2 + bx + c, a > 1: The ac-Method................................................................................................... 165

5.5 Factoring Differences of Squares, Differences of Cubes and Sums of Cubes ......................................................................................... 169

5.6 Solving Quadratic Equations by Factoring............................................................................................................................................... 174

5.7 Applications Involving Quadratic Equations ........................................................................................................................................... 178

Chapter 6: Rational Expressions and Equations ...........................................................................................................185

6.1 Simplifying Rational Expressions.............................................................................................................................................................186

6.2 Multiplying Rational Expressions.............................................................................................................................................................190

6.3 Dividing Rational Expressions..................................................................................................................................................................193

6.4 Adding and Subtracting Rational Expressions With Common Denominators .........................................................................................197

6.5 Least Common Denominators...................................................................................................................................................................202

6.6 Adding and Subtracting Rational Expressions With Different Denominators .........................................................................................205

6.7 Solving Rational Equations.......................................................................................................................................................................211

Chapter 7: Radical Expressions and Equations.............................................................................................................215

7.1 Multiplying and Simplifying Radical Expressions ...................................................................................................................................216

7.2 Dividing and Simplifying Radical Expressions ........................................................................................................................................221

7.3 Adding and Subtracting Radical Expressions ...........................................................................................................................................224

7.4 Rationalizing Denominators .....................................................................................................................................................................228

7.5 Solving Radical Equations ........................................................................................................................................................................233

Chapter 8: Quadratic Equations .....................................................................................................................................237

8.1 Solving Quadratic Equations: Square Root Principle ...............................................................................................................................238

8.2 Solving Quadratic Equations: Completing the Square..............................................................................................................................242

8.3 Solving Quadratic Equations: Quadratic Formula ....................................................................................................................................246

8.4 Graphing Quadratic Equations..................................................................................................................................................................250

8.5 More Applications Involving Quadratic Equations ..................................................................................................................................258

Appendix A: Review of Basic Math Skills ...................................................................................................................... 265

Skill Objective 1: Multiplying Fractions......................................................................................................................................................... 266

Skill Objective 2: Simplifying Fractions........................................................................................................................................................ 268

Skill Objective 3: Multiplying, Dividing, and Simplifying Fractions............................................................................................................ 271

Skill Objective 4: Adding and Subtracting Fractions With Common Denominators .................................................................................... 275

Skill Objective 5: Least Common Denominators........................................................................................................................................... 279

Skill Objective 6: Adding and Subtracting Fractions With Different Denominators .................................................................................... 284

Skill Objective 7: Multiplying Signed Numbers ............................................................................................................................................ 288

Skill Objective 8: Dividing Signed Numbers................................................................................................................................................. 290

Skill Objective 9: Adding Signed Numbers ................................................................................................................................................... 292

Skill Objective 10: Subtracting Signed Numbers........................................................................................................................................... 294

Appendix B: Answers to Odd-Numbered Exercises...................................................................................................... 299

Dedication To my beautiful wife, Laura. For over twenty years, she has held my hand while the lights have grown dim.

Chapter 1: One-Variable Linear Equations and Inequalities

Chapter 1 One-Variable Linear Equations and Inequalities

1.1 Solving Linear Equations: The Addition Principle of Equality 1.2 Solving Linear Equations: The Multiplication Principle of Equality 1.3 Solving Linear Equations: Combining Like Terms 1.4 Solving Linear Equations: The Distributive Property 1.5 Solving Linear Equations: Fractions and Decimals 1.6 Solving Linear Equations: A General Strategy 1.7 Solving Linear Inequalities 1.8 Solving Absolute Value Equations and Inequalities 1.9 Applications Involving One-Variable Linear Equations

Think of an equation as a scale.

Keep it balanced!

2


1.1 Solving Linear Equations: The Addition Principle of Equality

a. Determine whether a number is a solution of an equation.

b. Solve one-variable linear equations using the addition principle of equality.

What is an “Equation?”

An equation is a statement that claims that one mathematical expression is the same as, or is equal to, another mathematical expression. The two expressions are separated by an equal sign, =. If there is no equal sign, then it is just an expression, not an equation. Not all statements are true, and likewise, not all equations are true.

What is a “Solution” of an Equation?

Any value which, when substituted in for the variable in an equation, causes the equation to be true is called a solution of the equation. To “solve” an equation means to find ALL of its solutions. All of the solutions of an equation form the solution set of the equation.

Example 1

Determine whether the equation is true, false, or conditional.

a. 2 + 9 = 11 True

b. 5 – 1 = 3 False

c. x + 2 = 15 Conditional, the value of x is not known.

