INTEGERS AND INTRODUCTION TO ALGEBRA

c h a p t e r

2

119

© 2

010

McG

raw

-Hill

Co

mp

anie

s

INTRODUCTION

Of all the sciences, meteorology may be boththe least precise and the most talked about.Meteorologists study the weather and cli-mate. Together with geologists, they provideus with many predictions that play a partin our everyday decisions. Among the thingswe might look for are temperatures, rainfall,water level, and tide level.

On a day with unusual weather, we oftenbecome curious about the record for thatday or even the all-time record. Here are afew of those records:

• The record high temperature for the UnitedStates is 134�F in 1913 in Death Valley,California.

• The record low temperature for the U.S.is –80�F in 1971 in Prospect Creek Camp,Alaska.

• The greatest one day temperature drop inthe U.S. happened on Christmas Eve, 1924,in Montana. The temperature went from63�F during the day to –21�F at night.

The use of negative numbers for tempera-tures below zero is common for such num-bers, but there are many other applications.For example, the tide range in Delawarevaries between �10 feet and �5 feet. Weexamine these and several other climate-related applications in Sections 2.1, 2.2, 2.3,and 2.7.

INTEGERS ANDINTRODUCTION TO ALGEBRA

CHAPTER 2 OUTLINESection 2.1 Introduction to Integers page 121

Section 2.2 Addition of Integers page 132

Section 2.3 Subtraction of Integers page 139

Section 2.4 Multiplication of Integers page 145

Section 2.5 Division of Integers page 151

Section 2.6 Introduction to Algebra: Variables andExpressions page 158

Section 2.7 Evaluating Algebraic Expressions page 167

Section 2.8 Simplifying Algebraic Expressions page 177

Section 2.9 Introduction to Linear Equations page 186

Section 2.10 The Addition Property of Equality page 192

2

hut06236_ch02_A.qxd 9/22/08 9:55 AM 19

© 2

010

McG

raw

-Hill

Co

mp

anie

s

120

Name

Section Date

ANSWERS

1. See exercise

2. �4, �2, �1, 0, 1, 5

3. Max: 7; Min: �5

4. 5 5. 6

6. 6

7. 6

8. 6

9. 16

10. �23

11. x � 8

12.

13. No

14. Yes

15. �10

16. �1

17. �5

18. �3

19. �19

20. 12 21. 0

22. 21 23. 7

24. �3 25. 55

26. �7 27. 8w2t

28. �a2 � 4a � 3

w—17x

Pretest Chapter 2

This pretest will provide a preview of the types of exercises you will encounter in each sec-tion of this chapter. The answers for these exercises can be found in the back of the text. Ifyou are working on your own or are ahead of the class, this pretest can help you identify thesections in which you should focus more of your time.

[2.1] Represent the integers on the number line shown.

1. 6, �8, 4, �2, 10

2. Place the following data set in ascending order: 5, �2, �4, 0, �1, 1.

3. Determine the maximum and minimum of the following data set: �4, 1, �5,7, 3, 2.

Evaluate:

4. ⏐�5⏐ 5. ⏐6⏐ 6. ⏐11 � 5⏐

7. ⏐�11⏐�⏐5⏐ 8. ⏐4 � 5⏐�⏐6 � 3⏐

Find the opposite of each integer.

9. �16 10. 23

[2.6] Write each of the phrases using symbols.

11. 8 less than x

12. the quotient when w is divided by the product of x and 17

Identify which are expressions and which are not.

13. 7x � 5 � 11 14. 3x � 2(x � 1)

[2.2 to 2.5] Perform the indicated operations.

15. �7 � (�3) 16. 8 � (�9) 17. (�3) � (�2)

18. 8 � 11 19. �8 � 11 20. 9 � (�3)

21. 6 � (�6) 22. (�7)(�3) 23.

[2.7] Evaluate each expression.

24. 5 � 42 � 3 � 6 25. (45 � 3 � 5) � 52

26. If x � �2, y � 7, and w � �4, evaluate the expression

[2.8] Combine like terms.

27. 5w2t � 3w2t 28. 4a2 � 3a � 5 � 7a � 2 � 5a2

x2y

w.

�27 � 6

�3

�8 �6 �4 �2 0 2 4 6 8 10

hut06236_ch02_A.qxd 9/22/08 9:55 AM Page 120

121

© 2

010

McG

raw

-Hill

Co

mp

anie

s

2.1 Introduction to Integers

1. Represent integers on a number line2. Place a set of integers in ascending order3. Determine the extreme values of a data set4. Find the opposite of a given integer5. Evaluate expressions involving absolute value

When numbers are used to represent physical quantities (altitudes, temperatures, andamounts of money are examples), it may be necessary to distinguish between positive andnegative quantities. The symbols � and � are used for this purpose. For instance, the alti-tude of Mount Whitney is 14,495 ft above sea level (�14,495 ft).

