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ALGEBRA DEMYSTIFIED
Other Titles in the McGraw-Hill Demystified Series
Astronomy Demystified by Stan Gibilisco
Calculus Demystified by Steven G. Kra ntz
Physics Demystified by Stan Gibilisco
ALGEBRA DEMYSTIFIED
RHONDA HUETTENMUELLER
McGRAW-HILL
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otherwise.
DOI: 10.1036/0071412107
To all those who struggle with math
This page intentionally left blank.
vii
CONTENTS
Preface ix
CHAPTER 1 Fractions 1
CHAPTER 2 Introduction to Variables 37
CHAPTER 3 Decimals 55
CHAPTER 4 Negative Numbers 65
CHAPTER 5 Exponents and Roots 79
CHAPTER 6 Factoring 113
CHAPTER 7 Linear Equations 163
CHAPTER 8 Linear Applications 197

CHAPTER 9 Linear Inequalities 285
CHAPTER 10 Quadratic Equations 319
CHAPTER 11 Quadratic Applications 353
Appendix 417
Final Review 423
Index 437
For more information about this book, click here.
Copyright 2003 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
This page intentionally left blank.
ix
PREFACE
This book is designed to take the mystery out of algebra. Each section con-
tains exactly one new idea—unlike most math books, which cover several
ideas at once. Clear, brief explanations are followed by detailed examples.
Each section ends with a few Practice problems, most similar to the examples.
Solutions to the Practice problems are also given in great detail. The goal is
to help you understand the algebra concepts while building your skills and
confidence.
Each chapter ends with a Chapter Review, a multiple-choice test designed
to measure your mastery of the material. The Chapter Review could also be
used as a pretest. If you think you understand the material in a chapter, take
the Chapter Review test. If you answer all of the questions correctly, then
you can safely skip that chapter. When taking any multiple-choice test, work
the problems before looking at the answers. Sometimes incorrect answers
look reasonable and can throw you off. Once you have finished the book,
take the Final Review, which is a multiple-choice test based on material from
each chapter.
Spend as much time in each section as you need. Try not to rush, but do
make a commitment to learning on a schedule. If you find a concept difficult,
you might need to work the problems and examples several times. Try not to

jump around from section to section as most sections extend topics from
previous sections.
Not many shortcuts are used in this book. Does that mean you shouldn’t
use them? No. What you should do is try to find the shortcuts yourself. Once
you have found a method that seems to be a shortcut, try to figure out why it
works. If you understand how a shortcut works, you are less likely to use it
incorrectly (a common problem with algebra students).
Because many find fraction arithmetic difficult, the first chapter is devoted
almost exclusively to fractions. Make sure you understand the steps in this
chapter because they are the same steps used in much of the rest of the book.
For example, the steps used to compute
7
36
þ
5
16
are exactly those used to
compute
2x
x
2
þ x À 2
þ
6
x þ 2
.
Copyright 2003 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Even those who find algebra easy are stumped by word problems (also
called ‘‘applications’’). In this book, word problems are treated very care-
fully. Two important skills needed to solve word problems are discussed

earlier than the word problems themselves. First, you will learn how to
find quantitative relationships in word problems and how to represent
them using variables. Second, you will learn how to represent multiple quan-
tities using only one variable.
Most application problems come in ‘‘families’’—distance problems, work
problems, mixture problems, coin problems, and geometry problems, to
name a few. As in the rest of the book, exactly one topic is covered in
each section. If you take one section at a time and really make sure you
understand why the steps work, you will find yourself able to solve a great
many applied problems—even those not covered in this book.
Good luck.
R
HONDA HUETTENMUELLER
PREFACE
x
ACKNOWLEDGMENTS
I want to thank my husband and family for their patience during the many
months I worked on this project. I am also grateful to my students through
the years for their thoughtful questions. Finally, I want to express my appre-
ciation to Stan Gibilisco for his welcome advice.
This page intentionally left blank.
CHAPTER 1
Fractions
Fraction Multiplication
Multiplication of fractions is the easiest of all fraction operations. All you
have to do is multiply straight across—multiply the numerators (the top
numbers) and the denominators (the bottom numbers).
Example
2
3

