1
1.1 Numbers, Variables, and Expressions 1.2 Fractions 1.3 Exponents and Order of Operations 1.4 Real Numbers and the Number Line 1.5 Addition and Subtraction of Real Numbers 1.6 Multiplication and Division of Real Numbers 1.7 Properties of Real Numbers 1.8 Simplifying and Writing Algebraic Expressions

Introduction to Algebra

Unless you try to do something beyond what you have already mastered, you will never grow.
—RONALD E. OSBORN

ust over a century ago only about one in ten workers was in a professional, technical, or managerial occupation. Today this proportion is nearly one in three, and the study of mathematics is essential for anyone who wants to keep up with the technological changes that are occurring in nearly every occupation. Mathematics is the language of technology. In 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. It’s just a matter of time before the majority 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.

J

ISBN 1-256-49082-2

Source: A. Greenspan, “The Economic Importance of Improving Math-Science Education.”

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

2

CHAPTER 1 INTRODUCTION TO ALGEBRA

1.1

Numbers, Variables, and Expressions
Natural Numbers and Whole Numbers ● Prime Numbers and Composite Numbers ● Variables, Algebraic Expressions, and Equations ● Translating Words to Expressions

A LOOK INTO MATH N

Numbers are an important concept in every society. A number system once used in southern Africa consisted of only the numbers from 1 to 20. Numbers larger than 20 were named by counting groups of twenties. For example, the number 67 was called three twenties and seven. This base-20 number system would not work well in today’s technologically advanced world. In this section, we introduce two sets of numbers that are used extensively in the modern world: natural numbers and whole numbers.

Natural Numbers and Whole Numbers
NEW VOCABULARY n n n n n n n n n n n Natural numbers Whole numbers Product Factors Prime number Composite number Prime factorization Variable Algebraic expression Equation Formula

One important set of numbers is the set of natural numbers. These numbers comprise the counting numbers and may be expressed as follows. 1, 2, 3, 4, 5, 6, N Because there are infinitely many natural numbers, three dots are used to show that the list continues without end. A second set of numbers is called the whole numbers, and may be expressed as follows. 0, 1, 2, 3, 4, 5, N Whole numbers include the natural numbers and the number 0.

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 1.1 shows the number CRITICAL THINKING of apps on a student’s iPad for the first 5 months after buying the device. Note that both natural numbers and whole numbers are appropriate to describe these data because a fracGive an example from everytion of an app is not possible. day life of natural number or whole number use. iPad Apps

50 42 Number of Apps 40 30 20 10 0 Jan. Feb. Mar. Month Apr. 18 29 37

45

May

Figure 1.1

STUDY TIP
Bring your book, notebook, and a pen or pencil to every class. Write down major concepts presented 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.

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.1 NUMBERS, VARIABLES, AND EXPRESSIONS

3

Prime Numbers and Composite Numbers
When two natural numbers are multiplied, the result is another natural number. For example, 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 12. 3 factor #

4 factor =

12 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 only 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
• How do prime numbers differ from composite numbers?

A natural number greater than 1 that is not prime is a composite number. For example, the natural number 15 is a composite number because 3 # 5 = 15. In other words, 15 has factors other than 1 and itself. Every composite number can be written as a product of prime numbers. For example, we can use a factor tree such as the one shown in Figure 1.2 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
Suppose that you draw a tree diagram for the prime factorization of a prime number. Describe what the tree will look like.
2 2 2 2 2 2 2 2 2 3 60 30 15 5

Figure 1.2 Prime Factorization of 120

Figure 1.2 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 different than the one used earlier, the prime factors it reveals are the same.
120 10 2 2 5 5 3 3 2 12 4 2

ISBN 1-256-49082-2

No matter what tree is used, a prime factorization is always unique.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

4

CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 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 must be greater than 1. (c) The number 35 is composite because 5 and 7 are factors. It can be written as a product 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 to write 200 as a product of prime numbers, as shown in Figure 1.3. The factor tree reveals that 200 can be factored as 200 = 2 2 2 5 5.

200 2 2 2 2 2 2 2 2 2 5 100 50 25 5

#

# # # #

Now Try Exercises 13, 15, 17, 21
Figure 1.3 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 1.1 might help.

TABLE 1.1 Converting Feet to Inches

Feet Inches

1 12

2 24

3 36

4 48

5 60

6 72

7 84

However, this table is not helpful in converting 11 feet to inches. We could expand Table 1.1 into Table 1.2 to include 11 # 12 = 132, but expanding the table to accommodate every possible value for feet would be impossible.
TABLE 1.2 Converting Feet to Inches

Feet Inches

1 12

2 24

3 36

4 48

5 60

6 72

7 84

11 132

READING CHECK
• What is a variable? • Give one reason for using variables in mathematics.

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, ], or F, used to represent an unknown quantity. In the preceding example, the number of feet could be represented by the variable F, and the corresponding number of inches could be represented by the variable I. The number of inches in F feet is given by the algebraic expression 12 # F. That is, to calculate the number of inches in F feet, multiply F by 12. The relationship between feet F and inches I is shown using the equation or formula I = 12 # F.
ISBN 1-256-49082-2

A dot ( # ) is used to indicate multiplication because a multiplication sign ( * ) can be confused 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 )

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.1 NUMBERS, VARIABLES, AND EXPRESSIONS

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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 algebraic expression consists of numbers, variables, operation symbols such as + , - , # , and , , and grouping symbols such as parentheses. An equation is a mathematical statement that two algebraic expressions are equal. Equations always contain 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.

EXAMPLE 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.

#

4 4 x = = = 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 6 = 20 miles per gallon. In general, if a car travels M miles on G gallons of gasoline, 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.

EXAMPLE 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

# #

ISBN 1-256-49082-2

(c)

z 8 = = 4 y 2

Now Try Exercises 55, 57, 59

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 4

Evaluating formulas
Find the value of y for x = 15 and z = 10. x (c) y = 8xz (a) y = x - 3 (b) y = 5
Solution (a) Substitute 15 for x and then evaluate the right side of the formula to find y.

y = x - 3 = 12 (b) y = x 15 = = 3 5 5

Given formula Subtract.

= 15 - 3 Replace x with 15.

(c) y = 8xz = 8 # 15 # 10 = 1200
Now Try Exercises 61, 67, 69

Translating Words to Expressions
Many times in mathematics, algebraic expressions are not given; rather, we must write our own expressions. To accomplish this task, we often translate words to symbols. The symbols + , - , # , and , have special mathematical words associated with them. When two numbers are added, the result is called the sum. When one number is subtracted from another number, the result is called the difference. Similarly, multiplying two numbers results in a product and dividing two numbers results in a quotient. Table 1.3 lists many of the words commonly associated with these operations.
READING CHECK TABLE 1.3 Words Associated with Arithmetic Symbols
• Write three words associated with each of the symbols + , - , # , and , .

Symbol + -

Associated Words add, plus, more, sum, total subtract, minus, less, difference, fewer multiply, times, twice, double, triple, product divide, divided by, quotient

#
,

EXAMPLE 5

Translating words to expressions
Translate each phrase to an algebraic expression. Specify what each variable represents. (a) Four more than a number (b) Ten less than the president’s age (c) A number plus 10, all divided by a different number (d) The product of 6 and a number
Solution (a) If the number were 20, then four more than the number would be 20 + 4 = 24. If we let n represent the number, then four more than the number would be n + 4. (b) If the president’s age were 55, then ten less than the president’s age would be 55 - 10 = 45. If we let A represent the president’s age, then ten less would be A - 10.
ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.1 NUMBERS, VARIABLES, AND EXPRESSIONS

7

(c) Let x be the first number and y be the other number. Then the expression can be written as (x + 10) , y. Note that parentheses are used around x + 10 because it is “all” divided by y. This expression could also be written as x +y 10 . (d) If n represents the number, then the product of 6 and n is 6 # n or 6n.
Now Try Exercises 73, 77, 83, 85

N REAL-WORLD CONNECTION Cities are made up of large amounts of concrete and asphalt that heat up in the daytime from sunlight but do not cool off completely at night. As a result, urban areas tend to be warmer than surrounding rural areas. This effect is called the urban heat island and has been documented in cities throughout the world. The next example discusses the impact of this effect in Phoenix, Arizona.

EXAMPLE 6

Translating words to a formula
For each year after 1970, the average nighttime temperature in Phoenix has increased by about 0.1 C. (a) What was the increase in the nighttime temperature after 20 years, or in 1990? (b) Write a formula (or equation) that gives the increase T in average nighttime temperature x years after 1970. (c) Use your formula to estimate the increase in nighttime temperature in 2000.
Solution (a) The average nighttime temperature has increased by 0.1 C per year, so after 20 years the temperature increase would be 0.1 20 = 2.0 C. (b) To calculate the nighttime increase in temperature, multiply the number of years past 1970 by 0.1. Thus T = 0.1x, where x represents the number of years after 1970. (c) Because 2000 - 1970 = 30, let x = 30 in the formula T = 0.1x to get

#

T = 0.1(30) = 3. The average nighttime temperature increased by 3 C.
Now Try Exercise 91

Another use of translating words to formulas is in finding the areas of various shapes.

EXAMPLE 7

Finding the area of a rectangle
The area A of a rectangle equals its length L times its width W, as illustrated in the accompanying figure. (a) Write a formula that shows the relationship between these three quantities. (b) Find the area of a standard sheet of paper that is 8.5 inches wide and 11 inches long.

W

L

ISBN 1-256-49082-2

Solution (a) The word times indicates that the length and width should be multiplied. The formula is given by A = LW . (b) A = 11 8.5 = 93.5 square inches

#

Now Try Exercise 97

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

1.1

Putting It All Together
STUDY TIP
Putting It All Together gives a summary of important concepts in each section. Be sure that you have a good understanding of these concepts.

CONCEPT

COMMENTS

EXAMPLES

Natural Numbers Whole Numbers Products and Factors

Sometimes referred to as the counting numbers Includes the natural numbers and 0 When two numbers are multiplied, the result is called the product. The numbers being multiplied are called factors. A natural number greater than 1 whose only factors are itself and 1; there are infinitely many prime numbers. A natural number greater than 1 that is not a prime number; there are infinitely many composite numbers. Every composite number can be written as a product of prime numbers. This unique product is called the prime factorization. Represents an unknown quantity May consist of variables, numbers, operation symbols such as + , - , # , and , , and grouping symbols such as parentheses An equation is a statement that two algebraic expressions are equal. An equation always includes an equals sign. A formula is a special type of equation that expresses a relationship between two or more quantities.

1, 2, 3, 4, 5, p 0, 1, 2, 3, 4, p 6 factor #

7 factor =

42 product Prime Number

2, 3, 5, 7, 11, 13, 17, and 19 are the prime numbers less than 20. 4, 9, 25, 39, 62, 76, 87, 91, 100

Composite Number

Prime Factorization

60 = 2 # 2 # 3 # 5, 84 = 2 # 2 # 3 # 7

Variable Algebraic Expression

x, y, ], A, F, and T x + 3, x , 2] + 5, 12F, y x + y + ], x( y + 5) 2 + 3 = 5, x + 5 = 7, I = 12F, y = 0.1x I = 12F, y = 0.1x, A = LW, F = 3Y

Equation

Formula

ISBN 1-256-49082-2 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.1 NUMBERS, VARIABLES, AND EXPRESSIONS

9

1.1

Exercises
29. 32 31. 39 33. 294 35. 300 30. 100 32. 51 34. 175 36. 455

CONCEPTS AND VOCABULARY

1. The _____ numbers comprise the counting numbers. 2. A whole number is either a natural number or the number . 3. The factors of a prime number are itself and .

4. A natural number greater than 1 that is not a prime number is called a(n) _____ number. 5. Because 3 # 6 = 18, the numbers 3 and 6 are of 18. is a special type of equation that expresses 6. A(n) a relationship between two or more quantities. 7. A symbol or letter used to represent an unknown quantity is called a(n) . 8. Equations always contain a(n) .

Exercises 37–44: State whether the given quantity could accurately be described by the whole numbers. 37. The population of a country 38. The cost of a gallon of gasoline in dollars 39. A student’s grade point average 40. The Fahrenheit temperature in Antarctica 41. The number of apps on an iPad 42. The number of students in a class 43. The winning time in a 100-meter sprint 44. The number of bald eagles in the United States
ALGEBRAIC EXPRESSIONS, FORMULAS, AND EQUATIONS

9. When one number is added to another number, the result is called the . 10. When one number is multiplied by another number, the result is called the . 11. The result of dividing one number by another is called the . 12. The result of subtracting one number from another is called the .
PRIME NUMBERS AND COMPOSITE NUMBERS

Exercises 45–54: Evaluate the expression for the given value of x. 45. 3x 47. 9 - x 49. x 8 x = 5 x = 4 x = 32 x = 5 46. x + 10 48. 13x 50. 5 (x - 3) x = 8 x = 0 x = 8 x = 3 x = 2

Exercises 13–24: Classify the number as prime, composite, or neither. If the number is composite, write it as a product of prime numbers. 13. 4 15. 1 17. 29 19. 92 21. 225 23. 149
ISBN 1-256-49082-2

51. 3(x + 1)

52. 7(6 - x) 6 54. 3 - a b x

14. 36 16. 0 18. 13 20. 69 22. 900 24. 101

x 53. a b + 1 x = 6 2

Exercises 55–60: Evaluate the expression for the given values of x and y. 55. x + y 56. 5xy x 57. 6 # y 58. y - x 59. y(x - 2) 60. (x + y) - 5 x = 8, y = 14 x = 2, y = 3

Exercises 25–36: Write the composite number as a product of prime numbers. 25. 6 27. 12 26. 8 28. 20

x = 8, y = 4 x = 8, x = 5, y = 11 y = 3

x = 6, y = 3

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Exercises 61–64: Find the value of y for the given value of x. 61. y = x + 5 62. y = x # x 63. y = 4x 64. y = 2(x - 3) x = 0 x = 7 x = 7 x = 3

83. The product of a car’s speed and traveling time 84. The difference between 220 and a person’s heart rate 85. A number plus seven, all divided by a different number 86. One-fourth of a number increased by one-tenth of a different number
APPLICATIONS

Exercises 65–68: Find the value of F for the given value of ]. 65. F = ] - 5 66. F = ] 4 ] = 12 ] = 40 ] = 6 ] = 5

87. Dollars to Pennies Write a formula that converts D dollars to P pennies. 88. Quarters to Nickels Write a formula that converts Q quarters to N nickels. 89. Yards to Feet Make a table of values that converts y yards to F feet. Let y = 1, 2, 3, p , 7. Write a formula that converts y yards to F feet. 90. Gallons to Quarts Make a table of values that converts g gallons to Q quarts. Let g = 1, 2, 3, p , 6. Write a formula that converts g gallons to Q quarts. 91. NASCAR Speeds On the fastest speedways, some NASCAR drivers reach average speeds of 3 miles per minute. Write a formula that gives the number of miles M that such a driver would travel in x minutes. How far would this driver travel in 36 minutes? 92. NASCAR Speeds On slower speedways, some NASCAR drivers reach average speeds of 2 miles per minute. Write a formula that gives the number of miles M that such a driver would travel in x minutes. How far would this driver travel in 42 minutes? 93. Thinking Generally If there are 6 blims in every drog, is the formula that relates B blims and D drogs D = 6B or B = 6D? 94. Heart Beat The resting heart beat of a person is 70 beats per minute. Write a formula that gives the number of beats B that occur in x minutes. How many beats are there in an hour? 95. Cost of album downloads The table lists the cost C of downloading x albums. Write an equation that relates C and x. Albums (x) 1 2 3 4

30 67. F = ] 68. F = ] # ] # ]

Exercises 69–72: Find the value of y for the given values of x and ]. 69. y = 3x] 70. y = x + ] 71. y = x ] x = 2, ] = 0 x = 3, ] = 15

x = 9, ] = 3 x = 9, ] = 1

72. y = x - ]

TRANSLATING WORDS TO EXPRESSIONS

Exercises 73–86: Translate the phrase to an algebraic expression. State what each variable represents. 73. Five more than a number 74. Four less than a number 75. Three times the cost of a soda 76. Twice the cost of a gallon of gasoline 77. The sum of a number and 5 78. The quotient of two numbers 79. Two hundred less than the population of a town 80. The total number of dogs and cats in a city 81. A number divided by six 82. A number divided by another number

Cost (C) $12 $24 $36 $48
ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

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96. Gallons of Water The table lists the gallons G of water coming from a garden hose after m minutes. Write an equation that relates G and m. Minutes (m) Gallons (G) 1 4 2 8 3 12 4 16

WRITING ABOUT MATHEMATICS

99. Give an example in which the whole numbers are not sufficient to describe a quantity in real life. Explain your reasoning. 100. Explain what a prime number is. How can you determine whether a number is prime? 101. Explain what a composite number is. How can you determine whether a number is composite? 102. When are variables used? Give an example.

97. Area of a Rectangle The area of a rectangle equals its length times its width. Find the area of the rectangle shown in the figure.
9 ft 22 ft

98. Area of a Square A square is a rectangle whose length and width have equal measures. Find the area of a square with length 14 inches.

1.2

Fractions
Basic Concepts ● Simplifying Fractions to Lowest Terms ● Multiplication and Division of Fractions ● Addition and Subtraction of Fractions ● An Application

A LOOK INTO MATH N

Historically, natural and whole numbers have not been sufficient for most societies. Early on, the concept of splitting a quantity into parts was common, and as a result, fractions were developed. Today, fractions are used in many everyday situations. For example, there are four quarters in one dollar, so each quarter represents a fourth of a dollar. In this section we discuss fractions and how to add, subtract, multiply, and divide them.

Basic Concepts
NEW VOCABULARY n Lowest terms n Greatest common factor (GCF) n Basic principle of fractions n Multiplicative inverse or reciprocal n Least common denominator (LCD)

If we divide a circular pie into 6 equal slices, as shown in Figure 1.4, then each piece represents one-sixth of the pie and can be represented by the fraction 1 . Five slices of the pie 6 would represent five-sixths of the pie and can be represented by the fraction 5 . 6 The parts of a fraction are named as follows.
Numerator

Denominator

Sometimes we can represent a general fraction by using variables. The fraction a can b represent any fraction with numerator a and denominator b. However, the value of b cannot equal 0, which is denoted b 0. (The symbol means “not equal to.”)
ISBN 1-256-49082-2

NOTE: The fraction bar represents division. For example, the fraction 1 represents the 2 result when 1 is divided by 2, which is 0.5. We discuss this concept further in Section 1.4.
Figure 1.4

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

Q
Q

5 Q 6

Fraction bar

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CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 1

Identifying numerators and denominators
Give the numerator and denominator of each fraction. ac x - 5 6 (b) (c) (a) 13 b y + z
Solution (a) The numerator is 6, and the denominator is 13. (b) The numerator is ac, and the denominator is b. (c) The numerator is x - 5, and the denominator is y + z.
Now Try Exercise 2

Simplifying Fractions to Lowest Terms
Consider the amount of pizza shown in each of the three pies in Figure 1.5. The first pie is cut into sixths with three pieces remaining, the second pie is cut into fourths with two pieces remaining, and the third pie is cut into only two pieces with one piece remaining. In all three cases half a pizza remains.

3 6

2 4

1 2

Figure 1.5

READING CHECK
• How can you tell if a fraction is written in lowest terms?

Figure 1.5 illustrates that the fractions 3, 2 , and 1 are equal. The fraction 1 is in lowest 6 4 2 2 terms because its numerator and denominator have no factors in common, whereas the fractions 3 and 2 are not in lowest terms. In the fraction 3 , the numerator and denominator have 6 4 6 a common factor of 3, so the fraction can be simplified as follows. 3 1#3 = # 6 2 3 1 = 2
Factor out 3. a#c a = b#c b

STUDY TIP
A positive attitude is important. The first step to success is believing in yourself.

To simplify 3 , we used the basic principle of fractions: The value of a fraction is 6 unchanged if the numerator and denominator of the fraction are multiplied (or divided) by the same nonzero number. We can also simplify the fraction 2 to 1 by using the basic prin4 2 ciple of fractions. 2 1#2 1 = # = 4 2 2 2

READING CHECK
• What is the greatest common factor for two numbers?

When simplifying fractions, we usually factor out the greatest common factor of the numerator and the denominator. The greatest common factor (GCF) of two or more numbers is the largest factor that is common to those numbers. For example, to simplify 27 , we 36 first find the greatest common factor of 27 and 36. The numbers 3 and 9 are common factors of 27 and 36 because 3 # 9 = 27 and 3 # 12 = 36, and 9 # 3 = 27 and 9 # 4 = 36.

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

13

However, the greatest common factor is 9 because it is the largest number that divides evenly into both 27 and 36. The fraction 27 simplifies to lowest terms as 36 3#9 3 27 = # = . 36 4 9 4

SIMPLIFYING FRACTIONS
To simplify a fraction to lowest terms, factor out the greatest common factor c in the numerator and in the denominator. Then apply the basic principle of fractions: a#c a # c = b. b

NOTE: This principle is true because multiplying a fraction by c or 1 does not change the c value of the fraction.

The greatest common factor for two numbers is not always obvious. The next example demonstrates two different methods that can be used to find the GCF.

EXAMPLE 2

Finding the greatest common factor
Find the greatest common factor (GCF) for each pair of numbers. (a) 24, 60 (b) 36, 54
Solution (a) One way to determine the greatest common factor is to find the prime factorization of each number using factor trees, as shown in Figure 1.6.
24 2 2 2 2 2 2 12 6 3 2 2 2 2 2 3 60 30 15 5

Figure 1.6 Prime Factorizations of 24 and 60

The prime factorizations have two 2s and one 3 in common. 24 = 2 # 2 # 2 # 3 60 = 2 # 2 # 3 # 5
ISBN 1-256-49082-2

Thus the GCF of 24 and 60 is 2 # 2 # 3 = 12. (b) Another way to find the greatest common factor is to create a factor step diagram. Working downward from the top, the numbers in each step are found by dividing the two numbers in the previous step by their smallest common prime factor. The process continues until no common prime factor can be found. A factor step diagram for 36 and 54 is shown in Figure 1.7 on the next page.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Smallest common prime factor

2 3 3

36 18 6 2

54 27 9 3

Start: Divide 36 and 54 by 2. Divide 18 and 27 by 3. Divide 6 and 9 by 3. Stop: No common prime factors

Figure 1.7 Factor Step Diagram for 36 and 54

The greatest common factor is the product of the prime numbers along the side of the diagram. So the GCF of 36 and 54 is 2 # 3 # 3 = 18.
Now Try Exercises 15, 17

EXAMPLE 3

Simplifying fractions to lowest terms
Simplify each fraction to lowest terms. (a) 24 60 (b) 42 105

Solution (a) From Example 2(a), the GCF of 24 and 60 is 12. Thus CRITICAL THINKING
Describe a situation from everyday life in which fractions would be needed.

2 2 # 12 24 = # = . 60 5 12 5 (b) The prime factorizations of 42 and 105 are 42 = 2 # 3 # 7 and 105 = 5 # 3 # 7. The GCF of 42 and 105 is 3 # 7 = 21. Thus 2 2 # 21 42 = # = . 105 5 21 5
Now Try Exercises 29, 33

MAKING CONNECTIONS
Simplifying Fractions in Steps

Sometimes a fraction can be simplified to lowest terms in multiple steps. By using any common factor that is not the GCF, a new fraction in lower terms will result. This new fraction may then be simplified using a common factor of its numerator and denominator. If this process is continued, the result will be the given fraction simplified to lowest terms. The fraction in Example 3(b) could be simplified to lowest terms in two steps. 42 14 # 3 2#7 14 2 = # 3 = 35 = 5 # 7 = 5 105 35

Multiplication and Division of Fractions
ISBN 1-256-49082-2

Suppose we cut half an apple into thirds, as illustrated in Figure 1.8. Then each piece represents one-sixth of the original apple. One-third of one-half is described by the product
Figure 1.8

1 3

#1
2

=

1 . 6

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

15

This example demonstrates that the numerator of the product of two fractions is found by multiplying the numerators of the two fractions. Similarly, the denominator of the product of two fractions is found by multiplying the denominators of the two fractions. For example, the product of 2 and 5 is 7 3
Multiply numerators.

