Beginning and Intermediate Algebra with Applications & Visualization: Pearson New International Edition

1020 Pages·2013·32.883 MB·English

by  Gary K Rockswold, Terry A Krieger

Pearson New International Edition

Beginning and Intermediate Algebra with
Applications and Visualization
Gary K. Rockswold Terry A. Krieger
Third Edition
International_PCL_TP.indd 1 7/29/13 11:23 AM
ISBN 10: 1-292-02732-0
ISBN 13: 978-1-292-02732-6
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ISBN 10: 1-292-02732-0
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-02732-6
ISBN 13: 978-1-269-37450-7
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P E A R S O N C U S T O M L I B R AR Y
Table of Contents
1. Introduction to Algebra
Gary K. Rockswold/Terry A. Krieger 1
2. Linear Equations and Inequalities
Gary K. Rockswold/Terry A. Krieger 93
3. Graphing Equations
Gary K. Rockswold/Terry A. Krieger 165
4. Systems of Linear Equations in Two Variables
Gary K. Rockswold/Terry A. Krieger 263
5. Polynomials and Exponents
Gary K. Rockswold/Terry A. Krieger 319
6. Factoring Polynomials and Solving Equations
Gary K. Rockswold/Terry A. Krieger 383
7. Rational Expressions
Gary K. Rockswold/Terry A. Krieger 445
8. Introduction to Functions
Gary K. Rockswold/Terry A. Krieger 531
9. Systems of Linear Equations
Gary K. Rockswold/Terry A. Krieger 625
10. Radical Expressions and Functions
Gary K. Rockswold/Terry A. Krieger 667
11. Quadratic Functions and Equations
Gary K. Rockswold/Terry A. Krieger 751
12. Exponential and Logarithmic Functinos
Gary K. Rockswold/Terry A. Krieger 837
13. Conic Sections
Gary K. Rockswold/Terry A. Krieger 915
I
111999990000005788991193137
Appendix: Using the Graphing Calculator
Gary K. Rockswold/Terry A. Krieger 959
Appendix: Sets
Gary K. Rockswold/Terry A. Krieger 971
Appendix: Linear Programming
Gary K. Rockswold/Terry A. Krieger 981
Appendix: Synthetic Division
Gary K. Rockswold/Terry A. Krieger 989
Glossary
Gary K. Rockswold/Terry A. Krieger 993
Bibliography
Gary K. Rockswold/Terry A. Krieger 1001
Formulas from Geometry & Algebra
Gary K. Rockswold/Terry A. Krieger 1003
Index 1007
II
Introduction to Algebra
1 Numbers, Variables, and
Expressions
2 Fractions
3 Exponents and Order of
Operations
4 Real Numbers and the
Number Line
5 Addition and Subtraction
of Real Numbers
6 Multiplication and
Division of Real
Numbers
7 Properties of Real Num-
bers
8 Simplifying and Writing
Algebraic Expressions
NASA/Arena Creative/Shutterstock
J
Unless you try to do
something beyond
ust over a century ago only about one in ten workers was in a professional, tech-
what you have already
nical, or managerial occupation. Today this proportion is nearly one in three, and the
mastered, you will study of mathematics is essential for anyone who wants to keep up with the technologi-
never grow. cal changes that are occurring in nearly every occupation. Mathematics is the l anguage
of technology .
—RONALD E. OSBORN
I n the information age, mathematics is being used to describe human behavior in
areas such as economics, medicine, advertising, social networks, and Internet use. For
example, mathematics can help maximize the impact of advertising by analyzing social
networks such as Facebook. Today’s business managers need employees who not only
understand human behavior but can also describe that behavior using mathematics.
I t’s just a matter of time before the m ajority of the workforce will need the analytic
skills that are taught in mathematics classes every day. No matter what career path you
choose, a solid background in mathematics will provide you with opportunities to reach
your full potential in your vocation, income level, and lifestyle.
Source: A. Greenspan, “T he Eco-
nomic Importance of Improving
Math-Science Education. ”
From Chapter 1 of Beginning and Intermediate Algebra with Applications and Visualization. Third Edition. Gary K. Rockswold, Terry A.
Krieger. Copyright © 2013 by Pearson Education, Inc. All rights reserved. 1
INTRODUCTION TO ALGEBRA
1 Numbers, Variables, and Expressions
Natural Numbers and Whole Numbers ● P rime Numbers and Composite Numbers ●
Variables, Algebraic Expressions, and Equations ● Translating Words to Expressions