Example A

Determine whether the equation is true, false, or conditional.

a. 7 – 5 = 2

b. 4 + 6 = 9

c. x + 1 = 8

Example 2

Determine whether 7 is a solution of the equation x + 15 = 23.

Solution: x + 15 = 23 Start with the equation.

7 + 15 ? 23 Substitute 7 for x

22 23 False

Since the left and right sides differ, 7 is not a solution.

Example B

Determine whether 8 is a solution of the equation x + 15 = 23.

3


What Does “Equivalent Equation” Mean? Equations with the same solution set are called equivalent equations.

The Addition Principle of Equality For any real numbers a, b, and c,

if a = b, then a + c = b + c, (1) and

if a = b, then a – c = b – c. (2) This essentially means that the same value can be added to (or subtracted from) BOTH sides of an equation while preserving equality. The Addition Principle produces an equivalent equation. As a direct result of the Addition Principle, we can state the following rules:

if a + c = b, then a = b – c (3) and

if a – c = b, then a = b + c (4)

Example 3 Solve x + 5 = –7. Solution: Using rule (3), subtract 5 from both sides to isolate x.

5 77 512

xxx

Check: 5 712 5 ? 7

7 7

x

The solution is −12. ◄

Example C Solve x + 9 = –11.

Example 4 Solve 8 = x – 17 Solution: Using rule (4), add 17 to both sides to isolate x.

8 178 17

25

xxx

Check:

8 178 ? 25 178 8

x

The solution is 25. ◄

Example D Solve 15 = x – 12

4


Example 5

Solve 7 + x = –18

Solution: Using rule (4), add 7 to both sides to isolate x.

7 18

18 7

11

x

x

x

Check:

7 187 ( 11) ? 18

18 18

x

The solution is –11. ◄

Example E

Solve 3 + x = -15

Example 6

Solve x + 54

= 14

and graph the solution on a number line.

Solution: Using rule (3), subtract 54

from both sides to isolate x.

5 14 4

1 54 4

4 14

x

x

x

The solution is –1. ◄

Example F

Solve x + 78

= 116

and graph the solution on a number line.

5


Exercise Set 1.1

For exercises 1-26, solve the equation using the addition principle.

1) x – 2 = 3

2) x – 1 = 2

3) x + 7 = 9

4) x + 2 = 4

5) 9 = x + 6

6) 7 = x + 4

7) -10 = x – 29

8) -19 = x – 22

9) x – 28.79 = 0

10) x – 25.00 = 0

11) -7 + x = 18

12) -6 + x = 11

13) 17 = -24 + x

14) 10 = -17 + x

15) -17.0 = 13.2 + x

16) -17.1 = 29.2 + x

17) 14

+ x = 11

18) 14

+ x = 7

19) x + 59

= 89

20) x + 712

= 34

21) x – 16

= 56

22) x – 14

= 38

23) x + 112

= 132

24) x + 2 = 124

25) x – 720

= 14

26) x – 34

= 38

6


1.2 Solving Linear Equations: The Multiplication Principle of Equality a. Solve one-variable linear equations using the multiplication principle of equality.

The Multiplication Principle of Equality For any real numbers a, b, and c with c 0,

if a = b, then a • c = b • c, and

if a = b, then a c = b c. This means that both sides of an equation can be multiplied (or divided) by any nonzero number while preserving equality. As with the Addition Principle of Equality, using the Multiplication Principle of Equality results in an equivalent equation.

Example 1 Solve 7x = 77. Solution: Divide both sides of the equation by 7.

7x = 77 7 777 7x

x = 11 Check: 7x = 77 7(11) ? 77 77 = 77 The solution is 11. ◄

Example A Solve 11x = 77.

Example 2 Solve 8x = 56 Solution: Divide both sides of the equation by −8.

8x = 56 8 568 8x

x = –7 Check: 8x = 56 -8(-7) ? 56 56 = 56 The solution is −7. ◄

Example B Solve 7x = 56

7


Example 3 Solve x = 4 and graph the solution on a number line. Solution: A negative sign in front of x implies a multiplication by -1.

x = 4 (1)x = 4

Divide both sides of the equation by −1. 1 41 1x

x = 4 The solution is –4. ◄

Example C Solve x = 2 and graph the solution on a number line.

Example 4

Solve 4 205

x

Solution: Divide by 45

, which is the same as multiplying by 54

.

4 205

x

4205

4 45 5

x

5 4 5204 5 4

x

20x

5

51 4

x = –25

The solution is −25. ◄

Example D

Solve 5 156

x

:4 205

4 ( 25) 205

20 20

Check

x

8


Exercise Set 1.2

For exercise 1-46, solve using the multiplication principle.