The altitude of Death Valley is 282 ft below sea level (�282 ft).

On a given day the temperature in Chicago might be 10�F below zero (�10�F).

11010090876543210

–10–20

�282 ft

Death Valley

14,495 ft

Mount Whitney

2.1 OBJECTIVES


122 CHAPTER 2 INTEGERS AND INTRODUCTION TO ALGEBRA

An account could show a gain of $100 (�100) or a loss of $100 (�100).

These numbers suggest the need to extend the whole numbers to include both positivenumbers (like �100) and negative numbers (like �282).

To represent the negative numbers, we extend the number line to the left of zero andname equally spaced points.

Numbers corresponding to points to the right of zero are positive numbers. They arewritten with a positive (�) sign or with no sign at all.

�6 and 9 are positive numbers

Numbers corresponding to points to the left of zero are negative numbers. They are al-ways written with a negative (�) sign.

�3 and �20 are negative numbers

Read “negative 3.”

The positive and negative numbers, as well as zero, are called the real numbers.Here is the number line extended to include positive and negative numbers, and zero.

The numbers used to name the points shown on the number line are called the integers. Theintegers consist of the natural numbers, their negatives, and the number zero. We can rep-resent the set of integers by

{. . . , �3, �2, �1, 0, 1, 2, 3, . . .}

Represent the integers on the number line shown.

�3, �12, 8, 15, �7

�10 5�5 0 1510

15

�15

�12 �7 �3 8

Example 1 Representing Integers on the Number Line

�2 1�1 0

Negative numbers

32�3

Zero is neither positivenor negative

Positive numbers

© 2

010

McG

raw

-Hill

Co

mp

anie

s

NOTE On the number line, wecall zero the origin.

NOTE Braces { and } are usedto hold a collection of numbers.We call the collection a set. Thedots are called ellipses andindicate that the patterncontinues.

OBJECTIVE 1


INTRODUCTION TO INTEGERS SECTION 2.1 123

The set of numbers on the number line is ordered. The numbers get smaller moving tothe left on the number line and larger moving to the right.

When a group of numbers is written from smallest to largest, the numbers are said to be inascending order.

Place each group of numbers in ascending order.

(a) 9, �5, �8, 3, 7

From smallest to largest, the numbers are

�8, �5, 3, 7, 9 Note that this is the order in which the numbers appear on a number lineas we move from left to right.

(b) 3, �2, 18, �20, �13

From smallest to largest, the numbers are

�20, �13, �2, 3, 18

Example 2 Ordering Integers

10 32 4�3 �2 �1�4

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Represent the integers on a number line.

�1, �9, 4, �11, 8, 20

�10 5�5 0 1510�15 20

C H E C K Y O U R S E L F 1


(a) 12, �13, 15, 2, �8, �3 (b) 3, 6, �9, �3, 8


The least and greatest numbers in a group are called the extreme values. The leastnumber is called the minimum, and the greatest number is called the maximum.

For each group of numbers, determine the minimum and maximum values.

(a) 9, �5, �8, 3, 7

From our previous ordering of these numbers, we see that �8, the least number, is the min-imum, and 9, the greatest number, is the maximum.

(b) 3, �2, 18, �20, �13

�20 is the minimum and 18 is the maximum.

Example 3 Labeling Extreme Values

OBJECTIVE 2

OBJECTIVE 3



Each point on the number line corresponds to a real number.There are more points on thenumber line than integers. The real numbers include decimals, fractions, and other numbers.

Which of the real numbers, (a) 145, (b) �28, (c) 0.35, and (d) , are also integers?

(a) 145 is an integer.

(b) �28 is an integer.

(c) 0.35 is not an integer.

(d) is not an integer.�2

3

�2

3

Example 4 Identifying Real Numbers that Are Integers

© 2

010

McG

raw

-Hill

Co

mp

anie

s

For each group of numbers, determine the minimum and maximum values.

(a) 12, �13, 15, 2, �8, �3 (b) 3, 6, �9, �3, 8


Which of the real numbers are also integers?

�23 1,054 �0.23 0 �500 �45


Sometimes we refer to the negative of a number as its opposite. For a nonzero number, thiscorresponds to a point the same distance from the origin as the given number, but on theother side of zero. Example 5 illustrates this.