2 3

#5
7

=

2 # 5 Q 10 = . 3 # 7 Q 21
Multiply denominators.

NOTE: The word “of ” in mathematics often indicates multiplication. For example, the phrases “one-fifth of the cookies,” “twenty percent of the price,” and “half of the money” all suggest multiplication.

MULTIPLICATION OF FRACTIONS c The product of a and d is given by b

a b where b and d are not 0.

#c

d

=

ac , bd

EXAMPLE 4

Multiplying fractions
Multiply. Simplify the result when appropriate. (a) 4 5

#6
7

(b)

8 9

#3
4

(c) 3

#5
9

(d)

x y

#

z 3

Solution

(a) (b)

4 5 8 9

#6
7 4

= =

4#6 24 # 7 = 35 5 8#3 24 # 4 = 36 ; the GCF of 24 and 36 is 12, so 9 2 # 12 2 24 = # = . 36 3 12 3

#3

3 (c) Start by writing 3 as . 1 3

#5
9

=

3 1

#5
9

=

3#5 15 #9= 9 1

The GCF of 15 and 9 is 3, so 5#3 5 15 = # = . 9 3 3 3

ISBN 1-256-49082-2

(d)

x y

#z

3

=

x#z xz = y#3 3y

When we write the product of a variable and a number, such as y # 3, we typically write the number first, followed by the variable. That is, y # 3 = 3y.
Now Try Exercises 35, 39, 43, 47

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

16

CHAPTER 1 INTRODUCTION TO ALGEBRA

MAKING CONNECTIONS
Multiplying and Simplifying Fractions

When multiplying fractions, sometimes it is possible to change the order of the factors to rewrite the product so that it is easier to simplify. In Example 4(b) the product could be written as 8 9 Instead of simplifying fied first.

#3
4

=

3#8 3 #4=9 9

#8
4

=

1 3

#2
1

=

2 . 3 and 8 were simpli4

3 24 36 , which contains larger numbers, the fractions 9

EXAMPLE 5

Finding fractional parts
Find each fractional part. (a) One-fifth of two-thirds (b) Four-fifths of three-sevenths (c) Three-fifths of ten

Solution (a) The phrase “one-fifth of” indicates multiplication by one-fifth. The fractional part is

1 5 (b) (c)
4 5 3 5

#2
3

=

1#2 2 # 3 = 15 . 5

# = = # 10 = 3 # 10 = 30 5 1 5

3 7

4 5

# #

3 7

12 35

= 6

Now Try Exercises 49, 51, 53

N REAL-WORLD CONNECTION Fractions can be used to describe particular parts of the U.S. population. In the next application we use fractions to find the portion of the population that has completed 4 or more years of college.

EXAMPLE 6

Estimating college completion rates
8 About 17 of the U.S. population over the age of 25 has a high school diploma. About 25 of 20 those people have gone on to complete 4 or more years of college. What fraction of the U.S. population over the age of 25 has completed 4 or more years of college? (Source: U.S. Census Bureau.)

TABLE 1.4 Numbers and Their Reciprocals

Number 3
1 4 3 2 21 37

Reciprocal
1 3

Solution 8 We need to find 25 of 17 . 20 8 17 136 34 4 34 8 17 = = = = . 25 20 25 20 500 125 4 125

#

#

#

#

#

34 About 125 of the U.S. population over the age of 25 has completed 4 or more years of college.

4
2 3 37 21

Now Try Exercise 117

ISBN 1-256-49082-2

The multiplicative inverse, or reciprocal, of a nonzero number a is 1 . Table 1.4 lists a several numbers and their reciprocals. Note that the product of a number and its reciprocal is always 1. For example, the reciprocal of 2 is 1 , and their product is 2 # 1 = 1. 2 2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

17

N REAL-WORLD CONNECTION Suppose that a group of children wants to buy gum from a gum ball machine that costs a half dollar for each gum ball. If the caregiver for the children has 4 dollars, then the number of gum balls that can be bought equals the number of half dollars that there are in 4 dollars. Thus 8 gum balls can be bought. This calculation is given by 4 , 1 . To divide a number by a fraction, multiply the num2 ber by the reciprocal of the fraction.

4 ,

1 = 4 2 = = 4 1 8 1

#2
1 1

Multiply by the reciprocal of 1 . 2 Write 4 as 4 . 1 Multiply the fractions. a 1

#2

= 8

= a for all values of a.

Justification for multiplying by the reciprocal when dividing two fractions is a a c b b a , = = c b d c d d

#d #

a # d c b c a = = d 1 b c

# d. c These results are summarized as follows.

DIVISION OF FRACTIONS
For real numbers a, b, c, and d, with b, c, and d not equal to 0, c a a , = b d b

# d. c EXAMPLE 7

Dividing fractions
Divide. Simplify the result when appropriate. (a) 3 1 , 3 5 (b) 4 4 , 5 5 (c) 5 , 10 3 (d) y x , z 2

Solution (a) To divide 1 by 3 , multiply 1 by 5 , which is the reciprocal of 3 . 3 3 3 5 5 3 1 5 1 5 5 1 , = = = 3 5 3 3 3 3 9

#

# #

# (b) 4 , 4 = 4 # 5 = 4 # 5 = 20 = 1. Note that when we divide any nonzero number by 5 20 5 5 4 5 4 itself, the result is 1.
ISBN 1-256-49082-2

3 (c) 5 , 10 = 5 # 10 = 15 = 3 3 1 10 2 y x z x xz = # = (d) , z 2 2 y 2y

Now Try Exercises 59, 63, 67, 71

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

NOTE: In Example 7(c) the answer 3 is an improper fraction, which could be written as 2 the mixed number 1 1 . However, in algebra, fractions are often left as improper fractions. 2 TECHNOLOGY NOTE
When entering a mixed number into a calculator, it is usually easiest first to convert the mixed number to an improper fraction. For example, enter 2 2 as 8 . Otherwise, enter 2 2 as 2 + 2 . See 3 3 3 3 the accompanying figure.
8/ 3 2 2/3 2.666666667 2.666666667

EXAMPLE 8

Writing a problem
Describe a problem for which the solution could be found by dividing 5 by 1 . 6
Solution One possible problem could be stated as follows. If five pies are each cut into sixths, how many pieces of pie are there? See Figure 1.9.

Figure 1.9 Five Pies Cut into Sixths

Now Try Exercise 121

Addition and Subtraction of Fractions
FRACTIONS WITH LIKE DENOMINATORS Suppose that a person cuts a sheet of paper into eighths. If that person picks up two pieces and another person picks up three pieces, then together they have 3 5 2 + = 8 8 8 of a sheet of paper, as illustrated in Figure 1.10. When the denominator of one fraction is the same as the denominator of a second fraction, the sum of the two fractions can be found by adding their numerators and keeping the common denominator. Similarly, if someone picks up 5 pieces of paper and gives 2 away, then that person has 5 2 3 = 8 8 8 of a sheet of paper. To subtract two fractions with common denominators, subtract their numerators and keep the common denominator.

Figure 1.10

ADDITION AND SUBTRACTION OF FRACTIONS
To add or subtract fractions with a common denominator d, use the equations b a + b a + = d d d where d is not 0. and a b a - b = , d d d
ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

19

EXAMPLE 9

Adding and subtracting fractions with common denominators
Add or subtract as indicated. Simplify your answer to lowest terms when appropriate. (a) 12 5 + 13 13 (b) 11 5 8 8

Solution (a) Because the fractions have a common denominator, add the numerators and keep the common denominator.

5 12 5 12 17 + = = 13 13 13 13 (b) Because the fractions have a common denominator, subtract the numerators and keep the common denominator. 5 11 5 6 11 = = 8 8 8 8 The fraction 6 can be simplified to 3. 8 4
Now Try Exercise 75

FRACTIONS WITH UNLIKE DENOMINATORS Suppose that one person mows half a large lawn while another person mows a fourth of the lawn. To determine how much they mowed together, we need to find the sum 1 + 1 . See Figure 1.11(a). Before we can add 2 4 fractions with unlike denominators, we must write each fraction with a common denomina? tor. The least common denominator of 2 and 4 is 4. Thus we need to write 1 as 4 by multi2 plying the numerator and denominator by the same nonzero number.
1 + 1 2 4 (a)

1 1 = 2 2 = 2 4

#2
2

Multiply by 1. Multiply fractions.

Now we can find the needed sum. 1 2 1 3 1 + = + = 2 4 4 4 4
2 + 1 = 3 4 4 4 (b)

Figure 1.11

Together the two people mow three-fourths of the lawn, as illustrated in Figure 1.11(b). To add or subtract fractions with unlike denominators, we first find the least common denominator for the fractions. The least common denominator (LCD) for two or more fractions is the smallest number that is divisible by every denominator.

FINDING THE LEAST COMMON DENOMINATOR (LCD)
READING CHECK
ISBN 1-256-49082-2

STEP 1: STEP 2:

Find the prime factorization for each denominator. List each factor that appears in one or more of the factorizations. If a factor is repeated in any of the factorizations, list this factor the maximum number of times that it is repeated. The product of this list of factors is the LCD.

• What is the LCD and why is it needed?
STEP 3:

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

The next example demonstrates two different methods that can be used to find the LCD. In part (a), the LCD is found using the three-step method shown on the previous page, and in part (b), a factor step diagram is used instead.

EXAMPLE 10

Finding the least common denominator
Find the LCD for each set of fractions. (a) 5 3 , 6 4 (b) 7 5 , 36 54

Solution (a) STEP 1: For the fractions 5 and 3 the prime factorizations of the denominators are 6 4 6 = 2 3 and 4 = 2 2.

#

#

List the factors: 2, 2, 3. Note that, because the factor 2 appears a maximum of two times, it is listed twice. STEP 3: The LCD is the product of this list, or 2 # 2 # 3 = 12.
STEP 2:

NOTE: Finding an LCD is equivalent to finding the smallest number that each denominator divides into evenly. Both 6 and 4 divide into 12 evenly, and 12 is the smallest such number. Thus 12 is the LCD for 5 and 3 . 6 4

(b) The same factor step diagram that was used in Example 2(b) to find the GCF of 36 and 7 5 54 can be used to find the LCD for 36 and 54 ; however, the final step differs slightly. As in Example 2(b), we find the numbers in each step by dividing the two numbers in the previous step by their smallest common prime factor. The process continues until no common prime factor can be found, as shown in Figure 1.12.

Smallest common prime factor

2 3 3

36 18 6 2

54 27 9 3

Start: Divide 36 and 54 by 2. Divide 18 and 27 by 3. Divide 6 and 9 by 3. Stop: No common prime factors

Figure 1.12 Factor Step Diagram for Finding the LCD

The process for finding the LCD differs from that used to find the GCF in that we find the LCD by multiplying not only the numbers along the side of the diagram but also the numbers at the bottom of the diagram. The LCD is 2 # 3 # 3 # 2 # 3 = 108.
Now Try Exercises 79, 83

Once the LCD has been found, the next step in the process for adding or subtracting fractions with unlike denominators is to rewrite each fraction with the LCD.

EXAMPLE 11

Rewriting fractions with the LCD
Rewrite each set of fractions using the LCD. (a) 5 3 , 6 4 (b) 7 5 , 12 18

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

21

Solution (a) From Example 10(a) the LCD is 12. To write 5 with a denominator of 12, we multiply 6 the fraction by 1 in the form 2 . 2

5 6

#2
2

=

5#2 10 = 6#2 12 3#3 9 = 4#3 12

To write 3 with a denominator of 12, we multiply the fraction by 1 in the form 3 . 4 3 3 4

#3
3

=

9 Thus 5 can be rewritten as 10 and 3 can be rewritten as 12 . 6 12 4 7 (b) From Example 10(b) the LCD is 36. To write 12 with a denominator of 36, multiply the 3 5 fraction by 3 . To write 18 with a denominator of 36, multiply the fraction by 2 . 2

7 12

#3
3

=

7#3 21 = 12 # 3 36

and

5 18

#2
2

=

5#2 10 = 18 # 2 36

7 5 Thus 12 can be rewritten as 21 and 18 can be rewritten as 10 . 36 36

Now Try Exercises 89, 93

The next example demonstrates how the concepts shown in the last two examples can be used to add or subtract fractions with unlike denominators.

EXAMPLE 12

Adding and subtracting fractions with unlike denominators
Add or subtract as indicated. Simplify your answer to lowest terms when appropriate. (a) 3 5 + 6 4 (b) 7 5 12 18 (c) 1 2 7 + + 4 5 10

Solution (a) From Example 10(a) the LCD is 12. Begin by writing each fraction with a denominator of 12, as demonstrated in Example 11(a).

5 3 3 5 2 + = # + 6 4 6 2 4 10 9 = + 12 12 = = 10 + 9 12 19 12

#3
3

Change to LCD of 12. Multiply the fractions. Add the numerators. Simplify.

(b) Using Example 10(b) and Example 11(b), we perform the following steps. 5 5 7 # 3 7 = 12 18 12 3 18 21 10 = 36 36
ISBN 1-256-49082-2

#2
2

Change to LCD of 36. Multiply the fractions. Subtract the numerators. Simplify.

= =

21 - 10 36 11 36

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

(c) The LCD for 4, 5, and 10 is 20. 2 7 1 1 + + = 4 5 10 4 = = =

#5
5

+

2 5

#4
4

+

7 10

#2
2

Change to LCD of 20. Multiply the fractions. Add the numerators. Simplify.

8 14 5 + + 20 20 20 5 + 8 + 14 20 27 20

Now Try Exercises 97, 103, 107

An Application
The next example illustrates a situation where fractions occur in a real-world application.

EXAMPLE 13

Applying fractions to carpentry
A board measuring 35 3 inches is cut into four equal pieces, as depicted in Figure 1.13. Find 4 the length of each piece.

35 4 in.

3

Figure 1.13

Solution Begin by writing 353 as the improper fraction 143 (because 4 35 + 3 = 143). Because 4 4 the board is to be cut into four equal parts, the length of each piece should be

#

143 143 , 4 = 4 4
Now Try Exercise 111

#1
4

=

143 , or 16

8

15 inches. 16

1.2
Fraction

Putting It All Together
COMMENTS EXAMPLES

CONCEPT

The fraction has numerator a and denominator b. A fraction is in lowest terms if the numerator and denominator have no factors in common.

a b

The fraction has numerator xy and denominator 2.
ISBN 1-256-49082-2

xy 2

Lowest Terms

The fraction 3 is in lowest terms because 8 3 and 8 have no factors in common.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

23

CONCEPT

COMMENTS

EXAMPLES

Greatest Common Factor (GCF) Simplifying Fractions

The GCF of two numbers equals the largest number that divides into both evenly. Use the principle a a#c = b#c b to simplify fractions, where c is the GCF of the numerator and denominator.

The GCF of 12 and 18 is 6 because 6 is the largest number that divides into 12 and 18 evenly. The GCF of 24 and 32 is 8, so 24 3#8 3 = # = . 32 4 8 4 The GCF of 20 and 8 is 4, so 20 5#4 5 = # = . 8 2 4 2
3 The reciprocals of 5 and 4 are 1 and 4 , 3 5 1 respectively, because 5 # 5 = 1 and 3 # 4 4 3 = 1.

Multiplicative Inverse or Reciprocal Multiplication and Division of Fractions

The reciprocal of a is b , where a and b a b are not zero. The product of a number and its reciprocal is 1. a b

#c

d

=

ac bd

3 5 d c and

#4
9

=

12 4 = 45 15

a c a , = b d b Addition and Subtraction of Fractions with Like Denominators

#

3 6 3 , = 2 5 2

#5
6

=

15 5 = 12 4 and

a c a + c + = d d d a c a - c = d d d

3 4 3 + 4 7 + = = 5 5 5 5

17 11 17 - 11 6 1 = = = 12 12 12 12 2
5 7 The LCD of 12 and 18 is 36 because 36 is the smallest number that both 12 and 18 divide into evenly. 7 The LCD of 3 and 10 is 20. 4 7 3 5 7 3 + = # + 4 10 4 5 10

Least Common Denominator (LCD) Addition and Subtraction of Fractions with Unlike Denominators

The LCD of two fractions equals the smallest number that both denominators divide into evenly. First write each fraction with the least common denominator. Then add or subtract the numerators.

#2
2

= =

15 14 + 20 20 29 20

1.2
ISBN 1-256-49082-2

Exercises
2. In the fraction 11 the numerator is 21 . denominator is 3. In the fraction a , the variable b cannot equal b and the .

CONCEPTS AND VOCABULARY

1. A small pie is cut into 4 equal pieces. If someone eats 3 of the pieces, what fraction of the pie does the person eat? What fraction of the pie remains?

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

7 4. The fraction 12 is in terms because 7 and 12 have no factors in common.

37. 5 # 3 3 5 39. 5 # 18 6 25 41. 4 # 3 5 43. 2 # 3 8 45. 47. x y a b

38. 21 # 32 32 21
3 40. 7 # 14 9 7 9

5. (True or False?) The numerator of the product of two fractions is found by multiplying the numerators of the two fractions. 6. (True or False?) The denominator of the sum of two fractions is found by adding the denominators of the two fractions. ac = 7. bc 8. In the phrase “two-fifths of one-third,” the word of indicates that we the fractions 2 and 1 . 3 5 9. What is the reciprocal of a, provided a 10. To divide 11. 13. a b
3 4

42. 5

#

1 44. 10 # 100

#y x 46. 48.

x y 5 8

#y
]

#3
2

# 4x
5y

Exercises 49–54: Find the fractional part. 49. One-fourth of three-fourths 50. Three-sevenths of nine-sixteenths 51. Two-thirds of six 52. Three-fourths of seven 53. One-half of two-thirds 54. Four-elevenths of nine-eighths Exercises 55–58: Give the reciprocal of each number. 55. (a) 5 56. (a) 3 57. (a) 1 2 58. (a) 1 5 (b) 7 (b) 2 (b) 1 9 (b) 7 3 (c) 4 7 (c) 6 5
12 (c) 101

0?

by 5, multiply

3 4

by

.

#c

d

=

12. 14.

a c , = b d a c = b b

a c + = b b

LOWEST TERMS

Exercises 15–20: Find the greatest common factor. 15. 4, 12 17. 50, 75 19. 100, 60, 70 16. 3, 27 18. 45, 105 20. 36, 48, 72

(d) 9 8 (d) 3 8 (d) 31 17 (d) 63 29

(c) 23 64

Exercises 21–24: Use the basic principle of fractions to simplify the expression.

# 21. 3 # 4 5 4
23. 3 8 8 5

# 22. 2 # 7 9 7
24. 7 #16 16 3

Exercises 59–74: Divide and simplify to lowest terms when appropriate. 59. 1 , 1 2 3 61. 3 , 1 4 8 63. 4 , 4 3 3 65. 32 , 8 27 9 67. 10 , 5 6
9 69. 10 , 3

# #

#

60. 3 , 1 4 5
3 62. 6 , 14 7

Exercises 25–34: Simplify the fraction to lowest terms. 25. 4 8 27. 10 25 29. 12 36 31. 12 30 33. 19 76
4 26. 12 5 28. 20

64. 12 , 4 7 21
8 2 66. 15 , 25

68. 8 , 4 3 70. 32 , 16 27 72. 74. 3a 3 , c b x x , 3y 3

30. 16 24
60 32. 105

34. 17 51

71. 73.

a 2 , b b x x , y y

MULTIPLICATION AND DIVISION OF FRACTIONS

ISBN 1-256-49082-2

Exercises 35–48: Multiply and simplify to lowest terms when appropriate. 35. 3 # 1 4 5 36. 3 # 5 2 8

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.2 FRACTIONS

25

ADDITION AND SUBTRACTION OF FRACTIONS

Exercises 75–78: Add or subtract. Write each answer in lowest terms.
5 1 75. (a) 12 + 12 5 1 (b) 12 - 12

76. (a) 3 + 1 2 2
7 77. (a) 18 + 29 29 5 2 78. (a) 33 + 33

(b) 3 - 1 2 2
7 (b) 18 - 29 29 5 2 (b) 33 - 33

110. American Flag The blue rectangle containing the stars on an American flag is called the union. On an official American flag, the width of the union 7 should be 13 of the width of the flag. If an official flag has a width of 32 1 inches, what is the width of 2 the union? 111. Carpentry A board measuring 64 5 inches is cut in 8 half. Find the length of each half. 112. Cutting Rope A rope measures 15 1 feet and needs 2 to be cut in four equal parts. Find the length of each piece. 113. Geometry Find the area of the triangle shown with base 12 yards and height 3 yard. (Hint: The area 3 4 of a triangle equals half the product of its base and height.)
3 yd 4 1 3 yd
2

Exercises 79–88: Find the least common denominator.
2 79. 4, 15 9 3 81. 2, 15 5 1 80. 11, 1 2 8 82. 21, 3 7 5 84. 1, 12 9

83. 1, 5 6 8 85. 1, 1, 1 2 3 4
1 87. 1, 3, 12 4 8

86. 2, 2, 1 5 3 6
2 7 1 88. 15, 20, 30

Exercises 89–96: Rewrite each set of fractions with the least common denominator. 89. 1, 2 2 3
5 91. 7, 12 9 1 7 93. 16, 12

90. 3, 1 4 5
5 92. 13, 1 2 5 1 94. 18, 24

114. Geometry Find the area of the rectangle shown.
5 ft 4 1 ft 2

95.

1 3 5 3, 4, 6

96.

4 2 3 15 , 9 , 5

Exercises 97–108: Add or subtract. Write your answer in lowest terms.
3 97. 5 + 16 8 2 98. 1 + 15 9 25 24 7 8 4 5 7 8 1 4

115. Distance Use the map to find the distance between Smalltown and Bigtown by traveling through Middletown.
Middletown
1 3

99.

-

100. 102.

-

2 101. 11 + 35 14 5 1 103. 12 - 18

4 + 15

3 2 mi

4 4 mi Bigtown

7 9 104. 20 - 30

Smalltown

105. 107.

3 100 7 8

+
1 6

1 300

-

1 200

106. 108.

43 36 9 40

+ -

4 9 3 50

+ -

1 4 1 100

-

+

5 12

116. Distance An athlete jogs 13 miles, 5 3 miles, and 8 4 3 5 miles. In all, how far does the athlete jog? 8
4 117. Vegetarian Diets About 125 of U.S. adults follow a vegetarian diet. Within this part of the population, 1 200 is vegan. What fraction of the adult U.S. population is vegan? (Source: Vegetarian Times—2010.)

APPLICATIONS

109. American Flag According to Executive Order 10834, 9 the length of an official American flag should be 1 10 times the width. If an official flag has a width that measures 2 1 feet, find its length. 2
ISBN 1-256-49082-2

118. Women College Students In 2010, about 29 of all 50 college students were women. Of these women, about 13 were part-time students. What fraction of 20 college students were women who were part-time students? (Source: National Center for Education Statistics.)