Á
4
5
¼
2 Á 4
3 Á 5
¼
8
15
:
Practice
1:
7
6
Á
1
4
¼
2:
8
15
Á
6
5
¼
1
Copyright 2003 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
3:
5
3

Á
9
10
¼
4:
40
9
Á
2
3
¼
5:
3
7
Á
30
4
¼
Solutions
1:
7
6
Á
1
4
¼
7 Á 1
6 Á 4
¼
7

24
2:
8
15
Á
6
5
¼
8 Á 6
15 Á 5
¼
48
75
3:
5
3
Á
9
10
¼
5 Á 9
3 Á 10
¼
45
30
4:
40
9
Á
2

3
¼
40 Á 2
9 Á 3
¼
80
27
5:
3
7
Á
30
4
¼
3 Á 30
7 Á 4
¼
90
28
Multiplying Fractions and Whole Numbers
You can multiply fractions by whole numbers in one of two ways:
1. The numerator of the product will be the whole number times the
fraction’s numerator, and the denominator will be the fraction’s
denominator.
2. Treat the whole number as a fraction—the whole number over
one—then multiply as you would any two fractions.
CHAPTER 1 Fractions
2
Example
5 Á

2
3
¼
5 Á 2
3
¼
10
3
or
5 Á
2
3
¼
5
1
Á
2
3
¼
5 Á 2
1 Á 3
¼
10
3
Practice
1:
6
7
Á 9 ¼
2: 8 Á

1
6
¼
3: 4 Á
2
5
¼
4:
3
14
Á 2 ¼
5: 12 Á
2
15
¼
Solutions
1:
6
7
Á 9 ¼
6 Á 9
7
¼
54
7
or
6
7
Á
9

1
¼
6 Á 9
7 Á 1
¼
54
7
2: 8 Á
1
6
¼
8 Á 1
6
¼
8
6
or
8
1
Á
1
6
¼
8 Á 1
1 Á 6
¼
8
6
3: 4 Á
2

5
¼
4 Á 2
5
¼
8
5
or
4
1
Á
2
5
¼
4 Á 2
1 Á 5
¼
8
5
4:
3
14
Á 2 ¼
3 Á 2
14
¼
6
14
or
3

14
Á
2
1
¼
3 Á 2
14 Á 1
¼
6
14
CHAPTER 1 Fractions
3
5: 12 Á
2
15
¼
12 Á 2
15
¼
24
15
or
12
1
Á
2
15
¼
12 Á 2
1 Á 15

¼
24
15
Fraction Division
Fraction division is almost as easy as fraction multiplication. Invert (switch
the numerator and denominator) the second fraction and the fraction divi-
sion problem becomes a fraction multiplication problem.
Examples
2
3
Ä
4
5
¼
2
3
Á
5
4
¼
10
12
3
4
Ä 5 ¼
3
4
Ä
5
1

¼
3
4
Á
1
5
¼
3
20
Practice
1:
7
6
Ä
1
4
¼
2:
8
15
Ä
6
5
¼
3:
5
3
Ä
9
10

¼
4:
40
9
Ä
2
3
¼
5:
3
7
Ä
30
4
¼
6: 4 Ä
2
3
¼
7:
10
21
Ä 3 ¼
CHAPTER 1 Fractions
4
Solutions
1:
7
6
Ä

1
4
¼
7
6
Á
4
1
¼
28
6
2:
8
15
Ä
6
5
¼
8
15
Á
5
6
¼
40
90
3:
5
3
Ä

9
10
¼
5
3
Á
10
9
¼
50
27
4:
40
9
Ä
2
3
¼
40
9
Á
3
2
¼
120
18
5:
3
7
Ä

30
4
¼
3
7
Á
4
30
¼
12
210
6: 4 Ä
2
3
¼
4
1
Ä
2
3
¼
4
1
Á
3
2
¼
12
2
7:

10
21
Ä 3 ¼
10
21
Ä
3
1
¼
10
21
Á
1
3
¼
10
63
Reducing Fractions
When working with fractions, you are usually asked to ‘‘reduce the fraction
to lowest terms’’ or to ‘‘write the fraction in lowest terms’’ or to ‘‘reduce the
fraction.’’ These phrases mean that the numerator and denominator have no
common factors. For example,
2
3
is reduced to lowest terms but
4
6
is not.
Reducing fractions is like fraction multiplication in reverse. We will first use
the most basic approach to reducing fractions. In the next section, we will

learn a quicker method.
First write the numerator and denominator as a product of prime
numbers. Refer to the Appendix if you need to review how to find the
prime factorization of a number. Next collect the primes common to both
the numerator and denominator (if any) at beginning of each fraction. Split
each fraction into two fractions, the first with the common primes. Now the
fraction is in the form of ‘‘1’’ times another fraction.
CHAPTER 1 Fractions
5
Examples
6
18
¼
2 Á 3
2 Á 3 Á 3
¼
ð2 Á 3ÞÁ1
ð2 Á 3ÞÁ3
¼
2 Á 3
2 Á 3
Á
1
3
¼
6
6
Á
1
3

¼ 1 Á
1
3
¼
1
3
42
49
¼
7 Á 2 Á 3
7 Á 7
¼
7
7
Á
2 Á 3
7
¼ 1 Á
6
7
¼
6
7
Practice
1:
14
42
¼
2:
5

35
¼
3:
48
30
¼
4:
22
121
¼
5:
39
123
¼
6:
18
4
¼
7:
7
210
¼
8:
240
165
¼
9:
55
33
¼

10:
150
30
¼
CHAPTER 1 Fractions
6
Solutions
1:
14
42
¼
2 Á 7
2 Á 3 Á 7
¼
ð2 Á 7ÞÁ1
ð2 Á 7ÞÁ3
¼
2 Á 7
2 Á 7
Á
1
3
¼
14
14
Á
1
3
¼
1

3
2:
5
35
¼
5
5 Á 7
¼
5 Á 1
5 Á 7
¼
5
5
Á
1
7
¼
1
7
3:
48
30
¼
2 Á 2 Á 2 Á 2 Á 3
2 Á 3 Á 5
¼
ð2 Á 3ÞÁ2 Á 2 Á 2
ð2 Á 3ÞÁ5
¼
2 Á 3

2 Á 3
Á
2 Á 2 Á 2
5
¼
6
6
Á
8
5
¼
8
5
4:
22
121
¼
2 Á 11
11 Á 11
¼
11
11
Á
2
11
¼
2
11
5:
39

123
¼
3 Á 13
3 Á 41
¼
3
3
Á
13
41
¼
13
41
6:
18
4
¼
2 Á 3 Á 3
2 Á 2
¼
2
2
Á
3 Á 3
2
¼
9
2
7:
7

210
¼
7
2 Á 3 Á 5 Á 7
¼
7 Á 1
7 Á 2 Á 3 Á 5
¼
7
7
Á
1
2 Á 3 Á 5
¼
1
30
8:
240
165
¼
2 Á 2 Á 2 Á 2 Á 3 Á 5
3 Á 5 Á 11
¼
ð3 Á 5ÞÁ2 Á 2 Á 2 Á 2
ð3 Á 5ÞÁ11
¼
3 Á 5
3 Á 5
Á
2 Á 2 Á 2 Á 2

11
¼
15
15
Á
16
11
¼
16
11
9:
55
33
¼
5 Á 11
3 Á 11
¼
11 Á 5
11 Á 3
¼
11
11
Á
5
3
¼
5
3
10:
150

30
¼
2 Á 3 Á 5 Á 5
2 Á 3 Á 5
¼
ð2 Á 3 Á 5ÞÁ5
ð2 Á 3 Á 5ÞÁ1
¼
2 Á 3 Á 5
2 Á 3 Á 5
Á
5
1
¼
30
30
Á 5 ¼ 5
Fortunately there is a less tedious method for reducing fractions to their
lowest terms. Find the largest number that divides both the numerator
and the denominator. This number is called the greatest common divisor
(GCD) . Factor the GCD from the numerator and denominator and rewrite
the fraction. In the previous examples and practice problems, the product of
the common primes was the GCD.
CHAPTER 1 Fractions
7
Examples
32
48
¼
16 Á 2