(a) 5 The opposite of 5 is �5.

(b) �9 The opposite of �9 is 9.

Example 5 Find the Opposite of Each Number

Find the opposite of each number.

(a) 17 (b) �12


The absolute value of a number represents the distance of the point named bythe number from the origin on the number line.

Definition: Absolute Value

OBJECTIVE 4


INTRODUCTION TO INTEGERS SECTION 2.1 125

Because we think of distance as a positive quantity (or as zero), the absolute value ofa number is never negative.

The absolute value of 5 is 5. The absolute value of �5 is also 5.As a consequence of the definition, the absolute value of a positive number or zero is

itself. The absolute value of a negative number is its opposite.In symbols, we write

Read “the absolute Read “the absolute value of 5.” value of negative 5.”

The absolute value of a number does not depend on whether the number is to the rightor to the left of the origin, but on its distance from the origin.

(a) ⏐7⏐ � 7

(b) ⏐�7⏐ � 7

(c) �⏐�7⏐ � �7 This is the negative, or opposite, of the absolute value of negative 7.

Example 6 Simplifying Absolute Value Expressions

� 5 � � 5 and � �5 � � 5

5�5 0

5 units5 units

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Evaluate.

(a) ⏐8⏐ (b) ⏐�8⏐ (c) �⏐�8⏐


To determine the order of operation for an expression that includes absolute values,note that the absolute value bars are treated as a grouping symbol.

(a) ⏐�10⏐�⏐10⏐ � 10 � 10 � 20

(b) ⏐8 � 3⏐�⏐5⏐ � 5

(c) ⏐8⏐�⏐3⏐� 8 � 3 � 5 Evaluate the absolute values, then subtract.

Absolute value bars, like parentheses, serve as a set ofgrouping symbols, so do the operation inside first.

Example 7 Adding or Subtracting Absolute Values

Evaluate.

(a) ⏐�9⏐�⏐4⏐ (b) ⏐9 � 4⏐ (c) ⏐9⏐�⏐4⏐


OBJECTIVE 5



© 2

010

McG

raw

-Hill

Co

mp

anie

s

1. 2. (a) �13, �8, �3, 2, 12, 15

(b) �9, �3, 3, 6, 8

3. (a) minimum is �13; maximum is 15; (b) minimum is �9; maximum is 8

4. �23, 1054, 0, and �500 5. (a) �17; (b) 12

6. (a) 8; (b) 8; (c) �8. 7. (a) 13; (b) 5; (c) 5

�10 0�5 5

20

�20

�11�9 �1 4

�15 10 15 20

8

C H E C K Y O U R S E L F A N S W E R S

READING YOUR TEXT

The following fill-in-the-blank exercises are designed to assure that you understandthe key vocabulary used in this section. Each sentence comes directly from thesection. You will find the correct answers in Appendix C.

Section 2.1

(a) When numbers are used to represent physical quantities, it may be necessary to distinguish between positive and quantities.

(b) Numbers that correspond to points to the of zeroare negative numbers.

(c) When a set of numbers is written from smallest to largest, the numbers are said to be in order.

(d) The absolute value of a number depends on its fromthe origin.


© 2

010

McG

raw

-Hill

Co

mp

anie

s

Name

Section Date

ANSWERS

1. �400

2. �80

3. �200

4. �400

5. �25,000

6. �12,500

7. See exercise

8. See exercise

9. 5, 175, �234

10. �45, 700, �26

11. �7, �5, �1, 0, 2, 3, 8

12. �8, �2, �1, 0, 1, 6, 7

13. �11, �6, �2, 1, 4, 5, 9

14.�18, �15, �11, �5, 14, 20, 23

15. �7, �6, �3, 3, 6, 7

16.�15, �14, �13, 12,14, 15

17. Max: 15; Min: �6

18. Max: 11; Min: �4

19. Max: 21; Min: �15

20. Max: 22; Min: �31

21. Max: 5; Min: �2

22. Max: 7; Min: �10

SECTION 2.1 127

2.1 Exercises

Represent each quantity with an integer.

1. An altitude of 400 ft above sea level

2. An altitude of 80 ft below sea level

3. A loss of $200

4. A profit of $400

5. A decrease in population of 25,000

6. An increase in population of 12,500

[Objective 1]

Represent the integers on the number lines shown.

7. 5, �15, 18, �8, 3

8. �18, 4, �5, 13, 9

Which numbers in the sets are integers?

9. 10.