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

119. Accidental Deaths For the age group 15 to 24, motor vehicle accidents account for 31 of all acci42 31 dental deaths, and firearms account for 1260 of all accidental deaths. What fraction of all accidental deaths do vehicle accidents and firearms account for? (Source: National Safety Council.) 120. Illicit Drug Use For the age group 18 to 25, the fraction of people who used illicit drugs during their lifetime was 3 , whereas the fraction who used illicit 5 7 drugs during the past year was 20 . What fraction of

this population has used illicit drugs but not during the past year? (Source: Department of Health and Human
Services.)

WRITING ABOUT MATHEMATICS

121. Describe a problem in real life for which the solution could be found by multiplying 30 by 1 . 4 122. Describe a problem in real life for which the solution could be found by dividing 2 1 by 1 . 2 3

SECTIONS 1.1 AND 1.2

Checking Basic Concepts
7. Simplify each fraction to lowest terms. (a) 25 (b) 26 39 35 8. Give the reciprocal of 4 . 3 9. Evaluate each expression. Write each answer in lowest terms. (a) 2 # 3 (b) 5 , 10 3 4 6 3
3 1 (c) 10 + 10

1. Classify each number as prime, composite, or neither. If a number is composite, write it as a product of primes. (a) 19 (b) 28 (c) 1 (d) 180 2. Evaluate (x 10 2) for x = 3. + 3. Find y for x = 5 if y = 6x. 4. Translate the phrase “a number x plus five” into an algebraic expression. 5. Write a formula that converts F feet to I inches. 6. Find the greatest common factor. (a) 3, 18 (b) 40, 72

(d) 3 - 1 4 6

10. A recipe needs 12 cups of flour. How much flour 3 should be used if the recipe is doubled?

1.3

Exponents and Order of Operations
Natural Number Exponents ● Order of Operations ● Translating Words to Expressions

A LOOK INTO MATH N

If there are 15 energy drinks on a store shelf and 3 boxes of 24 drinks each in the storage room, then the total number of energy drinks is given by the expression 15 + 3 # 24. How would you find the total number of energy drinks? Is the total 15 3 # 24 18 # 24 432, or 15 + 3 ~ 24 15 + 72 87?

(add first, then multiply)

(multiply first, then add)

In this section, we discuss the order of operations agreement, which can be used to show that the expression 15 + 3 # 24 evaluates to 87.

NEW VOCABULARY n Exponential expression n Base n Exponent

Natural Number Exponents
In elementary school you learned addition. Later, you learned that multiplication is a fast way to add. For example, rather than adding 3 + 3 + 3 + 3, v 4 terms

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1.3 EXPONENTS AND ORDER OF OPERATIONS

27

we can multiply 3 # 4. Similarly, exponents represent a fast way to multiply. Rather than multiplying 3 # 3 # 3 # 3,
4

v 4 factors

we can evaluate the exponential expression 3 . We begin by discussing natural numbers as exponents.
READING CHECK
• What type of expression represents a fast way to multiply?

The area of a square equals the length of one of its sides times itself. If the square is 5 inches on a side, then its area is 5 # 5 = 52 = 25 square inches.
Q Base Q
2

Exponent

The expression 5 is an exponential expression with base 5 and exponent 2. Exponential expressions occur in a variety of applications. For example, suppose an investment doubles 3 times. Then, the calculation
3 2#2#2 3= 2 = 8

Factors

shows that the final value is 8 times as large as the original investment. For example, if $10 doubles 3 times, it becomes $20, $40, and finally $80, which is 8 times as large as $10. Table 1.5 contains examples of exponential expressions.
TABLE 1.5 Exponential Expressions

Repeated Multiplication 2#2#2#2 4#4#4 9#9
3 Squared
1 2

Exponential Expression 24 4
3

Q

Exponent 3

Base 2 4 9
1 2

Exponent 4 3 2 1 5

92

1121 2 b 5

b#b#b#b#b

b

Read 92 as “9 squared,” 43 as “4 cubed,” and 24 as “2 to the fourth power.” The terms squared and cubed come from geometry. If the length of a side of a square is 3, then its area is
(a)

3 # 3 = 32 = 9 square units, as illustrated in Figure 1.14(a). Similarly, if the length of an edge of a cube is 3, then its volume is 3 # 3 # 3 = 33 = 27

3 Cubed

ISBN 1-256-49082-2

cubic units, as shown in Figure 1.14(b).
NOTE: The expressions 32 and 33 can also be read as “3 to the second power” and “3 to the third power,” respectively. In general, the expression x n is read as “x to the nth power” or “the nth power of x.”

(b)

Figure 1.14

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28

CHAPTER 1 INTRODUCTION TO ALGEBRA

EXPONENTIAL NOTATION
The expression b n , where n is a natural number, means bn = b # b # b g b. 8

#

#

n factors

The base is b and the exponent is n.

EXAMPLE 1

Writing products in exponential notation
Write each product as an exponential expression. (a) 7 # 7 # 7 # 7 (b) 1 4

#1#1
4 4

(c) x # x # x # x # x

Solution (a) Because there are four factors of 7, the exponent is 4 and the base is 7. Thus 7 7 7 7 = 74 . (b) Because there are three factors of 1 , the exponent is 3 and the base is 1 . 4 4 Thus 1 1 1 = 1 1 2 3 . 4 4 4 4 (c) Because there are five factors of x, the exponent is 5 and the base is x. Thus x x x x x = x 5 .

# # #

# #

# # # #

Now Try Exercises 13, 15, 17

EXAMPLE 2

Evaluating exponential notation
Evaluate each expression. (a) 34 (b) 103 3 2 (c) a b 4

Solution (a) The exponential expression 34 indicates that 3 is to be multiplied times itself 4 times.

34 = 5 = 81 3#3#3#3
4 factors
CALCULATOR HELP
To evaluate an exponential expression with a calculator, see Appendix A (page AP-1).

(b) (c)

10 = 10 # 10 # 10 = 1000 2 9 1 3 2 = 3 # 3 = 16 4 4 4
3

Now Try Exercises 25, 27

EXAMPLE 3

Writing numbers in exponential notation
Use the given base to write each number as an exponential expression. Check your results with a calculator, if one is available. (a) 100 (base 10) (b) 16 (base 2) (c) 27 (base 3)
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Solution (a) 100 = 10 10 = 102 (b) 16 = 4 4 = 2 2 2 2 = 24 (c) 27 = 3 9 = 3 3 3 = 33

#

# #

# # # # #

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1.3 EXPONENTS AND ORDER OF OPERATIONS

29

10^2 2^4 3^3

100 16 27

These values are supported in Figure 1.15, where exponential expressions are evaluated with a calculator by using the “^” key. (Note that some calculators may have a different key for evaluating exponential expressions.)
Now Try Exercises 29, 33, 35

Figure 1.15

N REAL-WORLD CONNECTION Computer memory is often measured in bytes, with each byte capable of storing one letter of the alphabet. For example, it takes four bytes to store the word “math” in a computer. Bytes of computer memory are often manufactured in amounts equal to powers of 2, as illustrated in the next example.

EXAMPLE 4

Analyzing computer memory
In computer technology 1 K (kilobyte) of memory equals 210 bytes, and 1 MB (megabyte) of memory equals 220 bytes. Determine whether 1 K of memory equals one thousand bytes and whether 1 MB equals one million bytes.
Solution Figure 1.16 shows that 210 = 1024 and 220 = 1,048,576. Thus 1 K represents slightly more than one thousand bytes, and 1 MB represents more than one million bytes.

2^10 2^20

1024 1048576

Now Try Exercise 75

CRITICAL THINKING
Figure 1.16

One gigabyte of memory equals 230 bytes and is often referred to as 1 billion bytes. If you have a calculator available, determine whether 1 gigabyte is exactly 1 billion bytes.

Order of Operations
10 2 3

When the expression 10 - 2 # 3 is evaluated, is the result
4

8 # 3 = 24 or

10 - 6 = 4?

Figure 1.17

Figure 1.17 shows that a calculator gives a result of 4. The reason is that multiplication is performed before subtraction. Because arithmetic expressions may contain parentheses, exponents, absolute values, and several operations, it is important to evaluate these expressions consistently. (Absolute value will be discussed in Section 1.4.) To ensure that we all obtain the same result when evaluating an arithmetic expression, the following rules are used.

STUDY TIP
Ideas and procedures written in boxes like the one to the right are major concepts in this section. Be sure that you understand them.

ORDER OF OPERATIONS
Use the following order of operations. First perform all calculations within parentheses and absolute values, or above and below the fraction bar. 1. Evaluate all exponential expressions. 2. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right.

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30

CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 5

Evaluating arithmetic expressions
Evaluate each expression by hand. (a) 10 - 4 - 3 (b) 10 - (4 - 3) (c) 5 + 12 3 (d) 4 + 1 2 + 8

Solution (a) There are no parentheses, so we evaluate subtraction from left to right.

10

4 - 3 = 6 - 3 = 3

(b) Note the similarity between this part and part (a). The difference is the parentheses, so subtraction inside the parentheses must be performed first. 10 - (4 (c) We perform division before addition. 5 + 12 = 5 + 4 = 9 3 3) = 10 - 1 = 9

(d) Evaluate the expression as though both the numerator and the denominator have parentheses around them. 1) (4 5 1 4 + 1 = = = 2 + 8 (2 8) 10 2
Now Try Exercises 43, 45, 57

EXAMPLE 6

Evaluating arithmetic expressions
Evaluate each expression by hand. (a) 25 - 4 # 6 (b) 6 + 7 # 2 - (4 - 1) (c) 3 + 32 14 - 2 (d) 5 # 23 - (3 + 2)

Solution (a) Multiplication is performed before subtraction, so evaluate the expression as follows.

25 - 4 # 6 = 25 - 24 = 1

Multiply. Subtract.

(b) Start by performing the subtraction within the parentheses first and then perform the multiplication. Finally, perform the addition and subtraction from left to right. 6 + 7 # 2 - (4 - 1) = 6 + 7 # 2 - 3 Subtract within parentheses. = 6 + 14 - 3 = 20 - 3 = 17 (3 + 32) 3 + 32 = 14 - 2 (14 - 2) (3 + 9) = (14 - 2) = 12 12
Multiply. Add. Subtract.

(c) First note that parentheses are implied around the numerator and denominator.
Insert parentheses. Evaluate the exponent first. Add and subtract. Simplify.

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= 1

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1.3 EXPONENTS AND ORDER OF OPERATIONS

31

(d) Begin by evaluating the expression inside parentheses. 5 # 23 - (3 + 2) = 5 # 23 - 5 = 5#8 - 5 = 40 - 5 = 35
Now Try Exercises 59, 61, 63

Add within parentheses. Evaluate the exponent. Multiply. Subtract.

Translating Words to Expressions
Sometimes before we can solve a problem we must translate words into mathematical expressions. For example, if a cell phone plan allows for 500 minutes of call time each month and 376 minutes have already been used, then “five hundred minus three hundred seventy-six,” or 500 - 376 = 124, is the number of minutes remaining for the month.

EXAMPLE 7

Writing and evaluating expressions
Translate each phrase into a mathematical expression and then evaluate it. (a) Two to the fourth power plus ten (b) Twenty decreased by five times three (c) Ten cubed divided by five squared (d) Sixty divided by the quantity ten minus six
Solution (a) 24 + 10 = 2 2 2 2 + 10 = 16 + 10 = 26 (b) 20 - 5 3 = 20 - 15 = 5 103 1000 (c) 2 = = 40 25 5 (d) Here, the word “quantity” indicates that parentheses should be used.

# # #

#

60 , (10 - 6) = 60 , 4 = 15
Now Try Exercises 67, 69, 73

1.3

Putting It All Together
COMMENTS EXAMPLES n CONCEPT

Exponential Expression

If n is a natural number, then b equals b#b#b g b 8 n factors

#

#

5 72 43 k4

1

= = = =

5, 7 # 7 = 49, 4 # 4 # 4 = 64, and k#k#k#k

and is read “b to the nth power.”
ISBN 1-256-49082-2

Base and Exponent

The base in b n is b and the exponent is n.

74 has base 7 and exponent 4, and x 3 has base x and exponent 3. continued on next page

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32

CHAPTER 1 INTRODUCTION TO ALGEBRA

continued from previous page

CONCEPT

COMMENTS

EXAMPLES

Order of Operations

First, perform all calculations within parentheses and absolute values, or above and below a fraction bar. 1. Evaluate all exponential expressions. 2. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right.

10 , 5 + 3 # 7 = 2 + 21 = 23 27 , (4 - 1)2 = 27 , 32 = 27 , 9 = 3

1.3

Exercises
15. 1 # 1 # 1 # 1 2 2 2 2 . 17. a # a # a # a # a 19. (x + 3) # (x + 3) 20. (x - 4) # (x - 4) # (x - 4) Exercises 21–28: Evaluate each expression. 21. (a) 24 23. (a) 61 . 25. (a) 25 27. (a) 1 2 2 3 1 3 28. (a) 1 10 2
2

CONCEPTS AND VOCABULARY

16. 5 # 5 # 5 # 5 # 5 7 7 7 7 7 18. b # b # b # b

1. Exponents represent a fast way to
5

2. In the expression 2 , there are five factors of being multiplied. 3. In the expression 53 , the number 5 is called the and the number 3 is called the . 4. Use symbols to write “6 squared.” 5. Use symbols to write “8 cubed.” 6. When evaluating the expression 5 + 6 # 2, the result is because is performed before

(b) 42 (b) 16 (b) 103 (b) (b)

22. (a) 32 24. (a) 171 26. (a) 105

(b) 53 (b) 117 (b) 34

7. When evaluating the expression 10 - 23 , the result is because are evaluated before is performed. 8. The expression 10 - 4 - 2 equals subtraction is performed from to because .

1125 2 1421 3

9. (True or False?) The expressions 23 and 32 are equal. 10. (True or False?) The expression 52 equals 5 # 2.
NATURAL NUMBER EXPONENTS

Exercises 29–40: (Refer to Example 3.) Use the given base to write the number as an exponential expression. Check your result if you have a calculator available. 29. 8 31. 25 33. 49 35. 1000
1 37. 16 32 39. 243

(base 2) (base 5) (base 7) (base 10)

30. 9 32. 32 34. 81 36. 256
9 38. 25

(base 3) (base 2) (base 3)
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Exercises 11–20: Write the product as an exponential expression. 11. 3 # 3 # 3 # 3 13. 2 # 2 # 2 # 2 # 2 12. 10 # 10 14. 4 # 4 # 4

(base 4)

1 base 1 2 2 1 base 2 2 3

1 base 3 2 5 1 base 6 2 7

40. 216 343

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1.3 EXPONENTS AND ORDER OF OPERATIONS

33

ORDER OF OPERATIONS

APPLICATIONS

Exercises 41–64: Evaluate the expression by hand. 41. 5 + 4#6 42. 6#7 - 8

75. Flash Memory (Refer to Example 4.) Determine the number of bytes on a 512-MB memory stick. 76. iPod Memory Determine the number of bytes on a 60-GB video iPod. (Hint: One gigabyte equals 230 bytes.) 77. Population by Gender One way to measure the gender balance in a given population is to find the number of males for every 100 females in the population. In 1900, the western region of the United States was significantly out of gender balance. In this region, there were 128 males for every 100 females. (Source:
U.S. Census Bureau.)

43. 6 , 3 + 2 45. 100 - 50 5 47. 10 - 6 - 1 49. 20 , 5 , 2 51. 3 + 24 53. 4 # 23 55. (3 + 2)3 57. 59. 4 + 8 1 + 3 23 4 - 2

44. 20 - 10 , 5 46. 200 + 6 100 48. 30 - 9 - 5 50. 500 , 100 , 5 52. 10 - 32 + 1 54. 100 - 2 # 33 56. 5 # (3 - 2)8 - 5 58. 5 60. 3 + 1 3 - 1

(a) Find an exponent k so that 2k = 128. (b) During this time, how many males were there for every 25 females? 78. Solar Eclipse In early December 2048 there will be a total solar eclipse visible in parts of Botswana. Find an exponent k so that 2k = 2048. (Source: NASA.) 79. Rule of 72 Investors sometimes use the rule of 72 to determine the time required to double an investment. If 72 is divided by the annual interest rate earned on an investment, the result approximates the number of years needed to double the investment. For example, an investment earning 6% annual interest will double in value approximately every 72 , 6 = 12 years. (a) Approximate the number of years required to double an investment earning 9% annual interest. (b) If an investment of $10,000 earns 12% annual interest, approximate the value of the investment after 18 years. 80. Doubling Effect Suppose that a savings account containing $1000 doubles its value every 7 years. How much money will be in the account after 28 years?
WRITING ABOUT MATHEMATICS

10 - 32 2 # 42

61. 102 - (30 - 2 # 5) 1 4 5 + 4 63. a b + 2 3

62. 52 + 3 # 5 , 3 - 1 7 2 6 - 5 64. a b 9 3

TRANSLATING WORDS TO EXPRESSIONS

Exercises 65–74: Translate the phrase into a mathematical expression and then evaluate it. 65. Two cubed minus eight 66. Five squared plus nine 67. Thirty decreased by four times three 68. One hundred plus five times six 69. Four squared divided by two cubed 70. Three cubed times two squared 71. Forty divided by ten, plus two 72. Thirty times ten, minus three 73. One hundred times the quantity two plus three 74. Fifty divided by the quantity eight plus two

81. Explain how exponential expressions are related to multiplication. Give an example. 82. Explain why agreement on the order of operations is necessary.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Group Activity

Working with Real Data

Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond. Converting Temperatures To convert Celsius degrees C to Fahrenheit degrees F, use the formula F = 32 + 9C. 5 This exercise illustrates the importance of understanding the order of operations. (a) Complete the following table by evaluating the formula in the two ways shown. Celsius - 40 C 0 C 5 C 20 C 30 C 100 C 169 F 41 F F = (b) At what Celsius temperature does water freeze? At what Fahrenheit temperature does water freeze? (c) Which column gives the correct Fahrenheit temperatures? Why? (d) Explain why having an agreed order for operations in mathematics is necessary.

1 32

+ 92C 5

F = 32 + 1 9C 2 5

1.4

Real Numbers and the Number Line
Signed Numbers ● Integers and Rational Numbers ● Square Roots ● Real and Irrational Numbers ● The Number Line ● Absolute Value ● Inequality

A LOOK INTO MATH N

So far in this chapter, we have discussed natural numbers, whole numbers, and fractions. All of these numbers belong to a set of numbers called real numbers. In this section, we will see that real numbers also include integers and irrational numbers. Real-world quantities such as temperature, computer processor speed, height of a building, age of a fossil, and gas mileage are all described with real numbers.

Signed Numbers
The idea that numbers could be negative was a difficult concept for many mathematicians. As late as the eighteenth century, negative numbers were not readily accepted by everyone. After all, how could a person have - 5 oranges? However, negative numbers make more sense to someone working with money. If you owe someone 100 dollars (a debt), this amount can be thought of as - 100, whereas if you have a balance of 100 dollars in your checking account (an asset), this amount can be thought of as + 100. (The positive sign is usually omitted.) The opposite, or additive inverse, of a number a is - a. For example, the opposite of 25 is - 25, the opposite of - 5 is - ( - 5), or 5, and the opposite of 0 is 0 because 0 is neither positive nor negative. The following double negative rule is helpful in simplifying expressions containing negative signs.

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1.4 REAL NUMBERS AND THE NUMBER LINE

35

NEW VOCABULARY n n n n n n n n n n n n n n Opposite (additive inverse) Integers Rational number Square root Principal square root Real number Irrational numbers Approximately equal Average Origin Absolute value Less than/Greater than Less than or equal to Greater than or equal to

DOUBLE NEGATIVE RULE
Let a be any number. Then - ( - a) = a.

Thus - ( - 8) = 8 and - 1
READING CHECK

(

10) 2 = - (10) = - 10.

• How is the opposite of a number written?

EXAMPLE 1

Finding opposites (or additive inverses)
Find the opposite of each expression. (a) 13 (b) 4 7 (c) - ( - 7)

Solution (a) The opposite of 13 is - 13. (b) The opposite of - 4 is 4 . 7 7 (c) - ( - 7) = 7, so the opposite of - ( - 7) is - 7.
Now Try Exercises 17, 19, 21

NOTE: To find the opposite of an exponential expression, evaluate the exponent first. For example, the opposite of 24 is

- 24 = - (2 # 2 # 2 # 2) = - 16.

EXAMPLE 2

Finding an additive inverse (or opposite)
Find the additive inverse of - t, if t = - 2 . 3
Solution The additive inverse of - t is t = - 2 because - ( - t) = t by the double negative rule. 3
Now Try Exercise 27

Integers and Rational Numbers
In the opening section of this chapter we discussed natural numbers and whole numbers. Because these sets of numbers do not include negative numbers, fractions, or decimals, other sets of numbers are needed. The integers include the natural numbers, zero, and the opposites of the natural numbers. The integers are given by the following. p,
ISBN 1-256-49082-2

3,

2,

1, 0, 1, 2, 3, p p A rational number is any number that can be expressed as the ratio of two integers, q , where q 0. Rational numbers can be written as fractions, and they include all integers. Rational numbers may be positive, negative, or zero. Some examples of rational numbers are 2 , 3 3 - , 5 -7 , 1.2, and 3. 2

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36

CHAPTER 1 INTRODUCTION TO ALGEBRA

The numbers 1.2 and 3 are both rational numbers because they can be written as 12 and 3 . 10 1
STUDY TIP
The word “NOTE” is used to draw attention to important concepts that may otherwise be overlooked.
7 NOTE: The fraction - 7 can also be written as - 2 and - 7 . The position of the negative 2 2 sign does not affect the value of the fraction.

The fraction bar can be thought of as a division symbol. As a result, rational numbers have decimal equivalents. For example, 1 is equivalent to 1 , 2. The division 2 0.5 2 1.0

READING CHECK
• When a number is written in decimal form, how do we know if the number is a rational number?

shows that 1 = 0.5. In general, a rational number may be expressed in a decimal form that 2 either repeats or terminates. The fraction 1 may be expressed as 0.3, a repeating decimal, 3 and the fraction 1 may be expressed as 0.25, a terminating decimal. The overbar indicates 4 that 0.3 = 0.3333333p .
N REAL-WORLD CONNECTION Integers and rational numbers are used to describe quantities such as change in population. Figure 1.18 shows the change in population from 2000 to 2009 for selected U.S. cities. Note that both positive and negative numbers are used to describe these population changes. (Source: U.S. Census Bureau.)
Population Change 2000–2009
Milwaukee –7159

Cleveland –47,040 New York 383,603

Los Angeles 137,848

Houston 304,295

Figure 1.18

EXAMPLE 3

Classifying numbers
Classify each number as one or more of the following: natural number, whole number, integer, or rational number. (a) 12 4 (b) - 3 (c) 0 (d) 9 5

Solution (a) Because 12 = 3, the number 12 is a natural number, whole number, integer, and 4 4 rational number. (b) The number - 3 is an integer and a rational number but it is not a natural number or a whole number. (c) The number 0 is a whole number, integer, and rational number but not a natural number. (d) The fraction - 9 is a rational number because it is the ratio of two integers. It is not a 5 natural number, a whole number, or an integer.
Now Try Exercises 41, 43, 45, 47

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1.4 REAL NUMBERS AND THE NUMBER LINE

37

CALCULATOR HELP
To evaluate square roots with a calculator, see Appendix A (page AP-1).

Square Roots
The number b is a square root of a number a if b # b = a. Every positive number has one positive square root and one negative square root. For example, the positive square root of 9 is 3 because 3 # 3 = 9. The negative square root of 9 is - 3. (We will show that - 3 # ( - 3) = 9 in Section 1.6.) If a is a positive number, then the principal square root of a, denoted 1a, is the positive square root of a. For example, 125 = 5 because 5 # 5 = 25 and the number 5 is positive. Note that the principal square root of 0 is 0. That is, 10 = 0.