1) 3x = 12

2) 6x = 12

3) 40 = 8x

4) 48 = 8x

5) -x = 12

6) -x = -19

7) -2x = 8

8) -9x = 36

9) -12 = 3x

10) 30 = -5x

11) -8x = -32

12) -7x = -28

13) 8x = -88

14) 4x = -56

15) -28 = 2x

16) -55 = 5x

17) -108 = -6x

18) -144 = -8x

19) -2x = -24

20) -7x = -105

21) 14

x = 17

22) 17

x = 25

23) 19

x = 21

24) 17

x = 17

25) 23

x = 8

26) 47

x = - 8

27) 2 67 11

x

28) 1 12 3

x

29) 1 12 9

x

30) 1 13 5

x

31) 8 1621 9

x

32) 8 1635 21

x

33) 16 1627 15

x

34) 16 845 27

x

35) 294x

36) 162x

37) 195x

38) 228x

39) 112x

40) 164x

41) 7.3x = 29.2

42) 8.4x = 16.8

43) 9 = 4.5x

44) 13.8 = 4.6x

45) -9.5x = 38

46) -3.2x = 22.4

9


1.3 Solving Linear Equations: Combining Like Terms a. Solve equations in which like terms need to be combined.

Combining Like Terms With one-variable linear equations, there are only two types of terms: terms with variables and terms without variables, which are called constants. To combine constants, simply add or subtract as indicated. To combine variable terms, add or subtract the numbers in front of the variables, which are called coefficients, to obtain a term with the same variable. DO NOT change the variable.

Solving Equations: Combining Like Terms 1. If like terms appear on the same side of an equation, combine them, then 2. If like terms appear on opposite sides of the equation, use the addition principle to collect all like terms on the same side of the equation

and combine like terms again, then 3. Solve the equation using the multiplication principle.

Example 1 Solve 5x + 3x = 7 + 9 Solution: Combine like terms on each side of the equation.

5x + 3x = 7 + 9 8x = 16

Now, solve by dividing both sides by 8. 8x = 16

x = 2 ◄

Example A Solve 8x + 7x = 31 + 14

Example 2 Solve 3x + 7x = 19 – 9 Solution: Combine like terms on each side of the equation.

3x + 7x = 19 – 9 10x = 10

Now, solve by dividing both sides by 10. 10x = 10

x = 1 ◄

Example B Solve 14x + 8x = 50 − 6

10


Example 3

Solve -7x + 6x = -7 + 6

Solution: Combine like terms on both sides of the equation.

-7x + 6x = -7 + 6

-x = -1

Now, solve by dividing by -1.

-x = -1

x = 1 ◄

Example C

Solve -9x + 8x = -15 − 5

Example 4

Solve: 6x + 15 = 45

Solution: Isolate the x-term by subtracting 15 from both sides.

6x + 15 = 45

6x + 15 – 15 = 45 15

6x = 30

Now, solve for x by dividing by 6.

6x = 30

x = 5

Check: 6x + 15 = 45

6(5) + 15 ? 45

30 + 15 ? 45

45 = 45 The solution is 5. ◄

Example D

Solve: 11x + 10 = 43

11


Example 5

Solve 7x – 11 = 5x + 13

Solution: Add 11 to both sides of the equation.

7x – 11 + 11 = 5x + 13 + 11

7x = 5x + 24

Next, subtract 5x from both sides of the equation.

7x – 5x = 24

2x = 24

Divide both sides of the equation by 2.

x = 12 ◄

Example E

Solve 9x – 20 = 4x + 15

Example 6

Solve 6x – 8x + 16 = 7x – 14 + 5

Solution: Combine like terms on each side of the equation.

6x – 8x + 16 = 7x – 14 + 5

–2x + 16 = 7x – 9

Subtract 16 from both sides of the equation.

–2x = 7x – 25

Subtract 17x from both sides of the equation.

–9x = –25

Divide both sides by −9.

25 259 9

x

◄

Example F

Solve 5x – 9x + 11 = 3x – 13 + 7

12


Exercise Set 1.3

For exercises 1-80, solve the equation.