[Objective 2]


11. 3, �5, 2, 0, �7, �1, 8 12. �2, 7, 1, �8, 6, �1, 0

13. 9, �2, �11, 4, �6, 1, 5 14. 23, �18, �5, �11, �15, 14, 20

15. �6, 7, �7, 6, �3, 3 16. 12, �13, 14, �14, 15, �15

[Objective 3]

For each group of numbers, determine the maximum and minimum values.

17. 5, �6, 0, 10, �3, 15, 1, 8 18. 9, �1, 3, 11, �4, 2, 5, �2

19. 21, �15, 0, 7, �9, 16, �3, 11 20. �22, 0, 22, �31, 18, �5, 3

21. 3, 0, 1, �2, 5, 4, �1 22. 2, 7, �3, 5, �10, �5

��45, 0.35, 3

5, 700, �26��5, �

2

9, 175, �234, �0.64�

0�10�20

�18 �5

10

4 9

20

13

0�10�20

��1515 ��88

10

33 55

20

1818

Boost your GRADE atALEKS.com!

• Practice Problems • e-Professors• Self-Tests • Videos• NetTutor


© 2

010

McG

raw

-Hill

Co

mp

anie

s

128 SECTION 2.1

[Objective 4]

Find the opposite of each number.

23. 15 24. 18

25. 11 26. 34

27. �19 28. �5

29. �7 30. �54

[Objective 5]

Evaluate.

31. ⏐17⏐ 32. ⏐28⏐

33. ⏐�10⏐ 34. ⏐�7⏐

35. �⏐3⏐ 36. �⏐5⏐

37. �⏐�8⏐ 38. �⏐�13⏐

39. ⏐�2⏐�⏐3⏐ 40. ⏐4⏐�⏐�3⏐

41. ⏐�9⏐�⏐9⏐ 42. ⏐11⏐�⏐�11⏐

43. ⏐4⏐�⏐�4⏐ 44. ⏐5⏐�⏐�5⏐

45. ⏐15⏐�⏐8⏐ 46. ⏐11⏐�⏐3⏐

47. ⏐15 � 8⏐ 48. ⏐11 � 3⏐

49. ⏐�9⏐�⏐2⏐ 50. ⏐�7⏐�⏐4⏐

51. ⏐�8⏐�⏐�7⏐ 52. ⏐�9⏐�⏐�4⏐

Label each statement as true or false.

53. All whole numbers are integers.

54. All nonzero integers are real numbers.

ANSWERS

23. �15

24. �18

25. �11

26. �34

27. 19

28. 5

29. 7

30. 54

31. 17

32. 28

33. 10

34. 7

35. �3

36. �5

37. �8

38. �13

39. 5

40. 7

41. 18

42. 22

43. 0

44. 0

45. 7

46. 8

47. 7 48. 8

49. 11 50. 11

51. 1 52. 5

53. True 54. True


© 2

010

McG

raw

-Hill

Co

mp

anie

s

SECTION 2.1 129

55. All integers are whole numbers.

56. All real numbers are integers.

57. All negative integers are whole numbers.

58. Zero is neither positive nor negative.

Place absolute value bars in the proper location on the left side of the expression so that theequation is true.

59. (�6) � 2 � 8 60. (�8) � (�3) � 11

61. 6 � (�2) � 8 62. 8 � (�3) � 11

Represent each quantity with a real number.

63. Science and Medicine The erosion of 5 centimeters (cm) of topsoil from anIowa cornfield.

64. Science and Medicine The formation of 2.5 cm of new topsoil on the Africansavanna.

65. Business and Finance The withdrawal of $50 from a checking account.

66. Business and Finance The deposit of $200 in a savings account.

67. Science and Medicine The temperature change pictured.

––

2:00 P.M.

50°F

––

1:00 P.M.

60°F

2

ANSWERS

55. False

56. False

57. False

58. True

59. ⏐�6⏐� 2 � 8

60. ⏐�8⏐�⏐�3⏐ � 11

61. 6 �⏐�2⏐� 8

62. 8 �⏐(�3)⏐� 11

63. �5 cm

64. �2.5 cm

65. �50 dollars

66. �200 dollars

67. �10�


© 2

010

McG

raw

-Hill

Co

mp

anie

s

130 SECTION 2.1

68. Science and Medicine The temperature change indicated.



71. Business and Finance A country exported $90,000,000 more than it imported,creating a positive trade balance.

72. Business and Finance A country exported $60,000,000 less than it imported,creating a negative trade balance.

9080706050403020100

–10––30–40

9080

40302010

–

F

–30°F

2:00 P.M.1:00 P.M.