EXAMPLE 4

Calculating principal square roots
Evaluate each square root. Approximate your answer to three decimal places when appropriate. (a) 136 (b) 1100 (c) 15
Solution (a) 136 = 6 because 6 6 = 36 and 6 is positive. (b) 1100 = 10 because 10 10 = 100 and 10 is positive. (c) We can estimate the value of 15 with a calculator. Figure 1.19 reveals that 15 is approximately equal to 2.236. However, 2.236 does not exactly equal 15 because 2.236 2.236 = 4.999696, which does not equal 5.

#

#

#

2.236067977 2.236 2.236 4.999696

√ ( 5)

Figure 1.19

Now Try Exercises 49, 51, 53

Real and Irrational Numbers
If a number can be represented by a decimal number, then it is a real number. Every fraction has a decimal form, so real numbers include rational numbers. However, some real numbers cannot be expressed by fractions. They are called irrational numbers. The numbers 12, 115, and p are examples of irrational numbers. Every irrational number has a decimal representation that does not terminate or repeat.
NOTE: For any positive integer a, if 1a is not an integer then 1a is an irrational number.

Examples of real numbers include - 17, 4 , 5 1 - 23, 21 , 57.63, and 2 27.

is used Any real number may be approximated by a terminating decimal. The symbol to represent approximately equal. Each of the following real numbers has been approximated to two decimal places.
ISBN 1-256-49082-2

1 7

0.14, 2p

6.28, and

260

7.75

Figure 1.20 on the next page shows the relationships among the different sets of numbers. Note that each real number is either a rational number or an irrational number but not both. All natural numbers, whole numbers, and integers are rational numbers.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Real Numbers
Rational Numbers − 3, 1 , 4 = 0.4 5 3 9 Integers ..., −2, −1, 0, 1, 2, ... Whole Numbers 0, 1, 2, 3, 4, ... Natural Numbers 1, 2, 3, 4, ... Irrational Numbers −√17, p, √6

Figure 1.20 The Real Numbers

READING CHECK
• When a number is written in decimal form, how do we know if the number is an irrational number?

MAKING CONNECTIONS
Rational and Irrational Numbers

Both rational and irrational numbers can be written as decimals. However, rational numbers can be represented by either terminating or repeating decimals. For example, 1 = 0.5 is a 2 terminating decimal and 1 = 0.333 p is a repeating decimal. Irrational numbers are repre3 sented by decimals that neither terminate nor repeat.

EXAMPLE 5

Classifying numbers
Identify the natural numbers, whole numbers, integers, rational numbers, and irrational numbers in the following list. - 25, 9,
Solution

- 3.8,

249,

11 , and 4

- 41

Natural numbers: 9 and 149 = 7 Whole numbers: 9 and 149 = 7 Integers: 9, 149 = 7, and - 41 Rational numbers: 9, - 3.8, 149 = 7, 11, and - 41 4 Irrational number: - 15
Now Try Exercises 55, 59, 63

N REAL-WORLD CONNECTION Even though a data set may contain only integers, we often need decimals to describe it. For example, integers are used to represent the number of wireless subscribers in the United States for various years. However, the average number of subscribers over a longer time period may be a decimal. Recall that the average of a set of numbers is found by adding the numbers and then dividing by how many numbers there are in the set.

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1.4 REAL NUMBERS AND THE NUMBER LINE

39

EXAMPLE 6

Analyzing cell phone subscriber data
Table 1.6 lists the number of wireless subscribers, in millions, in the United States, for various years. Find the average number of subscribers for these years. Is the result a natural number, a rational number, or an irrational number?
TABLE 1.6 U.S. Wireless Subscribers

Year Subscribers
Source: CTIA.

1995 28

2000 97

2005 194

2010 292

Solution The average number of subscribers is

611 28 + 97 + 194 + 292 = = 152.75 million. 4 4 The average of these four natural numbers is an integer divided by an integer, which is a rational number. However, it is neither a natural number nor an irrational number.
Now Try Exercise 105

CRITICAL THINKING
Think of an example in which the sum of two irrational numbers is a rational number.

The Number Line
The real numbers can be represented visually by using a number line, as shown in Figure 1.21. Each real number corresponds to a unique point on the number line. The point associated with the real number 0 is called the origin. The positive integers are equally spaced to the right of the origin, and the negative integers are equally spaced to the left of the origin. The number line extends indefinitely both to the left and to the right.
Origin –Á2 –2 –1 0 1 2 1 5 4 2

Figure 1.21 A Number Line

Other real numbers can also be located on the number line. For example, the number can be identified by placing a dot halfway between the integers 0 and 1. The numbers - 12 - 1.41 and 5 = 1.25 can also be placed (approximately) on this number line. 4 (See Figure 1.21.)
1 2

EXAMPLE 7

Plotting numbers on a number line
Plot each real number on a number line. (a) 3 2 (b) 23 (c) p

ISBN 1-256-49082-2

–3 2 –2 –1 0 1

√3 2 3 4

Figure 1.22 Plotting Numbers

Solution (a) - 3 = - 1.5. Place a dot halfway between - 2 and - 1, as shown in Figure 1.22. 2 (b) A calculator gives 13 1.73. Place a dot between 1 and 2 so that it is about threefourths of the way toward 2, as shown in Figure 1.22. (c) p 3.14. Place a dot just past the integer 3, as shown in Figure 1.22.
Now Try Exercises 67, 73

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

40

CHAPTER 1 INTRODUCTION TO ALGEBRA

Absolute Value
⎥ −3⎥
-3 -2 -1

=
0

⎥ 3⎥
1 2 3

Figure 1.23

The absolute value of a real number equals its distance on the number line from the origin. Because distance is never negative, the absolute value of a real number is never negative. The absolute value of a real number a is denoted a and is read “the absolute value of a.” Figure 1.23 shows that the absolute values of - 3 and 3 are both equal to 3 because both have distance 3 from the origin. That is, - 3 = 3 and 3 = 3.

EXAMPLE 8

Finding the absolute value of a real number
Evaluate each expression. (a) 3.1 (b) - 7 (c) 0
Solution (a) 3.1 = 3.1 because the distance between the origin and 3.1 is 3.1. (b) - 7 = 7 because the distance between the origin and - 7 is 7. (c) 0 = 0 because the distance is 0 between the origin and 0.
Now Try Exercises 75, 77, 79

Our results about absolute value can be summarized as follows. eae eae a, if a is positive or 0. a, if a is negative.

Inequality
If a real number a is located to the left of a real number b on the number line, we say that a is less than b and write a 6 b. Similarly, if a real number b is located to the right of a real number a, we say that b is greater than a and write b 7 a. For example, - 3 6 2 because - 3 is located to the left of 2, and 2 7 - 3 because 2 is located to the right of - 3. See Figure 1.24.
3 * 2 or 2 +
–4 –3 –2 –1 0 1

3
2 3 4

Figure 1.24

NOTE: Any negative number is always less than any positive number, and any positive number is always greater than any negative number.

We say that a is less than or equal to b, denoted a … b, if either a 6 b or a = b is true. Similarly, a is greater than or equal to b, denoted a Ú b, if either a 7 b or a = b is true. Inequalities are often used to compare the relative sizes of two quantities. This can be illustrated visually as shown in the figure below.
Visualizing Inequality by Relative Size

READING CHECK
• Which math symbols are used to express inequality?
Ohio
ISBN 1-256-49082-2

<

Texas

>

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.4 REAL NUMBERS AND THE NUMBER LINE

41

EXAMPLE 9

Ordering real numbers
List the following numbers from least to greatest. Then plot these numbers on a number line. - 2, - p, 22, 0, and 2.5

Solution First note that - p - 3.14 6 - 2. The two negative numbers are less than 0, and the two positive numbers are greater than 0. Also, 12 1.41, so 12 6 2.5. Listing the numbers from least to greatest results in

- p,

- 2, 0,

22, and 2.5.

These numbers are plotted on the number line shown in Figure 1.25. Note that these numbers increase from left to right on the number line.
– √2 2.5

–4 –3 –2 –1 0 1 2 3 4

Figure 1.25

Now Try Exercise 103

1.4
Integers

Putting It All Together
COMMENTS EXAMPLES

CONCEPT

Include the natural numbers, their opposites, and 0 Include integers, all fractions q , where p and q 0 are integers, and all repeating and terminating decimals Any decimal numbers that neither terminate nor repeat; real numbers that are not rational Any numbers that can be expressed in decimal form; include the rational and irrational numbers To find the average of a set of numbers, find the sum of the numbers and then divide the sum by how many numbers there are in the set. A number line can be used to visualize the real number system. The point associated with the number 0 is called the origin. p p , - 2, - 1, 0, 1, 2, p - 3, 128, - 0.335, 0, 0.25 = 1 , and 6 4 0.3 = 1 3
1 2,

Rational Numbers

Irrational Numbers

p, 13, and 115

Real Numbers

p, 13, - 4, 0, - 10, 0.6 = 2, 1000, and 7 3 115 The average of 2, 6, and 10 is 2 + 6 + 10 18 = = 6. 3 3
Origin
-3 -2 -1 0 1 2 3

Average

Number Line
ISBN 1-256-49082-2

continued on next page

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

42

CHAPTER 1 INTRODUCTION TO ALGEBRA

continued from previous page

CONCEPT

COMMENTS

EXAMPLES

Absolute Value

The absolute value of a number a equals its distance on the number line from the origin. If a Ú 0, then a = a. If a 6 0, then a = - a. a is never negative. If a number a is located to the left of a number b on a number line, then a is less than b (written a 6 b). If a is located to the right of b, then a is greater than b (written a 7 b). Symbols of inequality include: 6 , 7, … , Ú , and .

17 = 17 - 12 = 12 0 = 0

Inequality

- 3 6 - 2 less than 6 7 4 -5 … -5 18 Ú 0 7 8 greater than less than or equal greater than or equal not equal

1.4

Exercises
13. The symbol are . is used to indicate that two numbers

CONCEPTS AND VOCABULARY

1. The opposite of the number b is

.

2. The integers include the natural numbers, zero, and the opposites of the numbers. 3. A number that can be written as q , where p and q are integers with q number. 0, is a(n) 4. If a number can be written in decimal form, then it is a(n) number. 5. If a real number is not a rational number, then it is a(n) number. 6. (True or False?) A rational number can be written as a repeating or terminating decimal. 7. (True or False?) An irrational number cannot be written as a repeating or terminating decimal. 8. Write 0.272727p using an overbar. 9. The decimal equivalent for 1 can be found by divid4 ing by . 10. The equation 4 # 4 = 16 indicates that 116 = . p 14. The origin on the number line corresponds to the number . 15. Negative numbers are located to the (left/right) of the origin on the number line. 16. The absolute value of a number a gives its distance on the number line from the .
SIGNED NUMBERS

Exercises 17–24: Find the opposite of each expression. 17. (a) 9 18. (a) - 6 19. (a) 2 3 20. (a) - 1 - 4 2 5 21. (a) - ( - 8) 22. (a) - 1 - ( - 2) 2 23. (a) a 24. (a) - b (b) - 9 (b) 6 (b) - 2 3 (b) - 1 - 4 2 -5 (b) - 1 - ( - 8) 2 (b) - ( - 2) (b) - a
ISBN 1-256-49082-2

11. The positive square root of a positive number is called the _____ square root. 12. The symbol are _____. is used to indicate that two numbers

(b) - ( - b)

25. Find the additive inverse of t, if - t = 6. 26. Find the additive inverse of - t, if t = - 4 . 5

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.4 REAL NUMBERS AND THE NUMBER LINE

43

27. Find the additive inverse of - b, if b = 1 . 2
5 28. Find the additive inverse of b, if - b = - 6 .

69. (a) 1 2 70. (a) 1.3 71. (a) - 10 72. (a) 5 73. (a) p 74. (a) 211
ABSOLUTE VALUE

(b) - 1 2 (b) - 2.5 (b) - 20 (b) 10 (b) 22 (b) - 25

(c) 2 (c) 0.7 (c) 30 (c) - 10 (c) - 23 (c) 24

NUMBERS AND THE NUMBER LINE

Exercises 29–40: Find the decimal equivalent for the rational number. 29. 1 4 31. 7 8 33. 3 2
1 35. 20

30. 3 5
3 32. 10 3 34. 50 3 36. 16

Exercises 75–82: Evaluate the expression. 75. 5.23 77. - 8 79. 2 - 2 76. p 78. - 22 80. 2 - 1 3 3 82. 3 - p

37. 2 3 39. 7 9

38. 2 9
5 40. 11

Exercises 41–48: Classify the number as one or more of the following: natural number, whole number, integer, or rational number. 41. 8 43. 16 4 45. 0 47. - 7 6 42. - 8 44. 5 7 46. - 15 31 48. - 10 5

81. p - 3

83. Thinking Generally Find b , if b is negative. 84. Thinking Generally Find - b , if b is positive.
INEQUALITY

Exercises 85–96: Insert the symbol 7 or 6 to make the statement true. 85. 5 87. - 5 89. - 1 3 91. - 1.9 93. - 8 95. - 2 7 -7 -2 3 - 1.3 3 -7 86. - 5 88. 3 5
1 90. - 10 2 5

Exercises 49–54: Evaluate the square root. Approximate your answer to three decimal places when appropriate. 49. 225 51. 249 53. 27 50. 281 52. 264 54. 211

7

0 - 6.2 -1 32

92. 5.1 94. 4 96. - 15

Exercises 55–66: Classify the number as one or more of the following: natural number, integer, rational number, or irrational number. 55. - 4.5 57. 3 7 59. 211 61. 8 4 63. 249 65. 1.8
ISBN 1-256-49082-2

56. p 58. 225 60. - 23 62. - 5 64. 3.3 66.
9 3

Exercises 97–100: Thinking Generally Insert the symbol 6 , = , or 7 to make the statement true. 97. If a number a is located to the right of a number b on the number line, then a b. 98. If b 7 0 and a 6 0, then b 99. If a Ú b, then either a 7 b or a 100. If a Ú b and b Ú a, then a b. a. b.

Exercises 67–74: Plot each number on a number line.* 67. (a) 0 68. (a) - 3 2 (b) - 2 (b) 3 2 (c) 3 (c) 0

Exercises 101–104: List the given numbers from least to greatest. 101. - 3, 0, 1, - 9, - 23 102. 4, - 23, 1, - 3, 3 2 2 2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

103. - 2, p, 1, - 3, 25 3 2
3 1 104. 9, 14, - 12, - 16, 27

106. Music Sales The table lists the percentage of recorded music purchased through digital downloads during selected years. Year 2005 6.0 2006 6.8 2007 12.0 2008 13.5

APPLICATIONS

Percent

105. Marriage Age The table lists the average age of females at first marriage during selected years. Year Age 2001 25.1 2003 25.3 2005 25.5 2007 25.9

Source: Recording Industry Association of America.

Source: U.S. Census Bureau.

(a) What was this percentage in 2006? (b) Mentally estimate the average percentage for this 4-year period. (c) Calculate the average percentage. Is your mental estimate in reasonable agreement with your calculated result?
WRITING ABOUT MATHEMATICS

(a) What was this age in 2005? (b) Mentally estimate the average marriage age for these selected years. (c) Calculate the average marriage age. Is your mental estimate in reasonable agreement with your calculated result?

107. What is a rational number? Is every integer a rational number? Why or why not? 108. Explain why 3 7 1 . Now explain in general how to 7 3 c determine whether a 7 d . Assume that a, b, c, and b d are natural numbers.

SECTIONS 1.3 AND 1.4

Checking Basic Concepts
8. Classify each number as one or more of the following: natural number, integer, rational number, or irrational number. (a) 10 (b) - 5 (c) 25 (d) - 5 2 6 9. Plot each number on the same number line. (a) 0 (b) - 3 (c) 2 (e) - 22 (d) 3 4 10. Evaluate each expression. (a) - 12 (b) 6 - 6 11. Insert the symbol 7 or 6 to make the statement true. (a) 4 9 (b) - 1.3 - 0.5 (c) - 3 -5 12. List the following numbers from least to greatest.

1. Write each product as an exponential expression. (a) 5 # 5 # 5 # 5 (b) 7 # 7 # 7 # 7 # 7 2. Evaluate each expression. (a) 23 (b) 104 (c)

1223 3

3. Use the given base to write the number as an exponential expression. (a) 64 (base 4) (b) 64 (base 2) 4. Evaluate each expression without a calculator. (a) 6 + 5 # 4 (b) 6 + 6 , 2 (c) 5 - 2 - 1 (d) 6 - 3 2 + 4 (e) 12 , (6 , 2) (f) 23 - 2 1 2 + 4 2 2 5. Translate the phrase “five cubed divided by three” to an algebraic expression. 6. Find the opposite of each expression. (a) - 17 (b) a 7. Find the decimal equivalent for the rational number. 3 (a) 20 (b) 5 8

23,

- 7, 0,

1 , 3

- 1.6, 32

ISBN 1-256-49082-2 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.5 ADDITION AND SUBTRACTION OF REAL NUMBERS

45

1.5

Addition and Subtraction of Real Numbers
Addition of Real Numbers ● Subtraction of Real Numbers ● Applications

A LOOK INTO MATH N

Addition and subtraction of real numbers occur every day at grocery stores, where the prices of various items are added to the total and discounts from coupons are subtracted. Even though prices are usually expressed in decimal form, the rules for adding and subtracting real numbers are the same no matter how the real numbers are expressed.

Addition of Real Numbers
NEW VOCABULARY n Addends n Sum n Difference

In an addition problem the two numbers added are called addends, and the answer is called the sum. For example, in the following addition problem the numbers 3 and 5 are the addends and the number 8 is the sum. 3
Addend

+

5
Addend

=

8
Sum

READING CHECK
• What is the answer to an addition problem called?

The opposite (or additive inverse) of a real number a is - a. When we add opposites, the result is 0. For example, 4 + ( - 4) = 0. In general, the equation a + ( - a) = 0 is true for every real number a.

EXAMPLE 1

Adding opposites
Find the opposite of each number and calculate the sum of the number and its opposite. (a) 45 (b) 22 (c) 1 2

Solution (a) The opposite of 45 is - 45. Their sum is 45 + ( - 45) = 0. (b) The opposite of 12 is - 12. Their sum is 12 + ( - 12) = 0. (c) The opposite of - 1 is 1 . Their sum is - 1 + 1 = 0. 2 2 2 2
Now Try Exercises 11, 13, 15

N REAL-WORLD CONNECTION When adding real numbers, it may be helpful to think of money. A positive number represents income, and a negative number indicates debt. The sum 9 + ( - 5) = 4 would represent being paid $9 and owing $5. In this case $4 would be left over. Similarly, the sum - 3 + ( - 6) = - 9 would represent debts of $3 and $6, resulting in a total debt of $9. To add two real numbers we can use the following rules.

ADDITION OF REAL NUMBERS
STUDY TIP
Do you know your instructor’s name? Do you know the location of his or her office and the hours when he or she is available for help? Make sure that you have the answers to these important questions so that you can get help when needed.

To add two real numbers with like signs, do the following. 1. Find the sum of the absolute values of the numbers. 2. Keep the common sign of the two numbers as the sign of the sum. To add two real numbers with unlike signs, do the following. 1. Find the absolute values of the numbers. 2. Subtract the smaller absolute value from the larger absolute value. 3. Keep the sign of the number with the larger absolute value as the sign of the sum.

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

46

CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 2

Adding real numbers
Find each sum by hand. 2 7 (a) - 2 + ( - 4) (b) - + 5 10 (c) 6.2 + ( - 8.5)

Solution (a) The numbers are both negative, so we add the absolute values - 2 and - 4 to obtain 6. The signs of the addends are both negative, so the answer is - 6. That is, - 2 + ( - 4) = - 6. If we owe $2 and then owe $4, the total amount owed is $6. (b) The numbers have opposite signs, so we subtract their absolute values to obtain

2 7 4 3 7 = = . 10 5 10 10 10
7 7 The sum is positive because 10 is greater than - 2 and 10 is positive. That is, 5 7 3 7 - 2 + 10 = 10 . If we spend $0.40 1 - 2 = - 0.4 2 and receive $0.70 1 10 = 0.7 2 , we 5 5 3 have $0.30 1 10 = 0.3 2 left. (c) 6.2 + ( - 8.5) = - 2.3 because - 8.5 is 2.3 more than 6.2 . If we have $6.20 and we owe $8.50, we are short $2.30.

Now Try Exercises 33, 39, 41

ADDING INTEGERS VISUALLY One way to add integers visually is to use the symbol to represent a positive unit and to use the symbol to represent a negative unit. Now adding opposites visually results in “zero,” as shown.

+

=

1 + (-1) = 0

For example, to add - 3 + 5, we draw three negative units and five positive units.
-3 + 5 = 2

Because the “zeros” add no value, the sum is two positive units, or 2.

EXAMPLE 3

Adding integers visually
Add visually, using the symbols and (a) 3 + 2 (b) - 6 + 4 (c) 2 + ( - 3) . (d) - 5 + ( - 2)

Solution (a) Draw three positive units and then draw two more positive units.
3 + 2 = 5

Because no zeros were formed, the sum is five positive units, or 5. (b) Draw six negative units and then draw four positive units.
-6 + 4 = -2

Ignoring the zeros that were formed, the sum is two negative units, or - 2. (c) Draw two positive and three negative units.
2 + (-3) = -1
ISBN 1-256-49082-2

The sum is - 1. (d) The sum is - 7.
Now Try Exercises 17, 19, 21, 23

-5 + (-2) = -7

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.5 ADDITION AND SUBTRACTION OF REAL NUMBERS

47

Another way to add integers visually is to use a number line. For example, to add 4 + ( - 3) start at 0 (the origin) and draw an arrow to the right 4 units long from 0 to 4. The number - 3 is a negative number, so draw an arrow 3 units long to the left, starting at the tip of the first arrow. See Figure 1.26a. The tip of the second arrow is at 1, which equals the sum of 4 and - 3.
Adding Integers on a Number Line
–3 4 –1 0 1 2 3 4 (a) 4 + ( - 3) = 1 5 –3 –2

–6 –5 –4 –3 –2 –1 0 1 (b) - 2 + ( - 3) = - 5

Figure 1.26

To find the sum - 2 + ( - 3), draw an arrow 2 units long to the left, starting at the origin. Then draw an arrow 3 units long to the left, starting at the tip of the first arrow, which is located at - 2. See Figure 1.26b. Because the tip of the second arrow coincides with - 5 on the number line, the sum of - 2 and - 3 is - 5.
READING CHECK
• What is the answer to a subtraction problem called?

Subtraction of Real Numbers
The answer to a subtraction problem is the difference. When subtracting two real numbers, changing the problem to an addition problem may be helpful.

SUBTRACTION OF REAL NUMBERS
For any real numbers a and b, a - b = a + ( - b). To subtract b from a, add a and the opposite of b.

EXAMPLE 4

Subtracting real numbers
Find each difference by hand. (a) 10 - 20 (b) - 5 - 2 (c) - 2.1 - ( - 3.2)
Solution (a) 10 20 = 10 ( 20) = - 10 (b) - 5 2 = -5 ( 2) = - 7 (c) - 2.1 ( 3.2) = - 2.1 3.2 = 1.1 (d) 1 1 32 = 1 3 = 2 + 3 = 5 2 4 2 4 4 4 4
Now Try Exercises 45, 47, 51, 53

(d) 1 2

1 -32 4

In the next example, we show how to add and subtract groups of numbers.
ISBN 1-256-49082-2

EXAMPLE 5

Adding and subtracting real numbers
Evaluate each expression by hand. (a) 5 - 4 - ( - 6) + 1 (b) 1 - 3 + 1 2 4 3 (c) - 6.1 + 5.6 - 10.1

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Solution (a) Rewrite the expression in terms of addition only, and then find the sum.