ck A LOOK INTO MATH N Numbers are an important concept in every society. A number system once used in south-
o
st ern Africa consisted of only the numbers from 1 to 20. Numbers larger than 20 were named
er
utt by counting groups of t wenties. For example, the number 67 was called three twenties
h
S and seven. This base-20 number system would not work well in today’s technologically
n/
ci advanced world. In this section, we introduce two sets of numbers that are used extensively
ar
M in the modern world: natural numbers and whole numbers.
z
s
Kir
Natural Numbers and Whole Numbers
NEW VOCABULARY One important set of numbers is the set of n atural numbers . These numbers comprise the
counting numbers and may be expressed as follows.
n N atural numbers
n W hole numbers 1, 2, 3, 4, 5, 6,N
n P roduct
n F actors B ecause there are infinitely many natural numbers, three dots are used to show that the list
n P rime number continues without end. A second set of numbers is called the w hole numbers , and may be
n C omposite number expressed as follows.
n P rime factorization 0, 1, 2, 3, 4, 5,N
n V ariable
n A lgebraic expression W hole numbers include the natural numbers and the number 0.
n E quation
n F ormula N REAL-WORLD CONNECTION Natural numbers and whole numbers can be used when data are
not broken into fractional parts. For example, the bar graph in Figure 1 shows the number of
CRITICAL THINKING apps on a student’s iPad for the first 5 months after buying the device. Note that both natu-
ral numbers and whole numbers are appropriate to describe these data because a fraction of
Give an example from every-
an app is not possible.
day life of natural number or
whole number use.
iPad Apps
50
45
42
ps 40 37
p
A
of 30 29
er
mb 20 18
u
N
10
0
Jan. Feb. Mar. Apr. May
Month
Figure 1
STUDY TIP
Bring your book, notebook, and a pen or pencil to every class. Write down major concepts pre-
sented by your instructor. Your notes should also include the meaning of words written in bold
type in the text. Be sure that you understand the meaning of these important words.
2
INTRODUCTION TO ALGEBRA
Prime Numbers and Composite Numbers
When two natural numbers are multiplied, the result is another natural number. For exam-
ple, multiplying the natural numbers 3 and 4 results in 12, a natural number. The result 12 is
called the product and the numbers 3 and 4 are factors of 1 2 .
#
3 4 = 12
factor factor product
#
NOTE: Products can be expressed in several ways. For example, the product 3 4 = 12
can also be written as 3 * 4 = 12 and 3(4) = 12.
A natural number greater than 1 that has o nly itself and 1 as natural number factors is a
prime number . The number 7 is prime because the only natural number factors of 7 are 1 and
7. The following is a partial list of prime numbers. There are infinitely many prime numbers.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
READING CHECK A natural number greater than 1 that is not prime is a c #omposite number . For example,
the natural number 15 is a composite number because 3 5 = 15 . In other words, 15 has
• How do prime numbers
factors other than 1 and itself.
differ from composite
Every composite number can be written as a product of prime numbers. For example,
numbers?
we can use a factor tree such as the one shown in Figure 2 to find the prime factors of the
composite number 120. Branches of the tree are made by writing each composite number as
a product that includes the smallest possible prime factor of the composite number.
120
CRITICAL THINKING
2 60
Suppose that you draw a tree
diagram for the prime factor-
2 2 30
ization of a prime number.
Describe what the tree will 2 2 2 15
look like.
2 2 2 3 5
Figure 2 Prime Factorization of 120
Figure 2 shows that the prime factorization of 120 is
# # # #
120 = 2 2 2 3 5.
Every composite number has a unique prime factorization, and it is customary to write the
prime factors in order from smallest to largest.
MAKING CONNECTIONS
Factor Trees and Prime Factorization
The prime factors of 120 can also be found using the following tree. Even though this tree is differ-
ent than the one used earlier, the prime factors it reveals are the same.
120