2

9080706050403020100

–10––30–40

9080

40302010

––30°F

0°F

2:00 P.M.1:00 P.M.

2

9080706050403020100

–10––30–40

–20°F

9080

40302010

––30°F

2:00 P.M.1:00 P.M.

2

ANSWERS

68. �10°

69. 20°

70. �30°

71. �90,000,000

72. �60,000,000


© 2

010

McG

raw

-Hill

Co

mp

anie

sANSWERS

73. �6; 8; 8; �2

74. �8; 9; 9; 3

75. �2; 6; 6; 0

76. �9; �9; �9; 0

77.

SECTION 2.1 131

For each group of numbers given in exercises 73 to 76, answer questions (a) to (d):

(a) Which number is smallest?

(b) Which number lies farthest from the origin?

(c) Which number has the largest absolute value?

(d) Which number has the smallest absolute value?

73. �6, 3, 8, 7, �2 74. �8, 3, �5, 4, 9

75. �2, 6, �1, 0, 2, 5 76. �9, 0, �2, 3, 6

77. Simplify each of the following:

�(�7) �(�(�7)) �(�(�(�7)))

Based on your answers, generalize your results.

Answers1. 400 or (�400) 3. �200 5. �25,000

7. 9. 5, 175, �234

11. �7, �5, �1, 0, 2, 3, 8 13. �11, �6, �2, 1, 4, 5, 915. �7, �6, �3, 3, 6, 7 17. Max: 15; Min: �6 19. Max: 21; Min: �1521. Max: 5; Min: �2 23. �15 25. �11 27. 19 29. 731. 17 33. 10 35. �3 37. �8 39. 5 41. 18 43. 045. 7 47. 7 49. 11 51. 1 53. True 55. False 57. False59. ⏐�6⏐� 2 � 8 61. 6 �⏐�2⏐� 8 63. �5 cm 65. �50 dollars67. �10� 69. 20� 71. �90,000,000 73. �6; 8; 8; �275. �2; 6; 6; 0 77.

0�10�20

��1515 ��88

10

33 55

20

1818

hut06236_ch02_A.qxd 9/22/08 9:55 AM 1

132

© 2

010

McG

raw

-Hill

Co

mp

anie

s

2.2 Addition of Integers

1. Add two integers with the same sign2. Add two integers with opposite signs3. Solve applications involving integers

In Section 2.1 we introduced the idea of negative numbers. Here we examine the four arith-metic operations (addition, subtraction, multiplication, and division) and see how those op-erations are performed when integers are involved. We start by considering addition.

An application may help. We will represent a gain of money as a positive number anda loss as a negative number.

If you gain $3 and then gain $4, the result is a gain of $7:

3 � 4 � 7

If you lose $3 and then lose $4, the result is a loss of $7:

�3 � (�4) � �7

If you gain $3 and then lose $4, the result is a loss of $1:

3 � (�4) � �1

If you lose $3 and then gain $4, the result is a gain of $1:

�3 � 4 � 1

The number line can be used to illustrate the addition of integers. Starting at the ori-gin, we move to the right for positive integers and to the left for negative integers.

(a) Add 3 � 4.

Start at the origin and move 3 units to the right. Then move 4 more units to the right to findthe sum. From the number line, we see that the sum is

3 � 4 � 7

(b) Add (�3) � (�4).

Start at the origin and move 3 units to the left. Then move 4 more units to the left to find thesum. From the number line, we see that the sum is

(�3) � (�4) � �7

0�7 �3

�4 �3

0 3 7

�3 �4

Example 1 Adding Integers

2.2 OBJECTIVES

OBJECTIVE 1


ADDITION OF INTEGERS SECTION 2.2 133

You have probably noticed a helpful pattern in the previous example. This pattern willallow you to do the work mentally without having to use the number line. Look at thefollowing rule.

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Add.

(a) (�4) � (�5) (b) (�3) � (�7)

(c) (�5) � (�15) (d) (�5) � (�3)


If two integers have the same sign, add their absolute values. Give the resultthe sign of the original integers.

Property: Adding Integers Case 1: Same Sign

We can use the number line to illustrate the addition of two integers. This time theintegers will have different signs.

(a) Add 3 � (�6).

First move 3 units to the right of the origin. Then move 6 units to the left.

3 � (�6) � �3

(b) Add �4 � 7.

This time move 4 units to the left of the origin as the first step. Then move 7 units to theright.

�4 � 7 � 3

�4

�7

4 30�

3

0 3�3

�6


Add.