5

4

(

6) + 1 = 5 = 8

(

4)

6 + 1

= 1 + 6 + 1 (b) Begin by rewriting the fractions with the LCD of 12. 3 1 6 1 + = 2 4 3 12 = 6 12 9 4 + 12 12 a 9 4 b + 12 12

= CRITICAL THINKING
Explain how subtraction of real numbers can be performed on a number line.

4 3 + 12 12

=

1 12

(c) The expression can be evaluated by changing subtraction to addition. - 6.1 + 5.6 10.1 = - 6.1 + 5.6 = - 10.6
Now Try Exercises 59, 65, 67

(

10.1)

= - 0.5 + ( - 10.1)

Applications
In application problems, some words indicate that we should add, while other words indicate that we should subtract. Tables 1.7 and 1.8 show examples of such words along with sample phrases.
TABLE 1.7 Words Associated with Addition TABLE 1.8 Words Associated with Subtraction

Words add plus more than sum total increased by

Sample Phrase add the two temperatures her age plus his age 5 cents more than the cost the sum of two measures the total of the four prices height increased by 3 inches

Words subtract minus fewer than difference less than decreased by

Sample Phrase subtract dues from the price his income minus his taxes 18 fewer flowers than shrubs the difference in their heights his age is 4 less than yours the weight decreased by 7

READING CHECK
• What words are associated with addition? • What words are associated with subtraction?

ISBN 1-256-49082-2

N REAL-WORLD CONNECTION Sometimes temperature differences are found by subtracting positive and negative real numbers. At other times, addition of positive and negative real numbers occurs at banks if positive numbers represent deposits and negative numbers represent withdrawals. The next two examples illustrate these situations.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.5 ADDITION AND SUBTRACTION OF REAL NUMBERS

49

EXAMPLE 6

Calculating temperature differences
The hottest outdoor temperature ever recorded in the shade was 136 F in the Sahara desert, and the coldest outside temperature ever recorded was - 129 F in Antarctica. Find the difference between these two temperatures. (Source: Guinness Book of Records.)
Solution The word difference indicates subtraction. We must subtract the two temperatures.

136 - ( - 129) = 136 + 129 = 265 F.
Now Try Exercise 87

EXAMPLE 7

Balancing a checking account
The initial balance in a checking account is $285. Find the final balance if the following represents a list of withdrawals and deposits: - $15, - $20, $500, and - $100.
Solution Find the sum of the five numbers.

285 + ( - 15) + ( - 20) + 500 + ( - 100) = 270 + ( - 20) + 500 + ( - 100) = 250 + 500 + ( - 100)
285 ( 15) ( 20) 500 ( 100) 650 285 15 20 500 10 0 650
Figure 1.27

= 750 + ( - 100) = 650 The final balance is $650. This result may be supported with a calculator, as illustrated in Figure 1.27. Note that the expression has been evaluated two different ways.
Now Try Exercise 83

TECHNOLOGY NOTE
Subtraction and Negation Calculators typically have different keys to represent subtraction and negation. Be sure to use the correct key. Many graphing calculators have the following keys for subtraction and negation.

π

D

Subtraction Negation

1.5
Addition

Putting It All Together
CONCEPT COMMENTS EXAMPLES

To add real numbers, use a number line or follow the rules found in the box on page 45.

-2 0.8 -1 7 -4 -3

+ + + + +

8 = 6 ( - 0.3) = 0.5 1 - 3 2 = - 1 +7 ( - 3) = - 4 7 7 4 = 0 4 + ( - 2) = - 1

ISBN 1-256-49082-2

Subtraction

To subtract real numbers, transform the problem to an addition problem by adding the opposite. a - b = a + ( - b)

6 - 8 = 6 + ( - 8) = - 2 - 3 - 4 = - 3 + ( - 4) = - 7 -1 - 1 -32 = -1 + 3 = 2 = 1 2 2 2 2 2 - 5 - ( - 5) = - 5 + 5 = 0 9.4 - ( - 1.2) = 9.4 + 1.2 = 10.6

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

1.5

Exercises
Exercises 33–44: Find the sum. . . 33. 5 + ( - 4) 35. - 1 + ( - 6) 37. 3 + 4 34. - 9 + 7 36. - 10 + ( - 23)
5 38. - 12 +

CONCEPTS AND VOCABULARY

1. The solution to an addition problem is the 2. When you add opposites, the sum is always

3. (True or False?) If two positive numbers are added, the sum is always a positive number. 4. (True or False?) If two negative numbers are added, the sum is always a negative number. 5. If two numbers with opposite signs are added, the sum has the same sign as the number with the larger . 6. The solution to a subtraction problem is the .

1 -12 2

1 -12 6
1 1 - 12 2

3 39. - 6 + 14 7

40. - 2 + 9

41. 0.6 + ( - 1.7) 43. - 52 + 86

42. 4.3 + ( - 2.4) 44. - 103 + ( - 134)

Exercises 45–56: Find the difference. 45. 5 - 8 47. - 2 - ( - 9) 49. 6 - 13 7 14
1 51. - 10 -

46. 3 - 5 48. - 10 - ( - 19) 50. - 5 - 1 6 6
5 1 - 11 2

7. When subtracting two real numbers, it may be helpful to change the problem to a(n) problem. 8. To subtract b from a, add the _____ of b to a. That is, a - b = a + . 9. The words sum, more, and plus indicate that should be performed. 10. The words difference, less than, and minus indicate that should be performed.
ADDITION AND SUBTRACTION OF REAL NUMBERS

1 -32 5

2 52. - 11 -

53. 0.8 - ( - 2.1) 55. - 73 - 91

54. - 9.6 - ( - 5.7) 56. 201 - 502

Exercises 57–70: Evaluate the expression. 57. 10 - 19 59. 19 - ( - 22) + 1 61. - 3 + 4 - 6 58. 5 + ( - 9) 60. 53 + ( - 43) - 10 62. - 11 + 8 - 10

Exercises 11–16: Find the opposite of the number and then calculate the sum of the number and its opposite. 11. 25 13. - 221 15. 5.63 12. - 1 2 14. - p 16. - 62

63. 100 - 200 + 100 - ( - 50) 64. - 50 - ( - 40) + ( - 60) + 80 65. 1.5 - 2.3 + 9.6 66. 10.5 - ( - 5.5) + ( - 1.5) 67. - 1 + 1 2 4 68. 1 4

Exercises 17–24: Refer to Example 3. Find the sum visually. 17. 1 + 3 19. 4 + ( - 2) 21. - 1 + ( - 2) 23. - 3 + 7 18. 3 + 1 20. - 4 + 6 22. - 2 + ( - 2) 24. 5 + ( - 6)

1 -32 4
+
3 1 - 20 2

1 -22 5

69. 4 - 9 - 1 - 7 70. - 5 - ( - 3) - - 6 + 8 Exercises 71–80: Write an arithmetic expression for the given phrase and then simplify it. 71. The sum of two and negative five 72. Subtract ten from negative six 73. Negative five increased by seven

Exercises 25–32: Use a number line to find the sum. 25. - 1 + 3 27. 4 + ( - 5) 29. - 10 + 20 31. - 50 + ( - 100) 26. 3 + ( - 1) 28. 2 + 6 30. 15 + ( - 5) 32. - 100 + 100

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.6 MULTIPLICATION AND DIVISION OF REAL NUMBERS

51

74. Negative twenty decreased by eight 75. The additive inverse of the quantity two cubed 76. Five minus the quantity two cubed 77. The difference between negative six and seven (Hint: Write the numbers for the subtraction problem in the order given.) 78. The difference between one-half and three-fourths 79. Six plus negative ten minus five 80. Ten minus seven plus negative twenty 81. Online Exploration Use the Internet to find the highest and lowest points in the continental United States, and then find the difference in their heights. 82. Online Exploration In 1972, residents of Loma, Montana, experienced the largest 24-hour temperature swing ever recorded in the United States. Use the Internet to find the starting and ending temperatures, and then write a subtraction equation that shows the difference.
APPLICATIONS

85. Football Stats A running back carries the ball five times. Find his total yardage if the carries were 9, - 2, - 1, 14, and 5 yards. 86. Tracking Weight A person weighs himself every Monday for four weeks. He gains one pound the first week, loses three pounds the second week, gains two pounds the third week, and loses one pound the last week. (a) Using positive numbers to represent weight gains and negative numbers to represent weight losses, write a sum that gives the total gain or loss over this four-week period. (b) If he weighed 170 pounds at the beginning of this process, what was his weight at the end? 87. Deepest and Highest The deepest point in the ocean is the Mariana Trench, which is 35,839 feet below sea level. The highest point on Earth is Mount Everest, which is 29,029 feet above sea level. What is the difference in height between Mount Everest and the Mariana Trench? (Source: The Guinness Book of Records.) 88. Greatest Temperature Ranges The greatest temperature range on Earth occurs in Siberia, where the temperature can vary between 98 F in the summer and - 90 F in the winter. Find the difference between these two temperatures. (Source: The Guinness Book of Records.)

83. Checking Account The initial balance in a checking account is $358. Find the final balance resulting from the following withdrawals and deposits: - $45, $37, $120, and - $240. 84. Savings Account A savings account has $1245 in it. Find the final balance resulting from the following withdrawals and deposits: - $189, $975, - $226, and - $876.

WRITING ABOUT MATHEMATICS

89. Explain how to add two negative numbers. Give an example. 90. Explain how to subtract a negative number from a positive number. Give an example.

1.6

Multiplication and Division of Real Numbers
Multiplication of Real Numbers ● Division of Real Numbers ● Applications

A LOOK INTO MATH N

Exam and quiz scores are sometimes displayed in fraction form. For example, a student who answers 17 questions correctly on a 20-question exam may see the fraction 17 writ20 ten at the top of the exam paper. If the instructor assigns grades based on percents, this fraction must be converted to a percent. Doing so involves division of real numbers. In this section, we discuss multiplication and division of real numbers.

Multiplication of Real Numbers
ISBN 1-256-49082-2

READING CHECK
• What is the answer to a multiplication problem called?

In a multiplication problem, the numbers multiplied are called the factors, and the answer is called the product. 7
Factor

#

4
Factor

=

28
Product

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52

CHAPTER 1 INTRODUCTION TO ALGEBRA

NEW VOCABULARY n n n n Dividend Divisor Quotient Reciprocal (multiplicative inverse)

Multiplication is a fast way to perform addition. For example, 5 # 2 = 10 is equivalent to finding the sum of five 2s, or 2 + 2 + 2 + 2 + 2 = 10. Similarly, the product 5 # ( - 2) is equivalent to finding the sum of five - 2s, or ( - 2) + ( - 2) + ( - 2) + ( - 2) + ( - 2) = - 10. Thus 5 # ( - 2) = - 10. In general, the product of a positive number and a negative number is a negative number. What sign should the product of two negative numbers have? To answer this question, consider the following patterns. What values should replace the question marks to continue the pattern? -4
Decrease by 1

# # # #

3 = - 12
Increase by 4

-4
Decrease by 1

2 = 1 = 0 =

-8
Increase by 4

-4
Decrease by 1

-4
Increase by 4

-4
Decrease by 1

0 ? ?

-4
Decrease by 1

# ( - 1) = # ( - 2) =

-4
STUDY TIP
Studying with other students can greatly improve your chances of success. Consider exchanging phone numbers or email addresses with some of your classmates so that you can find a time and place to study together. If possible, set up a regular meeting time and invite other classmates to join you.

Continuing the pattern results in - 4 # ( - 1) = 4 and - 4 # ( - 2) = 8. This pattern suggests that if we multiply two negative real numbers, the product is positive.

SIGNS OF PRODUCTS
The product of two numbers with like signs is positive. The product of two numbers with unlike signs is negative.

NOTE: Multiplying a number by - 1 results in the additive inverse (opposite) of the number. For example, - 1 4 = - 4, and - 4 is the additive inverse of 4. Similarly, - 1 ( - 3) = 3, and 3 is the additive inverse of - 3. In general, - 1 a = - a.

#

#

#

EXAMPLE 1

Multiplying real numbers
Find each product by hand. (a) - 9 # 7 (b) 2 # 5 (c) - 2.1( - 40) (d) ( - 2.5)(4)( - 9)( - 2) 3 9
Solution (a) The resulting product is negative because the factors - 9 and 7 have unlike signs. Thus - 9 7 = - 63. (b) The product is positive because both factors are positive.

#

2 3

#5
9

=

2#5 10 = 3#9 27

ISBN 1-256-49082-2

(c) As both factors are negative, the product is positive. Thus - 2.1( - 40) = 84. (d) ( - 2.5)(4)( - 9)( - 2) = ( - 10)( - 9)( - 2) = (90)( - 2) = - 180
Now Try Exercises 13, 23, 27, 29

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1.6 MULTIPLICATION AND DIVISION OF REAL NUMBERS

53

READING CHECK
• How can the number of negative factors be used to decide if a product is positive or negative?

MAKING CONNECTIONS
Multiplying More Than Two Negative Factors

Because the product of two negative numbers is positive, it is possible to determine the sign of a product by counting the number of negative factors. For example, the product - 3 # 4 # ( - 5) # 7 # ( - 6) is negative because there are an odd number of negative factors and the product 2 # ( - 1) # ( - 3) # ( - 5) # ( - 4) is positive because there are an even number of negative factors.

When evaluating expressions such as 52 (Square and then negate) and ( 5)2 (Negate and then square),

it is important to note that the first represents the opposite of five squared, while the second indicates that negative five should be squared. The next example illustrates the difference between the opposite of an exponential expression and a power of a negative number.

EXAMPLE 2

Evaluating real numbers with exponents
Evaluate each expression by hand. (a) ( - 4)2 (b) - 42 (c) ( - 2)3 (d) - 23
Solution (a) Because the exponent is outside of parentheses, the base of the exponential expression is - 4. The expression is evaluated as

(

4)2 = (

4)(

4) = 16.

(b) This is the negation of an exponential expression with base 4. Evaluating the exponent before negating results in - 42 = - (4)(4) = - 16. (c) ( - 2)3 = ( - 2)( - 2)( - 2) = - 8 (d) - 23 = - (2)(2)(2) = - 8
Now Try Exercises 35, 37, 39, 41

MAKING CONNECTIONS
Negative Square Roots

Because the product of two negative numbers is positive, ( - 3) # ( - 3) = 9. That is, - 3 is a square root of 9. As discussed in Section 1.4, every positive number has one positive square root and one negative square root. For a positive number a, the positive square root is called the principal square root and is denoted 1a. The negative square root is denoted - 1a. For example, 14 = 2 and - 14 = - 2.

Division of Real Numbers
READING CHECK
ISBN 1-256-49082-2

In a division problem, the dividend is divided by the divisor, and the result is the quotient. 20
Dividend

• What is the answer to a division problem called?

,

4
Divisor

=

5

Quotient

This division problem can also be written in fraction form as 20 = 5. Division of real 4 numbers can be defined in terms of multiplication and reciprocals. The reciprocal, or multiplicative inverse, of a real number a is 1 . The number 0 has no reciprocal. a

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

54

CHAPTER 1 INTRODUCTION TO ALGEBRA

DIVIDING REAL NUMBERS
TECHNOLOGY NOTE
Try dividing 5 by 0 with a calculator. On most calculators, dividing a number by 0 results in an error message.

For real numbers a and b with b

0, a = a b

# 1. b That is, to divide a by b, multiply a by the reciprocal of b.

NOTE: Division by 0 is undefined because 0 has no reciprocal.

EXAMPLE 3

Dividing real numbers
Evaluate each expression by hand. (a) - 16 ,
1 4 3 5

(b)

-8

-8 (c) - 36

(d) 9 , 0
The reciprocal of 1 is 4. 4 1 The reciprocal of - 8 is - 1. 8
1 The reciprocal of - 36 is - 36.

Solution 16 (a) - 16 , 1 = -1 4

(b)

3 5

#4 1

64 = -1 = - 64

-8

= 3 , ( 8) = 3 5 5
1 # (- 36 )

#(

1 8)

3 = - 40

-8 (c) - 36 = - 8

8 = 36 = 2 9

(d) 9 , 0 is undefined.
Now Try Exercises 49, 51, 59, 63

The number 0 has no reciprocal.

When determining the sign of a quotient, the following rules may be helpful. Note that these rules are similar to those for signs of products.

SIGNS OF QUOTIENTS
The quotient of two numbers with like signs is positive. The quotient of two numbers with unlike signs is negative.

NOTE: To see why a negative number divided by a negative number is a positive number, remember that division is a fast way to perform subtraction. For example, because

20 - 4 - 4 - 4 - 4 - 4 = 0, a total of five 4s can be subtracted from 20, so 20 = 5. Similarly, because 4 20 - ( a total of five 4) - ( 4) - ( 4) - ( 20, so
20 4

4) - ( = 5.

4) = 0,
ISBN 1-256-49082-2

4s can be subtracted from

N REAL-WORLD CONNECTION In business, employees may need to convert fractions and mixed numbers to decimal numbers. The next example illustrates this process.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.6 MULTIPLICATION AND DIVISION OF REAL NUMBERS

55

EXAMPLE 4

Converting fractions to decimals
Convert each measurement to a decimal number. (a) 2 3 -inch bolt (b) 15 -inch diameter (c) 1 1 -cup flour 8 16 3
Solution (a) First divide 3 by 8.

(b) Divide 15 by 16. 0.9375 16 15.0000 144 60 48 120 112 80 80 0 So, 15 = 0.9375. 16

(c) First divide 1 by 3. 0.333 N 3 1.000 9 10 9 10 9 1

0.375 8 3.000 24 60 56 40 40 0

So, 2 3 = 2.375. 8
Now Try Exercises 69, 71, 75

So, 1 1 = 1.3. 3

In the next example, numbers that are expressed as terminating decimals are converted to fractions.

EXAMPLE 5

Converting decimals to fractions
Convert each decimal number to a fraction in lowest terms. (a) 0.06 (b) 0.375 (c) 0.0025
Solution 6 (a) The decimal 0.06 equals six hundredths, or 100 . Simplifying this fraction gives

6 3#2 3 = = . 100 50 # 2 50
375 (b) The decimal 0.375 equals three hundred seventy-five thousandths, or 1000 . Simplifying this fraction gives

375 3 # 125 3 = # = . 1000 8 125 8
25 (c) The decimal 0.0025 equals twenty-five ten thousandths, or 10,000 . Simplifying this fraction gives

25 1 # 25 1 = = . 10,000 400 # 25 400
Now Try Exercises 81, 85, 87
ISBN 1-256-49082-2

FRACTIONS AND CALCULATORS (OPTIONAL) Many calculators have the capability to perform arithmetic on fractions and express the answer as either a decimal or a fraction. The next example illustrates this capability.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

56

CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 6

Performing arithmetic operations with technology
Use a calculator to evaluate each expression. Express your answer as a decimal and as a fraction. (a) 1 + 2 - 4 (b) 1 4 # 3 2 , 2 3 9 9 8 3 5
Solution (a) Figure 1.28(a) shows that

2 4 13 1 + = 0.28, or . 3 5 9 45 (b) Figure 1.28(b) shows that 1 4 # 3 2 , 2 = 0.25, or 1 . 9 8 3 4
( 1/3 ) ( 2/5 ) ( 4/9 ) .2888888889 ( 1/3 ) ( 2/5 ) ( 4/9 ) Frac 13/45
(a)

CALCULATOR HELP
To find fraction results with a calculator, see Appendix A (pages AP-1 and AP-2).

(( 4/9 ) ( 3/8 )) / ( 2 /3 ) .25 (( 4/9 ) ( 3/8 )) / ( 2 /3 ) Frac 1/4
(b)

Figure 1.28

Now Try Exercises 89, 91

NOTE: Generally it is a good idea to put parentheses around fractions when you are using a calculator.

MAKING CONNECTIONS
The Four Arithmetic Operations READING CHECK
• How are addition and subtraction related? • How are multiplication and division related?

If you know how to add real numbers, then you also know how to subtract real numbers because subtraction is defined in terms of addition. That is, a - b = a + ( - b). If you know how to multiply real numbers, then you also know how to divide real numbers because division is defined in terms of multiplication. That is, a = a b

# 1. b Applications
There are many instances when we may need to multiply or divide real numbers. Two examples are provided here.

EXAMPLE 7

Comparing top-grossing movies
ISBN 1-256-49082-2

Even though Avatar (2009) was an extremely successful film that set domestic box office records, it ranks only fourteenth among top-grossing movies of all time when calculated by using estimated total admissions. The total admissions for Avatar are 17 of the total admis25 sions for The Sound of Music (1965), which ranks third. (Source: Box Office Mojo.)

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.6 MULTIPLICATION AND DIVISION OF REAL NUMBERS

57

(a) If The Sound of Music had estimated total admissions of 142 million people, find the estimated total admissions for Avatar. (b) The top-grossing movie of all time is Gone With the Wind (1939), which had estimated total admissions of 202 million. How many more people saw Gone With the Wind than saw Avatar?
.

Solution (a) To find the total admissions for Avatar we multiply the real numbers 142 and 17 to 25 obtain 142 17 = 2414 97. 25 25 Total admissions for Avatar were about 97 million people. (b) The difference is 202 - 97 = 105 million people.

#

Now Try Exercise 97

EXAMPLE 8

Analyzing the federal budget
14 It is estimated that 125 of the federal budget is used to pay interest on loans. Write this fraction as a decimal. (Source: U.S. Office of Management and Budget.)

Solution One method for writing the fraction as a decimal is to divide 14 by 125, using long division. An alternative method is to multiply the fraction by 8 so that the denominator becomes 8 1000. Then write the numerator in the thousandths place in the decimal.

14 125
Now Try Exercise 99

#8
8

=

112 = 0.112 1000

1.6

Putting It All Together
CONCEPT COMMENTS EXAMPLES

Multiplication
ISBN 1-256-49082-2

The product of two numbers with like signs is positive, and the product of two numbers with unlike signs is negative.

6 # 7 = 42 6 # ( - 7) = - 42

Like signs Unlike signs

continued on next page

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

58

CHAPTER 1 INTRODUCTION TO ALGEBRA

continued from previous page

CONCEPT

COMMENTS

EXAMPLES

Division

For real numbers a and b, with b a = a b

0,

#

1 . b

42 = 7 6 - 42 = -7 6

Like signs Unlike signs

The quotient of two numbers with like signs is positive, and the quotient of two numbers with unlike signs is negative. Converting Fractions to Decimals Converting Terminating Decimals to Fractions The fraction a is equivalent to a , b, b where b 0. Write the decimal as a fraction with a denominator equal to a power of 10 and then simplify this fraction.