10 12
2 5 3 4
2 5 3 2 2
No matter what tree is used, a prime factorization is a lways unique.
3
INTRODUCTION TO ALGEBRA
E XAMPLE 1 Classifying numbers as prime or composite
Classify each number as prime or composite, if possible. If a number is composite, write it
as a product of prime numbers.
(a) 31 (b) 1 (c) 35 (d) 200
Solution
(a) The only factors of 31 are itself and 1, so the number 31 is prime.
(b) The number 1 is neither prime nor composite because prime and composite numbers
200 must be greater than 1.
(c) The number 35 is composite b#ecause 5 and 7 are factors. It can be written as a product
2 100 of prime numbers as 35 = 5 7.
(d) The number 200 is composite because 10 and 20 are factors. A factor tree can be used
2 2 50
to write 200 as a product of prime numbers#, as# sh#own# in Figure 3 . The factor tree
reveals that 200 can be factored as 200 = 2 2 2 5 5 .
2 2 2 25
Now Try Exercises 13, 15, 17, 21
2 2 2 5 5
Figure 3 Prime
Factorization of 200 Variables, Algebraic Expressions, and Equations
#
There #are 12 inches in 1 foot, so 5 feet equal 5 12 = 60 inches. Similarly, in 3 feet there
are 3 12 = 36 inches. To convert feet to inches frequently, Table 1 might help.
TABLE 1 Converting Feet to Inches
Feet 1 2 3 4 5 6 7
Inches 12 24 36 48 60 72 84
However, this table is not helpf#ul in converting 11 feet to inches. We could expand
Table 1 into Table 2 to include 11 12 = 132 , but expanding the table to accommodate
every possible value for feet would be impossible.
TABLE 2 Converting Feet to Inches
Feet 1 2 3 4 5 6 7 11
Inches 12 24 36 48 60 72 84 132
READING CHECK Variables are often used in mathematics when tables of numbers are inadequate. A
variable is a symbol, typically an italic letter such as x , y , z , or F , used to represent an
• What is a variable?
unknown quantity. In the preceding example, the number of feet could be represented by the
• Give one reason for using
variable F , and the corresponding number of inches could be represented# by the variable I .
variables in mathematics.
The number of inches in F feet is given by the algebraic expression 12 F . That is, to cal-
culate the number of inches in F feet, multiply F by 12. The relationship between feet F and
inches I is shown using the e quation or formula
#
I = 12 F.
#
A dot ( ) is used to indicate multiplication because a multiplication sign ( * ) can be con-
fused with the variable x . Many times the multiplication sign is omitted altogether. Thus all
three formulas
#
I = 12 F, I = 12F, and I = 12(F )
4
INTRODUCTION TO ALGEBRA
represent the same relationship between feet and inches. If we wish to find the number of
inches in 10 feet, for example, we can replace F in one of these formulas with the number
10. If we let F = 10 in the first formula,
#
I = 12 10 = 120.
That is, there are 120 inches in 10 feet.
More formally, #an a lgebraic expression consists of numbers, variables, operation sym-
bols such as +, -, , and ,, and grouping symbols such as parentheses. An equation is a
mathematical statement that two algebraic expressions are equal. Equations always con-
tain an equals sign. A formula is a special type of equation that expresses a relationship
between two or more quantities. The formula I = 12F states that to calculate the number of
inches in F feet, multiply F by 12.
E XAMPLE 2 Evaluating algebraic expressions with one variable
Evaluate each algebraic expression for x = 4 .
x
(a) x + 5 (b) 5x (c) 15 - x (d)
(x - 2)
Solution
(a) Replace x with 4 in the expression x + 5 to obtain 4 + 5 = 9 . #
(b) The expression 5x indicates multiplication of 5 and x . Thus 5x = 5 4 = 20 .
(c) 15 - x = 15 - 4 = 11
(d) Perform all arithmetic operations inside parentheses first.
x 4 4
= = = 2
(x - 2) (4 - 2) 2
Now Try Exercises 45, 47, 51, 53
N REAL-WORLD CONNECTION Some algebraic expressions contain more than one variable.
For example, if a car travels 120 miles on 6 gallons of gasoline, then the car’s mileage is
120 = 20 miles per gallon. In general, if a car travels M miles on G gallons of gasoline,
6
then its mileage is given by the expression M . Note that M contains two variables, M and G ,
G G
whereas the expression 12F contains only one variable, F .
E XAMPLE 3 Evaluating algebraic expressions with two variables
Evaluate each algebraic expression for y = 2 and z = 8 .
z
(a) 3yz (b) z - y (c)
y
Solution # #
(a) Replace y with 2 and z with 8 to obtain 3yz = 3 2 8 = 48 .
(b) z - y = 8 - 2 = 6
z 8
(c) = = 4
y 2
Now Try Exercises 55, 57, 59
5