(a) 7 � (�5) (b) 4 � (�8) (c) �1 � 16 (d) �7 � 3


NOTE This means that the sumof two positive integers ispositive and the sum of twonegative integers is negative.

OBJECTIVE 2



You have no doubt noticed that, in adding a positive integer and a negative integer,sometimes the sum is positive and sometimes it is negative. This depends on which of theintegers has the larger absolute value. This leads us to the second part of our addition rule.

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Add mentally.

(a) 5 � (�14) (b) �7 � (�8)

(c) �8 � 15 (d) 7 � (�8)


If two integers have different signs, subtract their absolute values, the smallerfrom the larger. Give the result the sign of the integer with the larger absolutevalue.

Property: Adding Integers Case 2: Different Signs

RECALL We first encounteredabsolute values in Section 2.1.

(a) � �12

Because the two integers have different signs, subtract the absolute values (19 � 7 � 12).The sum of 7 and �19 has the sign (�) of the integer with the larger absolute value, �19.

(b) � �6

Subtract the absolute values (13 � 7 � 6). The sum of �13 and 7 has the sign (�) of theinteger with the larger absolute value, �13.

�13 � 7

7 � (�19)


In Section 1.2, we discussed the commutative, associative, and additive identity prop-erties. There is another property of addition that we should mention.

Recall that every number has an opposite. It corresponds to a point the same distancefrom the origin as the given number but in the opposite direction.

The opposite of 9 is �9.The opposite of �15 is 15.

The additive inverse property states that the sum of any number and its opposite is 0.

�3

3

0 3

3

For any number a, there exists a number �a such that

a � (�a) � (�a) � a � 0

The sum of any number and its opposite, or additive inverse, is 0.

Property: Additive Inverse Property

NOTE The opposite of anumber is also called theadditive inverse of thatnumber.

NOTE 3 and �3 are opposites.

NOTE Here �a represents theopposite of the number a. If a ispositive, �a is negative. If a isnegative, �a is positive.


ADDITION OF INTEGERS SECTION 2.2 135

(a) � 0

(b) � 0�15 � 15

9 � (�9)


© 2

010

McG

raw

-Hill

Co

mp

anie

s

NOTE Later, we will show that�0 � 0; therefore, the oppositeof 0 is 0.

Add.

(a) (�17) � 17 (b) 12 � (�12)


When solving an application of integer arithmetic, the first step is to translate the phrase orstatement using integers. Example 5 illustrates this step.

Shanique has $250 in her checking account. She writes a check for $120 and makes adeposit of $90. What is the resulting balance?

First, translate the phrase using integers. Such problems will usually include some-thing that is represented by negative integers and something that is represented by positiveintegers. In this case, a check can be represented as a negative integer and a deposit as apositive integer. We have

250 � (�120) � 90

This expression can now be evaluated.

250 � (�120) � 90

� 130 � 90

� 220

The resulting balance is $220.

Example 5 An Application of the Addition of Integers

Translate the problem into an integer expression and then answer the question.

When Kirin awoke, the temperature was twelve degrees below zero, Fahrenheit.Over the next six hours, the temperature increased by seventeen degrees. What wasthe temperature at that time?


OBJECTIVE 3



© 2

010

McG

raw

-Hill

Co

mp

anie

s

1. (a) �9; (b) �10; (c) �20; (d) �8 2. (a) 2; (b) �4; (c) 15; (d) �4

3. (a) �9; (b) �15; (c) 7; (d) �1 4. (a) 0; (b) 0

5. �12 � 17; the temperature was 5�F.


READING YOUR TEXT


Section 2.2

(a) In Section 2.1, we introduced the idea of numbers.

(b) To add two numbers with different signs, their absolute values.

(c) The sum of any number and its opposite, or additive , is 0.

(d) The opposite of zero is .


© 2

010

McG

raw

-Hill

Co

mp

anie

s

Name

Section Date

ANSWERS

1. 9

2. 14

3. 16

4. 15

5. �5

6. �10

7. 6

8. 6

9. �9

10. �15

11. 0

12. 0

13. �1

14. �1

15. �2

16. �3

17. $128

18. $225

19. 120 lb/in.2

SECTION 2.2 137

2.2 Exercises

[Objectives 1 and 2]

Add.

1. 3 � 6 2. 5 � 9

3. 11 � 5 4. 8 � 7

5. (�2) � (�3) 6. (�1) � (�9)

7. 9 � (�3) 8. 10 � (�4)

9. �9 � 0 10. �15 � 0

11. 7 � (�7) 12. 12 � (�12)

13. 7 � (�9) � (�5) � 6 14. (�4) � 6 � (�3) � 0

15. 7 � (�3) � 5 � (�11) 16. �6 � (�13) � 16

[Objective 3]

In exercises 17 to 22, restate the problem using an expression involving integers and thenanswer the question.