The fraction 2 is equivalent to 0.2, 9 because 2 , 9 = 0.222 p . 0.55 = 11 # 5 11 55 = = 100 20 # 5 20

1.6

Exercises
MULTIPLICATION AND DIVISION OF REAL NUMBERS

CONCEPTS AND VOCABULARY

1. The solution to a multiplication problem is the _____. 2. The product of a positive number and a negative number is a(n) _____ number. 3. The product of two negative numbers is _____. 4. The solution to a division problem is the _____. 5. The reciprocal of a nonzero number a is _____. 6. Division by zero is undefined because zero has no _____. 7. To divide a by b, multiply a by the _____ of b. 8. In general, - 1 # a = _____. a 9. = a # _____ b 10. A negative number divided by a negative number is a(n) _____ number. 11. A negative number divided by a positive number is a(n) _____ number. 12. To convert 5 to a decimal, divide _____ by _____. 8

Exercises 13–32: Multiply. 13. - 3 # 4 15. 6 # ( - 3) 17. 0 # ( - 2.13) 19. - 6 # ( - 10) 21. - 1 2 23. 25. 27. 28. 29. 30. 31. 32. 14. - 5 # 7 16. 2 # ( - 1) 18. - 2 # ( - 7) 20. - 3 # ( - 1.7) # 0

5 # 1 - 22 22. - 3 # 1 - 12 2 4 4 4 -3 # 7 24. 5 # 1 - 15 2 7 3 8 26. 1000 # ( - 70) - 10 # ( - 20) - 0.5 # 100 - 0.5 # ( - 0.3) - 2 # 3 # ( - 4) # 5 - 3 # ( - 5) # ( - 2) # 10 -6 # 1 # 7 # 1 - 92 # 1 - 32 7 6 9 2 8 # 1 # 5 # - 5 8 1 - 72 -7

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.6 MULTIPLICATION AND DIVISION OF REAL NUMBERS

59

33. Thinking Generally Is the product given by the expression a # ( - a) # ( - a) # a # ( - a) positive or negative if a 7 0? 34. Thinking Generally Is the product given by the expression a # ( - a) # ( - a) # a # ( - a) positive or negative if a 6 0? Exercises 35–44: Evaluate the expression. 35. ( - 5)2 37. ( - 1)3 39. - 24 41. - ( - 2)
3

Exercises 81–88: Write the decimal number as a fraction in lowest terms. 81. 0.25 83. 0.16 85. 0.625 87. 0.6875 82. 0.8 84. 0.35 86. 0.0125 88. 0.21875

36. - 52 38. ( - 6)2 40. - ( - 4)2 42. 3 # ( - 3) 44. - 14
2

Exercises 89 –96: Use a calculator to evaluate each expression. Express your answer as a decimal and as a fraction. 89.

11 3

+ 52 , 1 6 2

90. 4 - 1 + 2 9 6 3 92. 4 - 7 # 2 4 94. 1 - 3 + 7 6 8 5 96. 3 4

43. 5 # ( - 2)3 Exercises 45–68: Divide. 45. - 10 , 5 47. - 20 , ( - 2)
12 49. -3

91. 4 , 2 # 7 3 4 5 93. 15 - 4 # 7 2 3 95. 17 + 3 , 8 40
APPLICATIONS

46. - 8 , 4 48. - 15 , ( - 3) 50. - 25 -5 52. 10 ,
0 54. - 5 0 56. - 2

# 16 + 12 2

51. - 16 , 1 2 53. 0 , 3 55. - 1 0 57. 1 , ( - 11) 2 59. 61. -4 5 -3
5 6

1 - 12 3

58. - 3 , ( - 6) 4 60.
7 8

97. Top-Grossing Movies (Refer to Example 7.) The Ten Commandments (1956) is the fifth top-grossing movie of all time. Find the total admissions for The Ten Commandments if they were 13 of the total 20 admissions for the top-grossing movie of all time, Gone With the Wind (1939), which had total admissions of 202 million. (Source: Box Office Mojo.) 98. Planet Climate Saturn has an average surface temperature of - 220°F. Neptune has an average surface temperature that is 3 times that of Saturn. 2 Find the average surface temperature on Neptune.
(Source: NASA.)

-7

,
1 2

1, 0

8 92

62. - 11 , 12 64. - 9 , 0

1 - 11 2 4 13 42

63. -

65. - 0.5 ,

1 2

66. - 0.25 , 68. 1 , 1.5 6

67. - 2 , 0.5 3

99. Uninsured Americans In 2010, the fraction of Americans who did not have health insurance 21 coverage was 125 . Write this fraction as a decimal.
(Source: U.S. Census Bureau.)

CONVERTING BETWEEN FRACTIONS AND DECIMALS

Exercises 69–80: Write the number as a decimal. 69. 1 2
3 71. 16

100. Uninsured Minnesotans In 2010, the fraction of Minnesotans who did not have health insurance 23 coverage was 250 . Write this fraction as a decimal.
(Source: U.S. Census Bureau.)

70. 3 4 72. 1 9 74. 2 1 4 76. 6 7 9
1 78. 6 12

WRITING ABOUT MATHEMATICS

73. 3 1 2
ISBN 1-256-49082-2

75. 5 2 3
7 77. 1 16

101. Division is a fast way to subtract. Consider the division problem - 6 , whose quotient represents the -2 number of - 2s in - 6. Using this idea, explain why the answer is a positive number. 102. Explain how to determine whether the product of three integers is positive or negative.

79. 7 8

80. 11 16

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

SECTIONS 1.5 and 1.6

Checking Basic Concepts
(b) - 10 + ( - 12) + 3 5. Find each product. (a) - 5 # ( - 7) (b) - 1 # 2 2 3

1. Find each sum. (a) - 4 + 4

# 1- 42 5
(c) ( - 5)2

2. Evaluate each expression. (b) - 1.2 - 5.1 + 3.1 (a) 2 - 1 - 2 2 3 9 3. Write an arithmetic expression for the given phrase and then simplify it. (a) The sum of negative one and five (b) The difference between four and negative three 4. The hottest temperature ever recorded at International Falls, Minnesota, was 99 F, and the coldest temperature ever recorded was - 46 F. What is the difference between these two temperatures?

6. Evaluate each expression. (b) 4 # ( - 2)3 (a) - 32 7. Evaluate each expression. (b) - 5 , (a) - 5 , 2 3 8 8. What is the reciprocal of - 7 ? 6 9. Simplify each expression. 10 10 (b) - 2 (c) - 10 (a) -2 2

1- 42 3

(d) - 10 -2

10. Convert each fraction or mixed number to a decimal number. (a) 3 (b) 3 7 8 5

1.7

Properties of Real Numbers
Commutative Properties ● Associative Properties ● Distributive Properties ● Identity and Inverse Properties ● Mental Calculations

A LOOK INTO MATH N

The order in which you perform actions is often important. For example, putting on your socks and then your shoes is not the same as putting on your shoes and then your socks. In mathematical terms, the action of putting on footwear is not commutative with respect to socks and shoes. However, the order in which you tie your shoes and put on a sweatshirt does not matter. So, these two actions are commutative. In mathematics some operations are commutative and others are not. In this section we discuss several properties of real numbers.

Commutative Properties
The commutative property for addition states that two numbers, a and b, can be added in any order and the result will be the same. That is, a + b = b + a. For example, if a person buys 4 DVDs and then buys 2 DVDs or first buys 2 DVDs and then buys 4 DVDs, as shown in Figure 1.29, the result is the same. Either way the person buys a total of 4 + 2 = 2 + 4 = 6 DVDs.

+

=

+
ISBN 1-256-49082-2

Figure 1.29 Commutative Property: 4 + 2 = 2 + 4

There is also a commutative property for multiplication. It states that two numbers, a and b, can be multiplied in any order and the result will be the same. That is, a # b = b # a.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.7 PROPERTIES OF REAL NUMBERS

61

NEW VOCABULARY n Commutative property for addition n Commutative property for multiplication n Associative property for addition n Associative property for multiplication n Distributive properties n Identity property of 0 n Additive identity n Identity property of 1 n Multiplicative identity

For example, if one person rolls 3 dice, each resulting in 6, and another person rolls 6 dice, each resulting in 3, as shown in Figure 1.30, then each person has rolled a total of 3 # 6 = 6 # 3 = 18.

=
Figure 1.30 Commutative Property: 3 # 6 = 6 # 3

We can summarize these results as follows.

COMMUTATIVE PROPERTIES
For any real numbers a and b, a + b = b + a
Addition

and

a # b = b # a.
Multiplication

EXAMPLE 1

Applying the commutative properties
Use a commutative property to rewrite each expression. (a) 15 + 100 (b) a # 8
Solution (a) By the commutative property for addition 15 + 100 can be written as 100 + 15. (b) By the commutative property for multiplication a 8 can be written as 8 a or 8a.

#

#

Now Try Exercises 13, 19

While there are commutative properties for addition and multiplication, the operations of subtraction and division are not commutative. Table 1.9 shows each of the four arithmetic operations along with examples illustrating whether or not each operation is commutative.
READING CHECK
• Which of the four arithmetic operations are commutative and which are not?

TABLE 1.9 Commutativity of Operations

Operation + -

Commutative? Yes No Yes No

Example 4 + 9 8#5 9 + 4 5#8 5 - 3 3 3 - 5 4 , 2 3 2 , 4

#
,

Associative Properties
The associative properties allow us to change how numbers are grouped. For example, if a person buys 1, 2, and 4 energy drinks, as shown in Figure 1.31, then the total number of drinks can be calculated either as (1
ISBN 1-256-49082-2

2) + 4 = 3 + 4 = 7

or as 1 + (2

4) = 1 + 6 = 7.

(

+

)

+

=

+

(

+

)

Figure 1.31 Associative Property: (1 + 2) + 4 = 1 + (2 + 4)

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In either case we obtain the same answer, 7 drinks, which is the result of the associative property for addition. We did not change the order of the numbers; we only changed how the numbers were grouped. There is also an associative property for multiplication, which can be illustrated by considering the total number of flowers shown in Figure 1.32, where 2 shelves hold 3 pots each, and each pot contains 4 flowers. The total number of flowers can be calculated either as (2 # 3) # 4 = 6 # 4 = 24 or as 2 # (3 # 4) = 2 # 12 = 24.

Figure 1.32

We can summarize these results as follows.

ASSOCIATIVE PROPERTIES
For any real numbers a, b, and c, (a + b) + c = a + (b + c)
Addition

and

(a # b) # c = a # (b # c).
Multiplication

NOTE: Sometimes we omit the multiplication dot. Thus a b = ab and 5 x y = 5x y.

#

# #

EXAMPLE 2

Applying the associative properties
Use an associative property to rewrite each expression. (a) (5 + 6) + 7 (b) x(y])
Solution (a) The given expression is equivalent to 5 + (6 + 7). (a) The given expression is equivalent to (x y)].
Now Try Exercises 21, 23

EXAMPLE 3

Identifying properties of real numbers
State the property that each equation illustrates. (a) 5 # (8y) = (5 # 8)y (b) 3 # 7 = 7 # 3 (c) x + y] = y] + x
Solution (a) This equation illustrates the associative property for multiplication because the grouping of the numbers has been changed. (b) This equation illustrates the commutative property for multiplication because the order of the numbers 3 and 7 has been changed. (c) This equation illustrates the commutative property for addition because the order of the terms x and y] has been changed.
Now Try Exercises 53, 55, 63

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Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.7 PROPERTIES OF REAL NUMBERS

63

While there are associative properties for addition and multiplication, the operations of subtraction and division are not associative. Table 1.10 shows each of the four arithmetic operations along with examples illustrating whether or not each operation is associative.
TABLE 1.10 Associativity of Operations

Operation
READING CHECK
• Which of the four arithmetic operations are associative and which are not?

Associative? Yes No Yes No

Example (3 + 6) + 7 (4 # 5) # 3 3 + (6 + 7) 4 # (5 # 3) (10 - 3) - 1 3 10 - (3 - 1) (16 , 8) , 2 3 16 , (8 , 2)

+ -

#
,

STUDY TIP
The information in Making Connections ties the current concepts to those studied earlier. By reviewing your notes often, you can gain a better understanding of mathematics.

MAKING CONNECTIONS
Commutative and Associative Properties

Both the commutative and associative properties work for addition and multiplication. However, neither property works for subtraction or division.

Distributive Properties
The distributive properties are used frequently in algebra to simplify expressions. Arrows are often used to indicate that a distributive property is being applied. 4(2 + 3) = 4 # 2 + 4 # 3 The 4 must be multiplied by both the 2 and the 3—not just the 2. The distributive property remains valid when addition is replaced with subtraction. 4(2 - 3) = 4 # 2 - 4 # 3 We illustrate a distributive property geometrically in Figure 1.33. Note that the area of one rectangle that is 4 squares by 5 squares is the same as the area of two rectangles: one that is 4 squares by 2 squares and another that is 4 squares by 3 squares. In either case the total area is 20 square units.
Distributive Property

4
ISBN 1-256-49082-2

=4

+4

2+3

2

3

Figure 1.33 4(2 + 3) = 4 # 2 + 4 # 3

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CHAPTER 1 INTRODUCTION TO ALGEBRA

DISTRIBUTIVE PROPERTIES
For any real numbers a, b, and c, a(b + c) = ab + ac and a(b - c) = ab - ac.

NOTE: Because multiplication is commutative, the distributive properties can also be written as

(b + c)a = ba + ca and (b - c)a = ba - ca.

EXAMPLE 4

Applying the distributive properties
Apply a distributive property to each expression. (a) 3(x + 2) (b) - 6(a - 2) (c) - (x + 7) (d) 15 - (b + 4)
Solution (a) Both the x and the 2 must be multiplied by 3.

3(x + 2) = 3 # x + 3 # 2 = 3x + 6 (b) 6(a - 2) = 6#a - ( 6) # 2 = - 6a + 12
In general, -a = -1 # a. Distributive property Multiply. Change subtraction to addition. Distributive property Multiply. Simplify.

(c) - (x + 7) = ( - 1)(x + 7) = ( - 1) # x + ( - 1) # 7 = -x - 7 (d) 15 - (b + 4) = 15 + ( - 1)(b + 4) = 15 + ( - 1) # b + ( - 1) # 4 = 15 - b - 4 = 11 - b

NOTE: To simplify the expression 15 - (b + 4), we subtract both the b and the 4. Thus we can quickly simplify the given expression to

15 - (b + 4) = 15 - b - 4.
Now Try Exercises 31, 35, 37, 41

EXAMPLE 5

Inserting parentheses using the distributive property
Use a distributive property to insert parentheses in the expression and then simplify the result. (a) 5a + 2a (b) 3x - 7x (c) - 4y + 5y
Solution (a) Because a is a factor in both 5a and 2a, we use the distributive property to write the a outside of parentheses.

5a + 2 a = (5 + 2) a Distributive property = 7a (b) 3x - 7x = (3 - 7) x = - 4 x (c) - 4y + 5y = ( - 4 + 5)y = 1y = y
Now Try Exercises 45, 47, 51

Simplify.

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1.7 PROPERTIES OF REAL NUMBERS

65

EXAMPLE 6

Identifying properties of real numbers
State the property or properties illustrated by each equation. (a) 4(5 - x) = 20 - 4 x (b) (4 + x) + 5 = x + 9 (c) 5] + 7] = 12] (d) x(y + ]) = ]x + yx
Solution (a) This equation illustrates the distributive property with subtraction.

4(5 - x) = 4 # 5 - 4 # x = 20 - 4 x (b) This equation illustrates the commutative and associative properties for addition. (4 + x) + 5 = (x + 4) + 5 = x + (4 + 5) = x + 9
Commutative property for addition Associative property for addition Simplify.

(c) This equation illustrates the distributive property with addition. 5z + 7z = (5 + 7)z = 12] (d) This equation illustrates a distributive property with addition and commutative properties for addition and multiplication. x(y + ]) = xy + x] = x] + xy = ]x + yx
Now Try Exercises 55, 57, 59, 61

Distributive property Commutative property for addition Commutative property for multiplication

Identity and Inverse Properties
The identity property of 0 states that if 0 is added to any real number a, the result is a. The number 0 is called the additive identity. Examples include - 4 + 0 = - 4 and 0 + 11 = 11. The identity property of 1 states that if any number a is multiplied by 1, the result is a. The number 1 is called the multiplicative identity. Examples include - 3 # 1 = - 3 and 1 # 8 = 8. We can summarize these results as follows.

IDENTITY PROPERTIES
For any real number a, a + 0 = 0 + a = a
Additive identity

and

a # 1 = 1 # a = a.
Multiplicative identity

ISBN 1-256-49082-2

READING CHECK
• Restate the identity property for 0 in your own words. • Restate the identity property for 1 in your own words.

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The additive inverse, or opposite, of a number a is - a. The number 0 is its own opposite. The sum of a number a and its additive inverse equals the additive identity 0. Thus - 5 + 5 = 0 and x + ( - x) = 0. The multiplicative inverse, or reciprocal, of a nonzero number a is 1 . The number 0 has a no multiplicative inverse. The product of a number and its multiplicative inverse equals the multiplicative identity 1. Thus - 5 # 1 - 4 2 = 1. 4 5

INVERSE PROPERTIES
For any real number a, a + ( - a) = 0 and - a + a = 0. 1 a
Additive inverse

For any nonzero real number a, a

#1 a = 1 and

# a = 1.

Multiplicative inverse

EXAMPLE 7

Identifying identity and inverse properties
State the property or properties illustrated by each equation. (a) 0 + xy = xy (b) 36 = 6 # 6 = 6 30 5 6 5 (c) x + ( - x) + 5 = 0 + 5 = 5 (d) 1 # 9y = 1 # y = y 9

Solution (a) This equation illustrates use of the identity property for 0. (b) Because 6 = 1, these equations illustrate how a fraction can be simplified by using the 6 identity property for 1. (c) These equations illustrate use of the additive inverse property and the identity property for 0. (d) These equations illustrate use of the multiplicative inverse property and the identity property for 1.
Now Try Exercises 67, 69, 71, 73

Mental Calculations
Properties of numbers can be used to simplify calculations. For example, to find the sum 4 + 7 + 6 + 3 we might apply the commutative and associative properties for addition to obtain (4 6) + (7 3) = 10 + 10 = 20.

Suppose that we are to add 128 + 19 mentally. One way is to add 20 to 128 and then subtract 1. 128 + 19 = 128 + (20 = 148 - 1 = 147 1)
19 = 20 - 1. Associative property
ISBN 1-256-49082-2

= (128 + 20) - 1

Add. Subtract.

The distributive property can be helpful when we multiply mentally. For example, to determine the number of people in a marching band with 7 columns and 23 rows we need to

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1.7 PROPERTIES OF REAL NUMBERS

67

find the product 7 # 23. To evaluate the product mentally, think of 23 as 20 + 3. 7 # 23 = 7(20 3)
23 = 20 + 3.

= 7 # 20 + 7 # 3 Distributive property = 140 + 21 = 161
Multiply. Add.

EXAMPLE 8

Performing calculations mentally
Use properties of real numbers to calculate each expression mentally. (a) 21 + 15 + 9 + 5 (b) 1 # 2 # 2 # 3 2 3 2 (c) 523 + 199 (d) 6 # 55
Solution (a) Use the commutative and associative properties to group numbers into pairs that sum to a multiple of 10.

21 + 15 + 9 + 5 = (21 1 2

9) + (15 2 3

5) = 30 + 20 = 50

(b) Use the commutative and associative properties to group numbers with their reciprocals.

#2#2#3
3 2

= a

1 2

# 2b #

a

# 3b
2

= 1#1 = 1

(c) Instead of adding 199, add 200 and then subtract 1. 523 + 200 6 # (50
Now Try Exercises 77, 81, 85, 91

1 = 723 - 1 = 722 5) = 300 + 30 = 330

(d) Think of 55 as 50 + 5 and then apply the distributive property.

CRITICAL THINKING
How could you quickly calculate 5283 - 198 without a calculator?

The next example illustrates how the commutative and associative properties for multiplication can be used together to simplify a product.

EXAMPLE 9

Finding the volume of a swimming pool
An Olympic swimming pool is 50 meters long, 25 meters wide, and 2 meters deep. The volume V of the pool is found by multiplying 50, 25, and 2. Use the commutative and associative properties for multiplication to calculate the volume of the pool mentally.

ISBN 1-256-49082-2

Solution Because 50 2 = 100 and multiplication by 100 is relatively easy, it may be convenient to order and group the multiplication as (50 2) 25 = 100 25 = 2500. Thus the pool contains 2500 cubic meters of water.

#

#

#

#

Now Try Exercise 103

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CHAPTER 1 INTRODUCTION TO ALGEBRA

1.7
Commutative

Putting It All Together
DEFINITION EXAMPLES

PROPERTY

For any real numbers a and b, a + b = b + a and a # b = b # a. 4 + 6 = 6 + 4 4#6 = 6#4 (3 + 4) + 5 = 3 + (4 + 5) (3 # 4) # 5 = 3 # (4 # 5) 5(x + 2) = 5x + 10 5(x - 2) = 5x - 10 5 + 0 = 5 and 5 # 1 = 5

Associative

For any real numbers a, b, and c, (a + b) + c = a + (b + c) (a # b) # c = a # (b # c). and

Distributive

For any real numbers a, b, and c, a(b + c) = ab + ac a(b - c) = ab - ac. and

Identity (0 and 1)

The identity for addition is 0, and the identity for multiplication is 1. For any real number a, a + 0 = a and a # 1 = a. The additive inverse of a is - a, and a + ( - a) = 0. The multiplicative inverse of a nonzero number a is 1 , a and a # 1 = 1. a

Inverse

8 + ( - 8) = 0

and

2 3

#3 2

= 1

1.7

Exercises
7. a(b + c) = ab + ac illustrates the _____ property. 8. a(b - c) = ab - ac illustrates the _____ property. 9. The equations a + 0 = a and 0 + a = a each illustrate the _____ property for _____. 10. The equations a # 1 = a and 1 # a = a each illustrate the _____ property for _____. 11. The additive inverse, or opposite, of a is _____ . 12. The multiplicative inverse, or reciprocal, of a nonzero number a is _____ .
ISBN 1-256-49082-2

CONCEPTS AND VOCABULARY

1. a + b = b + a illustrates the _____ property for _____. 2. a # b = b # a illustrates the _____ property for _____. 3. (a + b) + c = a + (b + c) illustrates the _____ property for _____. 4. (a # b) # c = a # (b # c) illustrates the _____ property for _____. 5. (True or False?) Addition and multiplication are commutative. 6. (True or False?) Subtraction and division are associative.