16 Á 3
¼
16
16
Á
2
3
¼ 1 Á
2
3
¼
2
3
45
60
¼
15 Á 3
15 Á 4
¼
15
15
Á
3
4
¼ 1 Á
3
4
¼
3
4

Practice
1:
12
38
¼
2:
12
54
¼
3:
16
52
¼
4:
56
21
¼
5:
45
100
¼
6:
48
56
¼
7:
28
18
¼
8:

24
32
¼
9:
36
60
¼
10:
12
42
¼
CHAPTER 1 Fractions
8
Solutions
1:
12
38
¼
2 Á 6
2 Á 19
¼
2
2
Á
6
19
¼
6
19
2:

12
54
¼
6 Á 2
6 Á 9
¼
6
6
Á
2
9
¼
2
9
3:
16
52
¼
4 Á 4
4 Á 13
¼
4
4
Á
4
13
¼
4
13
4:

56
21
¼
7 Á 8
7 Á 3
¼
7
7
Á
8
3
¼
8
3
5:
45
100
¼
5 Á 9
5 Á 20
¼
5
5
Á
9
20
¼
9
20
6:

48
56
¼
8 Á 6
8 Á 7
¼
8
8
Á
6
7
¼
6
7
7:
28
18
¼
2 Á 14
2 Á 9
¼
2
2
Á
14
9
¼
14
9
8:

24
32
¼
8 Á 3
8 Á 4
¼
8
8
Á
3
4
¼
3
4
9:
36
60
¼
12 Á 3
12 Á 5
¼
12
12
Á
3
5
¼
3
5
10:

12
42
¼
6 Á 2
6 Á 7
¼
6
6
Á
2
7
¼
2
7
Sometimes the greatest common divisor is not obvious. In these cases you
might find it easier to reduce the fraction in several steps.
Examples
3990
6762
¼
6 Á 665
6 Á 1127
¼
665
1127
¼
7 Á 95
7 Á 161
¼
95

161
644
2842
¼
2 Á 322
2 Á 1421
¼
322
1421
¼
7 Á 46
7 Á 203
¼
46
203
CHAPTER 1 Fractions
9
Practice
1:
600
1280
¼
2:
68
578
¼
3:
168
216
¼

4:
72
120
¼
5:
768
288
¼
Solutions
1:
600
1280
¼
10 Á 60
10 Á 128
¼
60
128
¼
4 Á 15
4 Á 32
¼
15
32
2:
68
578
¼
2 Á 34
2 Á 289

¼
34
289
¼
17 Á 2
17 Á 17
¼
2
17
3:
168
216
¼
6 Á 28
6 Á 36
¼
28
36
¼
4 Á 7
4 Á 9
¼
7
9
4:
72
120
¼
12 Á 6
12 Á 10

¼
6
10
¼
2 Á 3
2 Á 5
¼
3
5
5:
768
288
¼
4 Á 192
4 Á 72
¼
192
72
¼
2 Á 96
2 Á 36
¼
96
36
¼
4 Á 24
4 Á 9
¼
24
9

¼
3 Á 8
3 Á 3
¼
8
3
For the rest of the book, reduce fractions to their lowest terms.
Adding and Subtracting Fractions
When adding (or subtracting) fractions with the same denominators, add (or
subtract) their numerators.
CHAPTER 1 Fractions
10
Examples
7
9
À
2
9
¼
7 À 2
9
¼
5
9
8
15
þ
2
15
¼

8 þ 2
15
¼
10
15
¼
5 Á 2
5 Á 3
¼
2
3
Practice
1:
4
7
À
1
7
¼
2:
1
5
þ
3
5
¼
3:
1
6
þ

1
6
¼
4:
5
12
À
1
12
¼
5:
2
11
þ
9
11
¼
Solutions
1:
4
7
À
1
7
¼
4 À 1
7
¼
3
7

2:
1
5
þ
3
5
¼
1 þ 3
5
¼
4
5
3:
1
6
þ
1
6
¼
1 þ 1
6
¼
2
6
¼
1
3
4:
5
12

À
1
12
¼
5 À 1
12
¼
4
12
¼
1
3
5:
2
11
þ
9
11
¼
2 þ 9
11
¼
11
11
¼ 1
CHAPTER 1 Fractions
11