17. Business and Finance Amir has $100 in his checking account. He writes acheck for $23 and makes a deposit of $51. What is his new balance?

18. Business and Finance Olga has $250 in her checking account. She deposits $52and then writes a check for $77. What is her new balance?

19. Mechanical Engineering A pneumatic actuator is operated by a pressurized airreservoir. At the beginning of an operator’s shift, the pressure in the reservoir was126 pounds per square inch (lb/in.2). At the end of each hour, the operator recordedthe reservoir’s change in pressure. The values recorded (in lb/in.2) were a drop of12, a drop of 7, a rise of 32, a drop of 17, a drop of 15, a rise of 31, a drop of 4, anda drop of 14. What was the pressure in the tank at the end of the shift?




© 2

010

McG

raw

-Hill

Co

mp

anie

s

138 SECTION 2.2

20. Mechanical Engineering A diesel engine for an industrial shredder has an 18-quart (qt) oil capacity. When a maintenance technician checked the oil, it was7 qt low. Later that day, she added 4 qt to the engine. What was the oil level afterthe 4 qt were added?

21. Science and Medicine The lowest one-day temperature in Helena, Montana,was �21�F at night. The temperature increased by 25 degrees by noon. What wasthe temperature at noon?

22. Science and Medicine At 7 A.M., the temperature was �15�F. By 1 P.M., the tem-perature had increased by 18 degrees. What was the temperature at 1 P.M.?

23. In this chapter, it is stated that “every number has an opposite.” The opposite of 9is �9. This corresponds to the idea of an opposite in English. In English, an op-posite is often expressed by a prefix, for example, un- or ir-.

(a) Write the opposite of these words: unmentionable, uninteresting, irre-deemable, irregular, uncomfortable.

(b) What is the meaning of these expressions: not uninteresting, not irredeemable,not irregular, not unmentionable?

(c) Think of other prefixes that negate or change the meaning of a word to its op-posite. Make a list of words formed with these prefixes and write a sentencewith three of the words you found. Make a sentence with two words andphrases from parts (a) and (b).

What is the value of �[�(�5)]? What is the value of �(�6)? How doesthis relate to the given examples? Write a short description about this rela-tionship.

Answers1. 9 3. 16 5. �5 7. 6 9. �9 11. 0 13. �115. �2 17. $128 19. 120 lb/in.2 21. 4�F 23.

2

2

ANSWERS

20. 15 qt

21. 4°F

22. 3°F

23.


139

© 2

010

McG

raw

-Hill

Co

mp

anie

s

2.3 Subtraction of Integers

1. Find the difference of two integers2. Solve applications involving the subtraction of integers

To begin our discussion of subtraction when integers are involved, we can look back at aproblem using natural numbers. We know that

8 � 5 � 3

From our work in adding integers, we know that it is also true that

8 � (�5) � 3

Comparing these equations, we see that the results are the same. This leads us to an im-portant pattern. Any subtraction problem can be written as a problem in addition. Subtract-ing 5 is the same as adding the opposite of 5, or �5. We can write this fact as follows:

8 � 5 � 8 � (�5) � 3

This leads us to the following rule for subtracting integers.

2.3 OBJECTIVES

1. To rewrite the subtraction problem as an addition problem:

a. Change the subtraction operation to addition.b. Replace the integer being subtracted with its opposite.

2. Add the resulting integers as before.In symbols,

a � b � a � (�b)

Property: Subtracting Integers

Example 1 illustrates the use of this property while subtracting.

Change the subtraction symbol (�)to an addition symbol (�).

(a) � 15 � (�7)

Replace 7 with its opposite, �7.� 8

(b) � 9 � (�12) � �3

(c) � �6 � (�7) � �13

(d) Subtract 5 from �2. We write the statement as �2 � 5 and proceed as before:

�2 � 5 � �2 � (�5) � �7

�6 � 7

9 � 12

15 � 7

Example 1 Subtracting IntegersOBJECTIVE 1



The subtraction rule is used in the same way when the integer being subtracted is neg-ative. Change the subtraction to addition. Replace the negative integer being subtractedwith its opposite, which is positive. Example 2 illustrates this principle.

Change the subtractionto addition.

(a) � 5 � (�2) � 5 � 2 � 7

Replace �2 with its opposite, �2 or 2.

