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Developmental Algebra
Beginning and Intermediate Preparing for College Mathematics
By Paul Pierce
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Beginning and Intermediate
ALGEBRA
Preparing for College Mathematics
By Paul Pierce
Texas Tech University
Lubbock, 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 explanation without intent to infringe.
15 14 13 12 11
12345
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
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!
Chapter 1: One-Variable Linear Equations and Inequalities
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
Example A
Determine whether the equation is true, false, or conditional.
Determine whether the equation is true, false, or conditional.
a.
2 + 9 = 11
True
a.
7–5=2
b.
5–1=3
False
b.
4+6=9
c.
x + 2 = 15
Conditional, the value of x is not known.
c.
x+1=8
Example 2
Example B
Determine whether 7 is a solution of the equation x + 15 = 23.
Determine whether 8 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.
Chapter 1: One-Variable Linear Equations and Inequalities
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.
x  5  7
x  7  5
x  12
Check: x  5  7
The solution is −12. ◄
12  5 ?  7
7  7
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  x  17
8  17  x
25  x
Example D
Solve 15 = x – 12
Check:
8  x  17
8 ? 25  17
88
The solution is 25.
◄
Chapter 1: One-Variable Linear Equations and Inequalities
4
Example 5
Example E
Solve 7 + x = –18
Solve 3 + x = -15
Solution: Using rule (4), add 7 to both sides to isolate x.
7  x  18
x  18  7
x  11
Check:
7  x  18
7  (11) ?  18
18  18
The solution is –11.
◄
Example 6
Solve x +
Example F
5
1
=
and graph the solution on a number line.
4
4
Solution: Using rule (3), subtract
x
Solve x +
7
1
=
and graph the solution on a number line.
8
16
5
from both sides to isolate x.
4
5 1

4 4
x
1 5

4 4
x
4
 1
4
The solution is –1.
◄
Chapter 1: One-Variable Linear Equations and Inequalities
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)
1
+ x = 11
4
18)
1
+x=7
4
19)
x+
8
5
=
9
9
20)
x+
7
3
=
4
12
21)
x–
5
1
=
6
6
22)
x–
1
3
=
4
8
23)
x+1
24)
x+2= 2
25)
x–
7
1
= 
4
20
26)
x–
3
3
=
4
8
1
1
=3
2
2
1
4
Chapter 1: One-Variable Linear Equations and Inequalities
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
Example A
Solve 11x = 77.
7 x 77

7
7
Check:
7x = 77
7(11) ? 77
77 = 77
x = 11
The solution is 11.
◄
Example 2
Solve 8x = 56
Solution: Divide both sides of the equation by −8.
8x = 56
Example B
Solve 7x = 56
8 x 56

8
8
Check:
8x = 56
-8(-7) ? 56
56 = 56
x = –7
The solution is −7.
◄
Chapter 1: One-Variable Linear Equations and Inequalities
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.
Example C
Solve x = 2 and graph the solution on a number line.
1x
4

1 1
x = 4
The solution is –4.
◄
Example 4
Solve
Example D
4
x  20
5
Solution: Divide by
Solve
5
x  15
6
4
5
, which is the same as multiplying by .
5
4
4
x  20
5
4
x
5  20
4
4
5
5
Check :
4
x  20
5
4
( 25)  20
5
20  20
5 4
5
 x  20 
4 5
4
5
20 5
x

1
4
x = –25
The solution is −25.
◄
Chapter 1: One-Variable Linear Equations and Inequalities
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)
1
x = 17
4
32)
8
16
x
35
21
22)
1
x = 25
7
33)

16
16
x
27
15
23)

1
x = 21
9
34)

16
8
x
45
27
24)
1
 x = 17
7
35)
x
 29
4
25)
2
x=8
3
36)
x
 16
2
37)

x
 19
5
38)

x
 22
8
39)
x
 11
2
40)
x
 16
4
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
26)
27)

4
x=-8
7
2 6
 x
7 11
28)
1 1
 x
2 3
29)