1) 9x + 3x = 132

2) 11x + 2x = 117

3) 3x + 11x = 98

4) 4x + 9x = 104

5) 8x + 9x = 85

6) 6x + 3x = 45

7) 7x + 6x = 39

8) 11x + 8x = 228

9) 6x + 5x = 33

10) 6x + 4x = 120

11) -2x − 4x = 24

12) -3x − 6x = -63

13) -2x − 7x = 36

14) -7x − 7x = -70

15) -2x − 3x = -10

16) -6x − 4x = 30

17) -6x − 3x = 63

18) -6x − 3x = 45

19) -8x − 4x = -24

20) -3x − 5x = 32

21) 7x − 16x = 73 − 19

22) -3x − 2x = 21 − 16

23) -5x − 3x = 37 − 13

24) -5x + 2x = 33 − 12

25) -3x − 8x = 114 − 15

26) 9x − 16x = 58 − 16

27) 4x − 10x = 54 − 12

28) 5x − 7x = 16 − 14

29) 5x − 12x = -5 − 16

30) 2x − 5x = 4 − 19

31) 9x − x = 79 − 7

32) 7x − x = 25 − 7

33) 7x − x = 47 − 5

34) 9x − x = 41 − 9

35) 5x − x = 5 − 37

36) 3x − x = 3 − 17

37) 9x − x = 76 − 4

38) 6x + x = 31 − 3

39) 7x + x = 21 − 5

40) 3x + x = 13 – 5

41) 3x + 8 = 35

42) 8x + 7 = 87

43) 8x + 2 = 42

44) 3x + 10 = 22

45) 8x + 3 = 43

46) 8x + 9 = 33

47) 10x + 8 = 28

48) 2x + 6 = 24

49) 3x + 7 = 28

50) 3x + 2 = 23

51) 10x − 4 = 26

52) 4x − 4 = 24

53) 4x − 9 = 7

54) 4x − 2 = 10

55) 4x − 7 = 33

56) 2x − 8 = 6

57) 5x − 4 = 36

58) 10x − 8 = 82

59) 3x − 2 = 25

60) 8x − 3 = 13

61) 5x + 2 = 4x + 6

62) 6x + 4 = 2x + 20

63) 4x + 6 = 3x + 9

64) 8x + 6 = 2x + 54

65) 8x + 4 = 3x + 44

66) 7x + 3 = 5x + 7

67) 9x + 7 = 2x + 42

68) 6x + 8 = 4x + 18

69) 4x + 9 = 2x + 21

70) 8x + 4 = 3x + 39

71) 2x − 4 = 46 − 8x

72) 9x − 3 = 117 − 6x

73) 7x − 5 = 85 − 8x

74) 2x − 4 = 52 − 6x

75) 6x − 6 = 85 − 7x

76) 3x − 6 = 15 − 4x

77) 4x − 7 = 110 − 9x

78) 8x − 1 = 76 − 3x

79) 10x − 3 = 105 − 8x

80) 4x − 1 = 53 − 5x

13


1.4 Solving Linear Equations: The Distributive Property a. Solve equations containing parentheses using the distributive property.

The Distributive Property

When a number appears in front of a pair of parentheses, such as 5(x + 2), this represents a multiplication. To remove the parentheses, we use this distributive property:

a(b + c) = ab + ac

For example,

5(x + 2) = 5∙x + 5∙2 = 5x + 10

-5(x + 2) = (-5)∙x + (-5)∙2 = -5x – 10

Example 1

Solve 5(x + 2) = 25

Solution: Use the distributive property to remove the parentheses.

5(x + 2) = 25

5x + 10 = 25

Subtract 10 from both sides of the equation.

5x = 25 – 10

5x = 15

Divide both sides of the equation by 5.

x = 3 ◄

Example A

Solve: 11(x + 5) = 77

14


Example 2

Solve 7(x – 4) = 3(x + 4)

Solution: Use the distributive property to remove the parentheses.

7(x – 4) = 3(x + 4)

7x – 28 = 3x + 12

Add 28 to both sides of the equation.

7x = 3x + 40

Subtract 3x from both sides.

4x = 40

Divide by 4.

x = 10 ◄

Example B Solve 2(x – 6) = 9(x + 2)

Example 3 Solve 6x – 4(x + 3) = 5(x – 2) – 1 Solution: Remove parentheses using the distributive property. Note that the first parentheses are multiplied by –4, not just 4.

6x – 4(x + 3) = 5(x – 2) – 1 6x – 4x – 12 = 5x – 10 – 1

Combine like terms on each side of the equation. 2x – 12 = 5x – 11

Add 12 to both sides of the equation. 2x = 5x + 1

Subtract 5x from both sides of the equation. –3x = 1

Divide by –3.

13

x ◄

Example C

Solve 5x – 7(x + 5) = 4(x – 6) – 3

Documents

Developmental Algebra Beginning and Intermediate ... · c. x + 2 = 15 Conditional, the value of x is not known. Example A Determine whether the equation is true, false, or conditional