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1.7 PROPERTIES OF REAL NUMBERS

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PROPERTIES OF REAL NUMBERS

49. 3a - a 51. 13w - 27w

50. - 2 x + 5x 52. 25a - 21a

Exercises 13–20: Use a commutative property to rewrite the expression. Do not simplify. 13. - 6 + 10 15. - 5 # 6 17. a + 10 19. b # 7 14. 23 + 7 16. 25 # ( - 46) 18. b + c 20. a # 23

Exercises 53–66: State the property or properties that the equation illustrates. 53. x # 5 = 5x 55. (a + 5) + 7 = a + 12 56. (9 + a) + 8 = a + 17 57. 4(5 + x) = 20 + 4 x 59. x(3 - y) = 3x - xy 61. 6 x + 9x = 15x 63. 3 # (4 # a) = 12 a 65. - (t - 7) = - t + 7 66. ] # 5 - y # 6 = 5] - 6y
IDENTITY AND INVERSE PROPERTIES

54. 7 + a = a + 7

Exercises 21–28: Use an associative property to rewrite the expression. Do not simplify. 21. (1 + 2) + 3 23. 2 # (3 # 4) 25. (a + 5) + c 27. (x # 3) # 4 22. - 7 + (5 + 15) 24. (9 # ( - 4)) # 5 26. (10 + b) + a 28. 5 # (x # y)

58. 3(5 + x) = 3x + 15 60. - (u - v) = - u + v 62. 9x - 11x = - 2 x 64. (x # 3) # 5 = 15x

29. Thinking Generally Use the commutative and associative properties to show that a + b + c = c + b + a. 30. Thinking Generally Use the commutative and associative properties to show that a # b # c = c # b # a. Exercises 31–42: Use a distributive property to rewrite the expression. Then simplify the expression. 31. 4(3 + 2) 33. a(b - 8) 35. - 4(t - ]) 37. - (5 - a) 39. (a + 5)3 41. 12 - (a - 5) 32. 5(6 - 9) 34. 3(x + y) 36. - 1(a + 6) 38. 12 - (4u - b) 40. (x + y)7 42. 4 x - 2(3y - 5)

Exercises 67–76: State the property or properties that are illustrated. 67. 0 + x = x 69. 1 # a = a 71. 25 = 5 # 5 = 5 3 5 3 15
1 73. xy # xy = 1

68. 5x + 0 = 5x 70. 1 # (7 # a) = a 7 72. 50 = 5 # 10 = 5 40 4 10 4
1 74. a + b # (a + b) = 1

75. - xy] + xy] = 0
MENTAL CALCULATIONS

76. 1 + y

1- 12 y

= 0

43. Thinking Generally Use properties of real numbers to show that the distributive property can be extended as follows.* a # (b + c + d) = ab + ac + ad. 44. Thinking Generally Use properties of real numbers to show that the distributive property can be extended as follows.* a # (b - c - d) = ab - ac - ad. Exercises 45–52: Use the distributive property to insert parentheses in the expression and then simplify the result. 45. 6 x + 5x 47. - 4b + 3b 46. 4y - y 48. 2b + 8b

Exercises 77–92: Use properties of real numbers to calculate the expression mentally. 77. 4 + 2 + 9 + 8 + 1 + 6 78. 21 + 32 + 19 + 8 79. 45 + 43 + 5 + 7 80. 5 + 7 + 12 + 13 + 8 81. 129 + 49 83. 178 - 99 85. 6 # 15 87. 8 # 102 89. 1 # 1 # 1 # 2 # 2 # 2 2 2 2 91. 7 # 1 # 1 # 1 # 8 6 2 2 2 7 82. 87 + 99 84. 500 - 101 86. 4 # 56 88. 5 # 999 90. 1 # 4 # 7 # 2 # 5 2 5 3 4
4 92. 11 # 11 # 6 # 7 6 7 4

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MULTIPLYING AND DIVIDING BY POWERS OF 10 MENTALLY

93. Multiplying by 10 To multiply an integer by 10, attach one 0 to the number. For example, 10 # 23 = 230. Simplify each expression mentally. (a) 10 # 41 (b) 10 # 997 (d) - 14,000 # 10 (c) - 630 # 10 94. Multiplying by 10 To multiply a decimal number by 10, move the decimal point one place to the right. For example, 10 # 23.45 = 234.5. Simplify each expression mentally. (a) 10 # 101.68 (b) 10 # ( - 1.235) # 10 (d) 0.567 # 10 (c) - 113.4 # 0.0045 (f) - 0.05 # 10 (e) 10 95. Multiplying by Powers of 10 To multiply an integer by a power of 10 in the form 10 k = 100 p * ( 1)10, k zeros

(a) 78.89 , 100 (c) 5678 , 10,000 (e) - 101 , 100,000
APPLICATIONS

(b) 0.05 , 1000 (d) - 9.8 , 1000 (f) 7.8 , 100

99. Earnings Earning $100 one day and $75 the next day is equivalent to earning $75 the first day and $100 the second day. What property of real numbers does this example illustrate? 100. Leasing a Car An advertisement for a lease on a new car states that it costs $2480 down and $201 per month for 20 months. Mentally calculate the cost of the lease. Explain your reasoning. 101. Gasoline Mileage A car travels 198 miles on 10 gallons of gasoline. Mentally calculate the number of miles that the car travels on 1 gallon of gasoline. 102. Gallons of Water A wading pool is 50 feet long, 20 feet wide, and 1 foot deep. (a) Mentally determine the number of cubic feet in the pool. (Hint: Volume equals length times width times height.) (b) One cubic foot equals about 7.5 gallons. Mentally calculate the number of gallons of water in the pool. 103. Dimensions of a Pool A small pool of water is 13 feet long, 5 feet wide, and 2 feet deep. The volume V of the pool in cubic feet is found by multiplying 13 by 5 by 2. (a) To do this calculation mentally, would you multiply (13 # 5) # 2 or 13 # (5 # 2)? Why? (b) What property allows you to do either calculation and still obtain the correct answer? 104. Digital Images of Io The accompanying picture of Jupiter’s moon Io is a digital picture, created by using a rectangular pattern of small pixels. This image is 500 pixels wide and 400 pixels high, so the total number of pixels in it is 500 # 400 = 200,000 pixels.
(Source: NASA.)

attach k zeros to the number. Some examples of this are 100 # 45 = 4500, 1000 # 235 = 235,000, and 10,000 # 12 = 120,000. Simplify each expression mentally. (a) 1000 # 19 (b) 100 # ( - 451) (d) - 79 # 100,000 (c) 10,000 # 6 96. Multiplying by Powers of 10 To multiply a decimal number by a power of 10 in the form 10 k = 100 p * ( 1)10, k zeros

move the decimal point k places to the right. For example, 100 # 1.234 = 123.4. Simplify each given expression mentally. (a) 1000 # 1.2345 (b) 100 # ( - 5.1) (d) 0.567 # 10,000 (c) 45.67 # 1000 # 0.0005 (f) - 0.05 # 100,000 (e) 100 97. Dividing by 10 To divide a number by 10, move the decimal point one place to the left. For example, 78.9 , 10 = 7.89. Simplify each expression mentally. (a) 12.56 , 10 (b) 9.6 , 10 (c) 0.987 , 10 (d) - 0.056 , 10 (e) 1200 , 10 (f) 4578 , 10 98. Dividing by Powers of 10 To divide a decimal number by a power of 10 in the form 10k = 100 p * ( 1)10, k zeros

move the decimal point k places to the left. For example, 123.4 , 100 = 1.234. Simplify each expression mentally.

(a) Find the total number of pixels in an image 400 pixels wide and 500 pixels high. (b) Suppose that a picture is x pixels wide and y pixels high. What property states that it has the same number of pixels as a picture y pixels wide and x pixels high?

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1.8 SIMPLIFYING AND WRITING ALGEBRAIC EXPRESSIONS

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WRITING ABOUT MATHEMATICS

105. To determine the cost of tuition, a student tries to compute 16 # $96 with a calculator and gets $15,360. How do you know that this computation is not correct? 106. A student performs the following computation by hand. 20 - 6 - 2 + 8 , 4 , 2 20 - 4 + 8 , 2 20 - 4 + 4 20 Find any incorrect steps and explain what is wrong. What is the correct answer?

107. The computation 3 + 1020 - 1020 performed on a calculator gives a result of 0. (Try it.) What is the correct answer? Why is it important to know properties of real numbers even though you have a calculator? 108. Does the “distributive property” a + (b # c) (a + b) # (a + c) hold for all numbers a, b, and c? Explain your reasoning.

Group Activity

Working with Real Data

Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond. Winning the Lottery In the multistate lottery game Powerball, there are 120,526,770 possible number combinations, only one of which is the grand prize winner. The cost of a single ticket (one number combination) is $1. (Source: Powerball.com.) Suppose that a very wealthy person decides to buy tickets for every possible number combination to be assured of winning a $150 million grand prize. (a) If this individual could purchase one ticket every second, how many hours would it take to buy all of the tickets? How many years is this? (b) If there were a way for this individual to buy all possible number combinations quickly, discuss reasons why this strategy would probably lose money.

1.8

Simplifying and Writing Algebraic Expressions
Terms ● Combining Like Terms ● Simplifying Expressions ● Writing Expressions

A LOOK INTO MATH N

Filing taxes might be easier if several of the complicated tax formulas could be combined into a single formula. Furthermore, some of the large formulas might be easier to evaluate if they were simplified to be more concise. In mathematics, we combine like terms when we simplify expressions. In this section, we simplify algebraic expressions.

Terms
One way to simplify expressions is to combine like terms. A term is a number, a variable, or a product of numbers and variables raised to natural number powers. Examples include
NEW VOCABULARY
ISBN 1-256-49082-2

n Term n Coefficient n Like terms

4, ], 5x,

2 ], 5

- 4 xy,

- x 2, and 6 x 3y 4.

Terms

Terms do not contain addition or subtraction signs, but they can contain negative signs. The coefficient of a term is the number that appears in the term. If no number appears, then the coefficient is understood to be either - 1 or 1. Table 1.11 on the next page shows examples of terms and their corresponding coefficients.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

READING CHECK
• How do you identify the coefficient in a term?

TABLE 1.11 Terms and Coefficients

Term
1 2 xy

Coefficient
1 2

- ]2 12 - 5w 4

-1 12 -5

EXAMPLE 1

Identifying terms
Determine whether each expression is a term. If it is a term, identify its coefficient. (a) 51 (b) 5a (c) 2 x + 3y (d) - 3x 2
Solution (a) A number is a term. This term’s coefficient is 51. (b) The product of a number and a variable is a term. This term’s coefficient is 5. (c) The sum (or difference) of two terms is not a term. (d) The product of a number and a variable with an exponent is a term. This term’s coefficient is - 3.
Now Try Exercises 7, 9, 11, 13

STUDY TIP
One of the best ways to prepare for class is to read a section before it is covered by your instructor. Reading ahead gives you the chance to formulate any questions you might have about the concepts in the section.

MAKING CONNECTIONS
Factors and Terms

When variables and numbers are multiplied, they are called factors. For example, the expression 4xy has factors of 4, x, and y. When variables and numbers are added or subtracted, they are called terms. For example, the expression x - 5xy + 1 has three terms: x, - 5xy, and 1.

Combining Like Terms
READING CHECK
• Which property of real numbers is used to combine like terms?

Suppose that we have two boards with lengths 2 x and 3x, where the value of x could be any length such as 2 feet. See Figure 1.34. Because 2 x and 3x are like terms we can find the total length of the two boards by applying the distributive property and adding like terms. 2 x + 3x = (2 + 3)x = 5x The combined length of the two boards is 5x units.
Visualizing Like Terms

2x

3x

Figure 1.34 2x + 3x = 5x

We can also determine the difference between the lengths of the two boards by subtracting like terms. 3x - 2 x = (3 - 2)x = 1x = x The second board is x units longer than the first board.

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1.8 SIMPLIFYING AND WRITING ALGEBRAIC EXPRESSIONS

73

If two terms contain the same variables raised to the same powers, we call them like terms. We can add or subtract (combine) like terms but cannot combine unlike terms. For example, if one board has length 2 x and the other board has length 3y, then we cannot determine the total length other than to say that it is 2 x + 3y. See Figure 1.35. The terms 2 x and 3y are unlike terms and cannot be combined.

Visualizing Unlike Terms

READING CHECK
• How can you tell whether two terms are like terms?

2x

3y

Figure 1.35 2x + 3y

EXAMPLE 2

Identifying like terms
Determine whether the terms are like or unlike. (a) - 4m, 7m (b) 8 x 2, 8y 2 (c) 1 ], - 3] 2 (d) 5, - 4n 2
Solution (a) The variable in both terms is m (with power 1), so they are like terms. (b) The variables are different, so they are unlike terms. (c) The term - 3] 2 contains ] 2 = ] ], whereas the term 1 ] contains only ]. Thus they are 2 unlike terms. (d) The term 5 has no variable, whereas the term - 4n contains the variable n. They are unlike terms.

#

Now Try Exercises 19, 21, 23, 27

EXAMPLE 3

Combining like terms
Combine terms in each expression, if possible. (a) - 3x + 5x (b) - x 2 + 5x 2 (c) 1 y - 3y 3 2
Solution (a) Combine terms by applying a distributive property.

- 3x + 5x = ( - 3 + 5)x = 2 x (b) Note that - x can be written as - 1x 2 . They are like terms and can be combined. - x 2 + 5x 2 = ( - 1 + 5)x 2 = 4x 2 (c) They are unlike terms, so they cannot be combined.
Now Try Exercises 31, 39, 43
2

Simplifying Expressions
ISBN 1-256-49082-2

The area of a rectangle equals length times width. In Figure 1.36 on the next page the area of the first rectangle is 3x, the area of the second rectangle is 2 x, and the area of the third rectangle is x. The area of the last rectangle equals the total area of the three smaller rectangles. That is, 3x + 2 x + x = (3 + 2 + 1)x = 6 x. The expression 3x + 2 x + x can be simplified to 6 x.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CRITICAL THINKING
Use rectangles to explain how to add xy + 2xy. x Simplifying an Expression Visually
+ x + x = x

3

2

1

6

Figure 1.36 3x + 2x + x = 6x

EXAMPLE 4

Simplifying expressions
Simplify each expression. (a) 2 + x - 6 + 5x (b) 2y - (y + 3) (c) - 1.5x - 1.5

Solution (a) Combine like terms by applying the properties of real numbers.

2 + x - 6 + 5x = 2 + ( - 6) + x + 5x = 2 + ( - 6) + (1 + 5) x = - 4 + 6x (b) 2y - 1(y + 3) = 2y + ( - 1)y + ( - 1) # 3 = 2y - 1y - 3 = (2 - 1)y - 3 = y - 3 (c) - 1.5x - 1.5 = - 1.5 - 1.5 = 1#x = x
Now Try Exercises 47, 59, 75

Commutative property Distributive property Add. Distributive property Definition of subtraction Distributive property Subtract.

#x

1

Multiplication of fractions Simplify the fractions. Multiplicative identity

NOTE: The expression in Example 4(c) can be simplified directly by using the basic principle of fractions: ac = a . bc b

EXAMPLE 5

Simplifying expressions
Simplify each expression. (a) 7 - 2(5x - 3) (b) 3x 3 + 2 x 3 - x 3 (c) 3y 2 - z + 4y 2 - 2z
Solution (a) 7

(d)

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

2(5x

3) = 7 + ( = 7 + (

Change subtraction to addition. Distributive property Multiply. Combine like terms.
ISBN 1-256-49082-2

= 7 - 10 x + 6 = 13 - 10 x (b) 3x + 2 x - x = (3 + 2 - 1) x
3 3 3 3

Distributive property Add and subtract.

= 4x

3

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.8 SIMPLIFYING AND WRITING ALGEBRAIC EXPRESSIONS

75

(c) 3y 2

z + 4y 2

2z = 3y 2 + 4y 2 + (
2

1z) + (

2z) Commutative property
Distributive property Add. Subtraction of fractions Simplify fractions.

= (3 + 4)y + ( - 1 + ( - 2))] = 7y - 3]
2

(d)

12 x - 8 12 x 8 = 4 4 4 = 3x - 2

Now Try Exercises 57, 65, 71, 79

Recall that the commutative and associative properties of addition allow us to rearrange a sum in any order. For example, if we write the expression 2x 3 4x 10 as the sum 2x + ( 3 10) = 3) + ( 10). 4 x) 10,

the terms can be arranged and grouped as (2 x + ( (2 + ( 4 x)) + ( 3

Applying the distributive property and adding within each grouping results in 4))x + ( 2x 7.

Note that the terms in the result can be found directly by combining like (same color) terms in the given expression, where addition indicates that a term is positive and subtraction indicates that a term is negative. In the next example, we simplify an expression in this way.

EXAMPLE 6

Simplifying an expression directly
Simplify the expression 6 x + 5 - 2 x - 8.
Solution The like terms and their indicated signs are shown.

6x

5

2x

8

By combining like terms, the expression can be simplified as 4x
Now Try Exercise 53

3.

Writing Expressions
N REAL-WORLD CONNECTION In real-life situations, we often have to translate words to symbols. For example, to calculate federal and state income tax we might have to multiply taxable income by 0.15 and by 0.05 and then find the sum. If we let x represent taxable income, then the total federal and state income tax is 0.15x + 0.05x. This expression can be simplified with a distributive property.
ISBN 1-256-49082-2

0.15x + 0.05x = (0.15 + 0.05)x = 0.20 x Thus the total income tax on a taxable income of x = $20,000 would be 0.20(20,000) = $4000.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

EXAMPLE 7

Writing and simplifying an expression
A sidewalk has a constant width w and comprises several short sections with lengths 12, 6, and 5 feet, as illustrated in Figure 1.37. (a) Write and simplify an expression that gives the number of square feet of sidewalk. (b) Find the area of the sidewalk if its width is 3 feet.

6 ft 5 ft w

12 ft

Figure 1.37

Solution (a) The area of each sidewalk section equals its length times its width w. The total area of the sidewalk is

12w + 6w + 5w = (12 + 6 + 5)w = 23w. (b) When w = 3, the area is 23w = 23 # 3 = 69 square feet.
Now Try Exercise 89

1.8
Term

Putting It All Together
COMMENTS EXAMPLES

CONCEPT

A term is a number, variable, or product of numbers and variables raised to natural number powers. The coefficient of a term is the number that appears in the term. If no number appears, then the coefficient is either 1 or - 1. Like terms have the same variables raised to the same powers.

12, - 10, x, - y - 3x, 5], xy, 6x 2 10y 3, 3 x, 25xy] 2 4 12 x - xy - 4x Coefficient is 12 Coefficient is 1 Coefficient is - 1 Coefficient is - 4

Coefficient of a Term

Like Terms

5m and - 6m x 2 and - 74 x 2 2 xy and - xy
ISBN 1-256-49082-2

Combining Like Terms

Like terms can be combined by using a distributive property.

5x + 2 x = (5 + 2)x = 7x y - 3y = (1 - 3)y = - 2y 8 x 2 + x 2 = (8 + 1)x 2 = 9x 2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

1.8 SIMPLIFYING AND WRITING ALGEBRAIC EXPRESSIONS

77

1.8

Exercises
Exercises 31–46: Combine terms, if possible. 31. - 4x + 7x 33. 19y - 5y 35. 28a + 13a 37. 11] - 11] 39. 5x - 7y 41. 5 + 5y 43. 5x 2 - 2 x 2 45. 8y - 10y + y 32. 6 x - 8 x 34. 22] + ] 36. 41b - 17b 38. 4y + 4y 40. 3y + 3] 42. x + x 2 44. 25] 3 - 10] 3 46. 4 x 2 + x 2 - 5x 2

CONCEPTS AND VOCABULARY

1. A(n) is a number, a variable, or a product of numbers and variables raised to natural number powers. 2. The number 7 in the term 7x 2y is called the _____ of the term. 3. When variables and numbers are multiplied, they are called _____. When they are added or subtracted, they are called _____. 4. If two terms contain the same variables raised to the same powers, they are (like/unlike) terms. 5. We can add or subtract (like/unlike) terms. 6. We can combine like terms in an expression by applying a(n) _____ property.
LIKE TERMS

SIMPLIFYING AND WRITING EXPRESSIONS

Exercises 47–82: Simplify the expression. 47. 5 + x - 3 + 2x 49. - 3 + ] - 3] + 5 4 4 51. 4y - y + 8y 53. - 3 + 6] + 2 - 2] 55. - 2(3] - 6y) - ] 57. 2 - 3 (4 x + 8) 4 59. - x - (5x + 1) 61. 1 - 1 (x + 1) 3 63. 3 (x + y) - 1 (x - 1) 5 5 64. - 5(a + b) - (a + b) 65. 0.2 x 2 + 0.3x 2 - 0.1x 2 67. 2 x 2 - 3x + 5x 2 - 4 x 69. a + 3b - a - b 71. 8 x 3 + 7y - x 3 - 5y 73. 8x 8 - 3y -y 66. 32] 3 - 52] 3 + 20] 3
1 68. 5 y 2 - 4 + 12 y 2 + 3 6

Exercises 7–16: Determine whether the expression is a term. If the expression is a term, identify its coefficient. 7. 91 9. - 6b 11. x + 10 13. x
2

48. x - 5 - 5x + 7 50. 4 ] - 100 + 200 - 1 ] 3 3 52. 14] - 15] - ] 54. 19a - 12a + 5 - 6 56. 6 1 1 a - 1 b 2 - 3b 2 6 58. - 5 - (5x - 6) 60. 2 x - 4(x + 2) 62. - 3 - 3(4 - x)

8. - 12 10. 9] 12. 20 - 2y 14. 4 x
3

15. 4 x - 5

16. 5] + 6 x

Exercises 17–28: Determine whether the terms are like or unlike. 17. 6, - 8 19. 5x, - 22 x 21. 14, 14a 23. 18 x, 18y 25. x 2, - 15x 2 27. 3x 2, 1 x 5 18. 2 x, 19 20. 19y, - y 22. - 33b, - 3b 24. - 6a, - 6b 26. y, 19y 28. 12y 2, - y 2

70. 2] 2 - ] - ] 2 + 3] 72. 4y - 6] + 2y - 3] 74. - 0.1y - 0.1 2x 7x

29. Thinking Generally Are the terms 4ab and - 3ba like or unlike? 30. Thinking Generally Are the terms - xy] 2 and 3y] 2x like or unlike?

ISBN 1-256-49082-2

75.

76.

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78

CHAPTER 1 INTRODUCTION TO ALGEBRA

77.

- 108] - 108 9x - 6 3 14] + 21 7

78.

3xy - 6 xy 18y + 9 9 15x - 20 5

(b) Find the area of the street if its width is 42 feet. 90. Sidewalk Dimensions A sidewalk has a constant width w and comprises several short sections having lengths 12, 14, and 10 feet. (a) Write a simplified expression that gives the number of square feet of sidewalk. (b) Find the area of the sidewalk if its width is 5 feet. 91. Snowblowers Two snowblowers are being used to clear a driveway. The first blower can remove 20 cubic feet per minute, and the second blower can remove 30 cubic feet per minute. (a) Write a simplified expression that gives the number of cubic feet of snow removed in x minutes. (b) Find the total number of cubic feet of snow removed in 48 minutes. (c) How many minutes would it take to remove the snow from a driveway that measures 30 feet by 20 feet, if the snow is 2 feet deep? 92. Winding Rope Two motors are winding up rope. The first motor can wind up rope at 2 feet per second, and the second motor can wind up rope at 5 feet per second. (a) Write a simplified expression that gives the length of rope wound by both motors in x seconds. (b) Find the total length of rope wound up by the two motors in 3 minutes. (c) How many minutes would it take to wind up 2100 feet of rope by using both motors?
WRITING ABOUT MATHEMATICS

79.

80.

81.

82.

Exercises 83–88: Translate the phrase into a mathematical expression and then simplify. Use the variable x. 83. The sum of five times a number and six times the same number 84. The sum of a number and three times the same number 85. The sum of a number squared and twice the same number squared 86. One-half of a number minus three-fourths of the same number 87. Six times a number minus four times the same number 88. Two cubed times a number minus three squared times the same number
APPLICATIONS

89. Street Dimensions (Refer to Example 7.) A street has a constant width w and comprises several straight sections having lengths 600, 400, 350, and 220 feet.
350

93. The following expression was simplified incorrectly. 3(x - 5) + 5x 3x - 5 + 5x 3x + 5x - 5 (3 + 5)x - 5 8x - 5 Find the error and explain what went wrong. What should the final answer be?

400

220

w

600

94. Explain how to add like terms. What property of real numbers is used?

(a) Write a simplified expression that gives the square footage of the street.

ISBN 1-256-49082-2 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

CHAPTER 1 SUMMARY

79

SECTIONS 1.7 and 1.8

Checking Basic Concepts
7. Determine whether the terms are like or unlike. (a) - 3x, - 3] (b) 4x 2, - 2 x 2 8. Combine like terms in each expression. (a) 5] + 9] (b) 5y - 4 - 8y + 7 9. Simplify each expression. (a) 2y - (5y + 3) (b) - 4(x + 3y) + 2(2 x - y) 35x 2 20 x (d) (c) 20 x2 10. Write “the sum of three times a number and five times the same number” as a mathematical expression with the variable x and then simplify the expression.

1. Use a commutative property to rewrite each expression. (a) y # 18 (b) 10 + x 2. Use an associative and a commutative property to simplify 5 # (y # 4). 3. Simplify each expression. (a) 10 - (5 + x) (b) 5(x - 7) 4. State the property that the equation 5x + 3x = 8x illustrates. 5. Simplify - 4xy + 4xy. 6. Mentally evaluate each expression. (a) 32 + 17 + 8 + 3 (b) 5 # 7 # 6 # 8 6 8 5 (c) 567 - 199

CHAPTER 1
SECTION

Summary
1, 2, 3, 4, p 0, 1, 2, 3, p Two numbers that are multiplied are called factors and the result is called the product.
Example:

1.1

.