30)
1
1
 x
3
5
31)
8
16
x
21
9
1
1
x
2
9
Chapter 1: One-Variable Linear Equations and Inequalities
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
Chapter 1: One-Variable Linear Equations and Inequalities
10
Example 3
Example C
Solve -7x + 6x = -7 + 6
Solve -9x + 8x = -15 − 5
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 4
Example D
Solve: 6x + 15 = 45
Solve: 11x + 10 = 43
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.
◄
Chapter 1: One-Variable Linear Equations and Inequalities
11
Example 5
Example E
Solve 7x – 11 = 5x + 13
Solve 9x – 20 = 4x + 15
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 6
Example F
Solve 6x – 8x + 16 = 7x – 14 + 5
Solve 5x – 9x + 11 = 3x – 13 + 7
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.
x
25 25

9
9
◄
Chapter 1: One-Variable Linear Equations and Inequalities
12
Exercise Set 1.3
For exercises 1-80, solve the equation.
1)
9x + 3x = 132
21)
7x − 16x = 73 − 19
41)
3x + 8 = 35
61)
5x + 2 = 4x + 6
2)
11x + 2x = 117
22)
-3x − 2x = 21 − 16
42)
8x + 7 = 87
62)
6x + 4 = 2x + 20
3)
3x + 11x = 98
23)
-5x − 3x = 37 − 13
43)
8x + 2 = 42
63)
4x + 6 = 3x + 9
4)
4x + 9x = 104
24)
-5x + 2x = 33 − 12
44)
3x + 10 = 22
64)
8x + 6 = 2x + 54
5)
8x + 9x = 85
25)
-3x − 8x = 114 − 15
45)
8x + 3 = 43
65)
8x + 4 = 3x + 44
6)
6x + 3x = 45
26)
9x − 16x = 58 − 16
46)
8x + 9 = 33
66)
7x + 3 = 5x + 7
7)
7x + 6x = 39
27)
4x − 10x = 54 − 12
47)
10x + 8 = 28
67)
9x + 7 = 2x + 42
8)
11x + 8x = 228
28)
5x − 7x = 16 − 14
48)
2x + 6 = 24
68)
6x + 8 = 4x + 18
9)
6x + 5x = 33
29)
5x − 12x = -5 − 16
49)
3x + 7 = 28
69)
4x + 9 = 2x + 21
10)
6x + 4x = 120
30)
2x − 5x = 4 − 19
50)
3x + 2 = 23
70)
8x + 4 = 3x + 39
11)
-2x − 4x = 24
31)
9x − x = 79 − 7
51)
10x − 4 = 26
71)
2x − 4 = 46 − 8x
12)
-3x − 6x = -63
32)
7x − x = 25 − 7
52)
4x − 4 = 24
72)
9x − 3 = 117 − 6x
13)
-2x − 7x = 36
33)
7x − x = 47 − 5
53)
4x − 9 = 7
73)
7x − 5 = 85 − 8x
14)
-7x − 7x = -70
34)
9x − x = 41 − 9
54)
4x − 2 = 10
74)
2x − 4 = 52 − 6x
15)
-2x − 3x = -10
35)
5x − x = 5 − 37
55)
4x − 7 = 33
75)
6x − 6 = 85 − 7x
16)
-6x − 4x = 30
36)
3x − x = 3 − 17
56)
2x − 8 = 6
76)
3x − 6 = 15 − 4x
4x − 7 = 110 − 9x
17)
-6x − 3x = 63
37)
9x − x = 76 − 4
77)
57)
5x − 4 = 36
78)
8x − 1 = 76 − 3x
18)
-6x − 3x = 45
38)
6x + x = 31 − 3
58)
10x − 8 = 82
79)
10x − 3 = 105 − 8x
19)
-8x − 4x = -24
39)
7x + x = 21 − 5
59)
3x − 2 = 25
80)
4x − 1 = 53 − 5x
20)
-3x − 5x = 32
40)
3x + x = 13 – 5
60)
8x − 3 = 13
Chapter 1: One-Variable Linear Equations and Inequalities
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
Example A
Solve 5(x + 2) = 25
Solve: 11(x + 5) = 77
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
◄
Chapter 1: One-Variable Linear Equations and Inequalities
14
Example 2
Example B
Solve 7(x – 4) = 3(x + 4)
Solve 2(x – 6) = 9(x + 2)
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 3
Example C
Solve 6x – 4(x + 3) = 5(x – 2) – 1
Solve 5x – 7(x + 5) = 4(x – 6) – 3
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.
x
1
3
◄
Chapter 1: One-Variable Linear Equations and Inequalities