NUMBERS , VARIABLES , AND EXPRESSIONS

Sets of Numbers Natural Numbers

Whole Numbers Products and Factors

3
Factor

#

5
Factor

=

15
Product

Prime and Composite Numbers Prime Number

A natural number greater than 1 whose only natural number factors are itself and 1
Examples: 2, 3, 5, 7, 11, 13, and 17

Composite Number and Prime Factorization

A natural number greater than 1 that is not prime; a composite number can be written as a product of two or more prime numbers.
Examples: 24 = 2

#2#2#3

and

18 = 2 # 3 # 3

Important Terms Variable

A symbol or letter that represents an unknown quantity
Examples: a, b, F, x, and y

Algebraic Expression

Consists of numbers, variables, arithmetic symbols, and grouping symbols
Examples: 3x + 1

and 5(x + 2) - y

ISBN 1-256-49082-2

Equation

A statement that two algebraic expressions are equal; an equation always contains an equals sign.
Examples: 1 + 2 = 3

and ] - 7 = 8

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

80

CHAPTER 1 INTRODUCTION TO ALGEBRA

Formula

A special type of equation that expresses a relationship between two or more quantities.
Examples: P = 2l + 2w

and A = lw

SECTION

1.2

.

FRACTIONS

Fractions and Lowest Terms Fraction

A fraction has a numerator, a denominator, and a fraction bar.
Numerator R
Example:

4 d Fraction bar 9

Denominator Q

Lowest Terms

A fraction is in lowest terms if the numerator and denominator have no factors in common.
Example: The fraction 5 is in lowest terms because 3 and 5 have no factors in
3

common.

Simplifying Fractions
Example: 35 = 7
20 4

# #

5 5

= 4 7

a#c b#c

=

a b

Multiplicative Inverse, or Reciprocal The reciprocal of a nonzero number a is 1 . a
Examples: The reciprocal of - 5 is - 5 , and the reciprocal of y is 2 x , provided x
1 2x y

0 and y

0.

Multiplication and Division

a b
Examples: 4
3

#c

d

=

ac bd

and

a c a , = b d b

#d c b and d are nonzero.

b, c, and d are nonzero.

#7 5

= 21 20

# and 3 , 7 = 3 # 5 = 3 # 5 = 15 4 4 7 4 7 28 5 a c a + c + = b b b a c a - c = b b b

Addition and Subtraction with Like Denominators

and

Examples: 5 + 5 =

3

1

3 + 1 5

= 4 5

and 3 - 1 = 3 - 1 = 2 5 5 5 5

Addition and Subtraction with Unlike Denominators Write the expressions with the LCD. Then add or subtract the numerators, keeping the denominator unchanged.
Examples: 9 + 6 = 9
2 9 2 1

- 1 6
.

#2 2 = 2 #2 9 2

2

3 7 4 + 1 # 3 = 18 + 18 = 18 6 3 3 4 1 - 1 # 3 = 18 - 18 = 18 6 3

LCD is 18.

SECTION

1.3

EXPONENTS AND ORDER OF OPERATIONS
3 d Exponent Base S 5

Exponential Expression
ISBN 1-256-49082-2

Example: 34 = 3

# 3 # 3 # 3 = 81

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

CHAPTER 1 SUMMARY

81

Order of Operations Use the following order of operations. First perform all calculations within parentheses and absolute values, or above and below the fraction bar.

1. Evaluate all exponential expressions. 2. Do all multiplication and division from left to right. 3. Do all addition and subtraction from left to right.
NOTE: Negative signs are evaluated after exponents, so - 24 = - 16.
Examples: 100 - 52

# 2 = 100 - 25 # 2
= 100 - 50 = 50

and

4 + 2 5 - 3

#4=6#4 2 = 3#4
= 12

SECTION

1.4

.

REAL NUMBERS AND THE NUMBER LINE

Opposite or Additive Inverse The opposite of the number a is - a.
Examples: The opposite of 5 is - 5, and the opposite of - 8 is - ( - 8) = 8.

Sets of Numbers Integers

p , - 3, - 2, - 1, 0, 1, 2, 3, p where p and q 0 are integers; rational numbers can be written as decimal numbers that either repeat or terminate.
Examples: - 4 , 5, - 7.6, 3 , and 3
3 6 1 p q,

Rational Numbers

Real Numbers

Numbers that can be written as decimal numbers
Examples: - 4 , 5, - 7.6, 3 , p, 23, and - 27
3 6

Irrational Numbers

Real numbers that are not rational
Examples: p, 23, and - 27

Average To calculate the average of a set of numbers, find the sum of the numbers and then divide the sum by how many numbers there are in the set.
Example: The average of 4, 5, 20, and 11 is

4 + 5 + 20 + 11 40 = = 10. 4 4 We divide by 4 because we are finding the average of 4 numbers.
The Number Line
Origin
-3 -2 -1 0 1 2 3

The origin corresponds to the number 0.
Absolute Value If a is positive or 0, then a = a, and if a is negative, then a = - a. The absolute value of a number is never negative.
Examples:

ISBN 1-256-49082-2

5 = 5,

- 5 = 5, and

0 = 0

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

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CHAPTER 1 INTRODUCTION TO ALGEBRA

Inequality If a is located to the left of b on the number line, then a 6 b. If a is located to the right of b on the number line, then a 7 b.
Examples: - 3 6 6

and

-1 7 -2

SECTION

1.5

.

ADDITION AND SUBTRACTION OF REAL NUMBERS

Addition of Opposites

a + ( - a) = 0
Examples: 3 + ( - 3) = 0

and

-5 + 5 = 0 7 7

Addition of Real Numbers To add two real numbers with like signs, do the following.

1. Find the sum of the absolute values of the numbers. 2. Keep the common sign of the two numbers as the sign of the sum. To add two real numbers with unlike signs, do the following. 1. Find the absolute values of the numbers. 2. Subtract the smaller absolute value from the larger absolute value. 3. Keep the sign of the number with the larger absolute value as the sign of the sum.
Examples: - 4 + 5 = 1,

3 + ( - 7) = - 4,

- 4 + ( - 2) = - 6,

and

8 + 2 = 10

Subtraction of Real Numbers For any real numbers a and b, a - b = a + ( - b).
Examples: 5 - 9 = 5 + ( - 9) = - 4

and

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

SECTION

1.6

.

MULTIPLICATION AND DIVISION OF REAL NUMBERS

Important Terms Factors

Numbers multiplied in a multiplication problem
Example: 5 and 4 are factors of 20 because 5

# 4 = 20.

Product

The answer to a multiplication problem
Example: The product of 5 and 4 is 20.

Dividend, Divisor, and Quotient

If a = c, then a is the dividend, b is the divisor, and c is the quotient. b
Example: In the division problems 30 , 5 = 6

and 30 = 6, 30 is the 5 dividend, 5 is the divisor, and 6 is the quotient. 0,

Dividing Real Numbers For real numbers a and b with b

a = a b
Examples:

# 1. b and 5 , 0 is undefined.

8 8 = 1 1 2

#2
1

= 16,

14 ,

2 14 = 3 1

#3
2

=

42 = 21, 2

Signs of Products or Quotients The product or quotient of two numbers with like signs is positive. The product or quotient of two numbers with unlike signs is negative.
Examples: - 4

ISBN 1-256-49082-2

#6=

- 24,

- 2 # ( - 5) = 10,

- 18 6

= - 3, and

- 4 , ( - 2) = 2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

CHAPTER 1 SUMMARY

83

Writing Fractions as Decimals To write the fraction a as a decimal, divide b into a. b
Example: 9 = 0.4 because division of 4 by 9 gives the repeating decimal 0.4444 p .
4

Writing Decimals as Fractions Write the decimal as a fraction with a denominator equal to a power of 10 and then simplify this fraction.
Example: 0.45 = 100 = 20
SECTION

45

9

#

# 55

9 = 20

1.7

.

PROPERTIES OF REAL NUMBERS

Important Properties Commutative

a + b = b + a and a # b = b # a
Examples: 3 + 4 = 4 + 3

and

- 6 # 3 = 3 # ( - 6)

Associative

(a + b) + c = a + (b + c)

and (a # b) # c = a # (b # c)

Examples: (2 + 3) + 4 = 2 + (3 + 4)

(2 # 3) # 4 = 2 # (3 # 4)

Distributive

a(b + c) = ab + ac

and a(b - c) = ab - ac

Examples: 3(x + 5) = 3

#x+3#5 4(5 - 2) = 4 # 5 - 4 # 2
Additive identity is 0. Multiplicative identity is 1.

Identity

a + 0 = 0 + a = a a#1 = 1#a = a

Examples: 5 + 0 = 0 + 5 = 5 and

1 # ( - 4) = - 4 # 1 = - 4

Inverse

a + ( - a) = 0 and - a + a = 0 a # 1 = 1 and 1 # a = 1 a a
Examples: 5 + ( - 5) = 0

and 1 # 2 = 1 2

NOTE: The commutative and associative properties apply to addition and multiplication but not to subtraction and division.
SECTION

1.8

.

SIMPLIFYING AND WRITING ALGEBRAIC EXPRESSIONS

Important Terms Term

A term is a number, a variable, or a product of numbers and variables raised to natural number powers.
Examples: 5,

- 10x, 3xy, and x 2

Coefficient

The number portion of a term
Examples: The coefficients for the terms 3xy, - x 2, and - 7 are 3, - 1, and

- 7, respectively. Like Terms Terms containing the same variables raised to the same powers; their coefficients may be different.
Examples: The following pairs are like terms:

1 5x and - x; 6x 2 and - 2x 2; 3y and - y. 2
ISBN 1-256-49082-2

Combining Like Terms To add or subtract like terms, apply a distributive property.
Examples: 4 x + 5x = (4 + 5)x = 9x

and 5y - 7y = (5 - 7)y = - 2y

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CHAPTER 1 INTRODUCTION TO ALGEBRA

CHAPTER 1
SECTION 1.1

Review Exercises
22. Simplify each fraction to lowest terms. 9 (b) 36 (a) 12 60 Exercises 23–30: Multiply and then simplify the result to lowest terms when appropriate. 23. 3 # 5 4 6
9 25. 2 # 10 3

Exercises 1–6: Classify the number as prime, composite, or neither. If the number is composite, write it as a product of prime numbers. 1. 29 3. 108 5. 0 2. 27 4. 91 6. 1

24. 1 # 4 2 9 26. 12 # 22 11 23 28. 2 # 9 3 30. 2 # 9x 3 4y

27. 4 # 5 8 x 29. 3 # 6 x

Exercises 7–10: Evaluate the expression for the given values of x and y. 7. 2 x - 5 10 8. 7 x 9. 9x - 2y 10. 2x x - y x = 4 x = 5 x = 2, y = 3

31. Find the fractional part: one-fifth of three-sevenths. 32. Find the reciprocal of each number. 5 (a) 8 (b) 1 (c) 19 (d) 3 2 Exercises 33–38: Divide and then simplify to lowest terms when appropriate. 33. 3 , 1 2 6 35. 8 , 2 3 37. x , 3 y y
9 34. 10 , 7 5

x = 6, y = 4

36. 3 , 6 4 38. 4x , 9x 3y 5

Exercises 11–14: Find the value of y for the given values of x and ]. 11. y = x - 5 12. y = x] + 1 13. y = 4(x - ]) 14. y = x + ] 4 x = 12 x = 2, x = 7, x = 14, ] = 3 ] = 5 ] = 10

Exercises 39 and 40: Find the least common denominator for the fractions.
5 39. 1, 12 8 3 1 40. 14, 21

Exercises 41–46: Add or subtract and then simplify to lowest terms when appropriate.
3 2 41. 15 + 15

42. 5 - 3 4 4
6 3 44. 11 - 22

Exercises 15–18: Translate the phrase into an algebraic expression. State precisely what each variable represents when appropriate. 15. Three squared increased by five 16. Two cubed divided by the quantity three plus one 17. The product of three and a number 18. The difference between a number and four
SECTION 1.2

43. 11 - 1 12 8 45. 2 - 1 + 1 3 2 4
SECTION 1.3

46. 1 + 2 - 1 6 3 9

Exercises 47–52: Write the expression as an exponential expression. 47. 5 # 5 # 5 # 5 # 5 # 5 49. x # x # x # x # x 51. (x + 1) # (x + 1) 52. (a - 5) # (a - 5) # (a - 5)
ISBN 1-256-49082-2

48. 7 # 7 # 7 6 6 6 50. 3 # 3 # 3 # 3

Exercises 19 and 20: Find the greatest common factor. 19. 15, 35 20. 12, 30, 42

21. Use the basic principle of fractions to simplify each expression. # (a) 5 # 7 (b) 3a 8 7 4a

53. Use multiplication to rewrite each expression, and then evaluate the result. (a) 43 (b) 72 (c) 81 54. Find a natural number n such that 2n = 32.

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

CHAPTER 1 REVIEW EXERCISES

85

Exercises 55–66: Evaluate the expression by hand. 55. 7 + 3 # 6 57. 24 , 4 , 2 59. 18 , 6 - 2 61. 9 - 32 63. 24 - 8 + 4 2 65. 7 - 4 + 6 2 + 3
SECTION 1.4

Exercises 85–96: Evaluate the expression. 85. 5 + ( - 4) 87. 11 # ( - 4) 89. 11 , ( - 4) 91. - 5 9 93. - 1 3 86. - 9 - ( - 7) 88. - 8 # ( - 5) 90. - 4 , 4 7 92. - 1 + 2 94.
4 5

56. 15 - 5 - 3 58. 30 - 15 , 3 60. 4 18 5 + 62. 23 - 8 64. 32 - 4(5 - 3) 66. 33 - 23

1- 12 3

1 - 32 4

# 1- 62 7
1- 32 8

-7

95. - 3 , 2

96. 3 , ( - 0.5) 8

Exercises 67 and 68: Find the opposite of each expression. 67. (a) - 8 68. (a) - 1 - 3 2 7 (b) - ( - ( - 3)) (b) - 2 -5

Exercises 97 and 98: Write an arithmetic expression for the given phrase and then simplify. 97. Three plus negative five 98. Subtract negative four from two Exercises 99 and 100: Write the fraction or mixed number as a decimal. 99. 7 9 100. 2 1 5

Exercises 69 and 70: Find the decimal equivalent for the rational number. 69. (a) 4 5 70. (a)
5 9 3 (b) 20 7 11

(b)

Exercises 71–76: Classify the number as one or more of the following: natural number, whole number, integer, rational number, or irrational number. 71. 0 73. - 7 75. p 72. - 5 6 74. 217 76. 3.4

Exercises 101 and 102: Write the decimal number as a fraction in lowest terms. 101. 0.6
SECTION 1.7

102. 0.375

Exercises 103 and 104. Use a commutative property to rewrite the expression. Do not simplify. 103. 3 + 16 104. 14 # ( - x)

77. Plot each number on the same number line. (a) 0 (b) - 2 (c) 5 4 78. Evaluate each expression. (a) - 5 (b) - p (c) 4 - 4

Exercises 105 and 106: Use an associative property to rewrite the expression by changing the parentheses. Do not simplify. 105. - 4 + (1 + 3) 106. (x # y) # 5

79. Insert the symbol 7 or 6 to make each statement true. (a) - 5 4 (b) - 1 -5 2 2 (c) - 3 -9 (d) - 8 -1 80. List the numbers 13, - 3, 3, - 2, and p - 1 from 3 least to greatest.
SECTIONS 1.5 AND 1.6

Exercises 107 and 108: Use a distributive property to rewrite the expression. Then simplify the expression. 107. 5(x + 12) 108. - (a - 3)

Exercises 109–112: State the property or properties that are illustrated. 109. y + 0 = y 111. 1 # 4 = 1 4 110. b # 1 = b 112. - 3a + 3a = 0

Exercises 81 and 82: (Refer to Example 3 in Section 1.5.) Find the sum visually.
ISBN 1-256-49082-2

81. - 5 + 9 4

82. 4 + ( - 7)

Exercises 113–122: State the property that the equation illustrates. 113. z # 3 = 3z 114. 6 + (7 + 5x) = (6 + 7) + 5x

Exercises 83 and 84: Use a number line to find the sum. 83. - 1 + 2 1 84. - 2 + ( - 3)

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

86

CHAPTER 1 INTRODUCTION TO ALGEBRA

115. 2(5x - 2) = 10 x - 4 116. 5 + x + 3 = 5 + 3 + x 117. 1 # a = a 118. 3 # (5x) = (3 # 5)x

146. Area of a Triangle Find the area of the triangle shown.

4 ft 8 ft

119. 12 - (x + 7) = 12 - x - 7 120. a + 0 = a 122. - 5 121. - 5x + 5x = 0 = 1

# 1- 12 5

Exercises 123–128: Use properties of real numbers to evaluate the expression mentally. 123. 7 + 9 + 12 + 8 + 1 + 3 124. 500 - 199 125. 25 # 99 127. 54.98 * 10
SECTION 1.8

147. Gallons to Pints There are 8 pints in 1 gallon. Let G = 1, 2, 3, p , 6, and make a table of values that converts G gallons to P pints. Write a formula that converts G gallons to P pints. 148. Text Messages The table lists the cost C of sending x text messages after the number of text messages included in a monthly plan is reached. Write an equation that relates C and x. Texts (x) Cost (C ) 1 $0.05 2 $0.10 3 $0.15 4 $0.20

126. 4581 + 1999 128. 4356 , 100

Exercises 129–132: Determine whether the expression is a term. If the expression is a term, identify its coefficient. 129. 55x 131. 9xy + 2] 130. - xy 132. x - 7

149. Aging in the United States In 2050, about 1 of the 5 1 population will be age 65 or over and about 20 of the population will be age 85 or over. Estimate the fraction of the population that will be between the ages of 65 and 85 in 2050. (Source: U.S. Census Bureau.) 150. Rule of 72 (Refer to Exercise 79 in Section 1.3.) If an investment of $25,000 earns 9% annual interest, approximate the value of the investment after 24 years. 151. Carpentry A board measures 5 3 feet and needs to 4 be cut in five equal pieces. Find the length of each piece.

Exercises 133–144: Simplify the expression. 133. - 10 x + 4x 135. 3x 2 + x 2 137. - 1 + 3 ] - ] + 5 2 2 2 138. 5(x - 3) - (4x + 3) 139. 4x 2 - 3 + 5x 2 - 3 141. 35a 7a 140. 3x 2 + 4x 2 - 7x 2 142. 0.5c 0.5 134. 19] - 4] 136. 7 + 2 x - 6 + x

152. Distance Over four days, an athlete jogs 3 1 miles, 8 4 3 miles, 6 1 miles, and 1 5 miles. How far does the 8 4 8 athlete jog in all? 153. Checking Account The initial balance in a checking account is $1652. Find the final balance resulting from the following sequence of withdrawals and deposits: - $78, - $91, $256, and - $638. 154. Temperature Range The highest temperature ever recorded in Amarillo, Texas, was 108 F and the lowest was - 16 F. Find the difference between these two temperatures. (Source: The Weather Almanac.) 155. Top-Grossing Movies Titanic (1997) is the sixth top-grossing movie of all time. Find the total admissions for Titanic if they were 16 of the total admissions 25 for the top-grossing movie of all time, Gone With the Wind (1939), which had total admissions of 202 million. (Source: Box Office Mojo.)

15y + 10 143. 5
APPLICATIONS

24 x - 60 144. 12

145. Painting a Wall Two people are painting a large wall. The first person paints 3 square feet per minute while the second person paints 4 square feet per minute. (a) Write a simplified expression that gives the total number of square feet the two people can paint in x minutes. (b) Find the number of square feet painted in 1 hour. (c) How many minutes would it take for them to paint a wall 8 feet tall and 21 feet wide?

ISBN 1-256-49082-2

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

CHAPTER 1 TEST

87

CHAPTER 1

Test

Step-by-step test solutions are found on the Chapter Test Prep Videos available via the Video Resources , and on (search “RockswoldBeginAlg” and click on “Channels”). on DVD, in

1. Classify the number as prime or composite. If the number is composite, write it as a product of prime numbers. (a) 29 (b) 56 2. Evaluate the expression 5x for x = - 3. 2x - 1

15. Write 0.75 as a fraction in lowest terms. 16. State the property or properties that each equation illustrates. (b) 12 # (3x) = 36 x (a) 6 x - 2 x = 4x (c) 4 + x + 8 = 12 + x 17. Use properties of real numbers to evaluate 17 # 102 mentally. 18. Simplify each expression. (a) 5 - 5z + 7 + z (b) 12 x - (6 - 3x) (c) 5 - 4(x + 6) + 15x 3

3. Translate the phrase “four squared decreased by three” to an algebraic expression. Then find the value of the expression. 4. Simplify 24 to lowest terms. 32 5. Find the fractional part: two-tenths of one-sixth.
5 6. Find the least common denominator for 3 and 12. 8

7. Evaluate each expression. Write your answer in lowest terms. 3 (a) 5 + 1 (b) 5 - 15 (c) 3 # 10 8 8 9 5 21 (d) 6 , 8 5
5 (e) 12 + 4 9

(f) 10 , 5 13

8. Write y # y # y # y as an exponential expression. 9. Evaluate each expression. (a) 6 + 10 , 5 (b) 43 - (3 - 5 # 2) (d) 11 - 1 + 3 (c) - 62 - 6 + 4 2 6 - 4
7 10. Write 20 as a decimal.

19. Mowing a Lawn Two people are mowing a lawn. The first person has a riding mower and can mow 4 3 acres per hour; the second person has a push mower and can mow 1 acre per hour. 4 (a) Write a simplified expression that gives the total number of acres that the two people mow in x hours. (b) Find the total acreage that they can mow in an 8-hour work day. 20. A wire 7 4 feet long is to be cut in 3 equal parts. How 5 long should each part be? 21. Cost Equation The table lists the cost C of buying x tickets to a hockey game. Tickets (x) Cost (C) 3 $39 4 $52 5 $65 6 $78

11. Classify the number as one or more of the following: natural number, whole number, integer, rational number, or irrational number. (a) - 1 (b) 25 12. Plot each number on the same number line. (c) 27 (a) - 2 (b) 1 3 13. Insert the symbol 7 or 6 to make each statement true. (a) 2 0 -5 (b) - 1 14. Evaluate the expression. (a) - 5 , 5 6 (b) - 7 # ( - 3)

(a) Find an equation that relates C and x. (b) What is the cost of 17 tickets? 22. The initial balance for a savings account is $892. Find the final balance resulting from withdrawals and deposits of - $57, $150, and - $345.

ISBN 1-256-49082-2 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

88

CHAPTER 1 INTRODUCTION TO ALGEBRA

CHAPTER 1

Extended and Discovery Exercises
Complete the following magic square having 4 rows and 4 columns by arranging the numbers 1 through 16 so that each row, column, and diagonal sums to 34. The four corners will also sum to 34. 2 5 6 4 1 13

1. Arithmetic Operations Insert one of the symbols + , - , # , or , in each blank to obtain the given answer. Do not use any parentheses. 2 3 4 6 7 2 3 4 6 7 2 3 4 6 7 2 = 0 3 = 10 4 = 1 6 = 36 7 = 63

2. Magic Squares The following square is called a “magic square” because it contains each of the numbers from 1 to 9 and the numbers in each row, column, and diagonal sum to 15. 8 1 6 3 5 7 4 9 2

ISBN 1-256-49082-2 Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.