(b) � 7 � (�8) � 7 � 8 � 15

(c) � �9 � 5 � �4

(d) Subtract �4 from �5. We write

�5 � (�4) � �5 � 4 � �1

�9 � (�5)

7 � (�8)

5 � (�2)

Example 2 Subtracting Integers

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Subtract.

(a) 18 � 7 (b) 5 � 13 (c) �7 � 9 (d) �2 � 7


Subtract.

(a) 8 � (�2) (b) 3 � (�10)

(c) �7 � (�2) (d) 7 � (�7)


Susanna’s checking account shows a balance of $285. She has discovered that a deposit for$47 was accidently recorded as a check for $47. Write an integer expression that representsthe correction on the balance. Then find the corrected balance.

285 � (�47) � (47)

Subtract the check and then add the deposit.

285 � (�47) � (47) � 285 � 47 � 47 � 379

The corrected balance is $379.

Example 3 An Application of the Subtraction of Integers

It appears that Marshal, a running back, gained 97 yards in the last game. A closerinspection of the statistics revealed that a 9-yard gain had been recorded as a9-yard loss. Write an integer expression that represents the corrected yards gainedand then find that number.


OBJECTIVE 2


SUBTRACTION OF INTEGERS SECTION 2.3 141

Using Your Calculator to Addand Subtract Integers

Your scientific (or graphing) calculator has a key that makes a number negative. This key

is different from the “subtraction” key. The negative key is marked as either or .With a scientific calculator, this key is pressed after the number you wish to make negative isentered. All of the instructions in this section assume that you have a scientific calculator.

Enter each of the following into your calculator.

(a) �24

24

(b) �(�(�(�12)))

12 ��

��

Example 4 Entering a Negative Integer into the Calculator

(�)��

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Enter each number into your calculator.

(a) �36 (b) �(�(�6))


Find the sum for each pair of integers.

(a) 256 � (�297)

256 297

Your display should read �41.

(b) �312 � (�569)

312 569


��

��


Find the sum for each pair of integers.

(a) �368 � 547 (b) �596 � (�834)


Find the difference for �356 � (�469).

356 469

Your display should read 113.

��

Example 6 Subtracting Integers

NOTE The 12 changes betweenpositive and negative in thedisplay. The final display is 12,because there are an evennumber of negative signs infront of the 12.



© 2

010

McG

raw

-Hill

Co

mp

anie

s

Find each difference.

(a) 349 � (�49) (b) �294 � (�137)


READING YOUR TEXT


Section 2.3

(a) Any subtraction problem can be written as a problem in .

(b) To rewrite a subtraction problem as an addition problem, change the sub-traction operation to addition and replace the integer being subtracted with its .

(c) The opposite of �2 is .

(d) The calculator key that makes a number negative is different from the key.

1. (a) 11; (b) �8; (c) �16; (d) �9 2. (a) 10; (b) 13; (c) �5; (d) 14

3. 97 � (�9) � 9 � 115 yards 4. (a) �36; (b) �6 5. (a) 179; (b) �1,430

6. (a) 398; (b) �157



© 2

010

McG

raw

-Hill

Co

mp

anie

s

Name

Section Date

ANSWERS

1. 8

2. 14

3. 37

4. 47

5. �2

6. �5

7. �21

8. �216

9. �8

10. �23

11. �23

12. �20

13. 16

14. 12

15. 19

16. 13

17. �12

18. �17

19. 8

20. 5

21. 82° � 12° � 70°F

22.853 � (�70) � 70$993 is the balance

23.947 � 86 � (�86)$775 is the balance

24.

SECTION 2.3 143

2.3 Exercises

[Objective 1]

Subtract.

1. 21 � 13 2. 36 � 22 3. 82 � 45

4. 103 � 56 5. 8 � 10 6. 14 � 19

7. 24 � 45 8. 136 � 352 9. �5 � 3

10. �15 � 8 11. �9 � 14 12. �8 � 12

13. 5 � (�11) 14. 7 � (�5) 15. 7 � (�12)

16. 3 � (�10) 17. �36 � (�24) 18. �28 � (�11)

19. �19 � (�27) 20. �11 � (�16)

[Objective 2]

For exercises 21 to 23, write an integer expression that describes the situation. Then answerthe question.

21. Science and Medicine The temperature at noon on a June day was 82�F. It fellby 12 degrees in the next 4 h. What was the temperature at 4:00 P.M.?

22. Business and Finance Jason’s checking account shows a balance of $853. Hehas discovered that a deposit of $70 was accidently recorded as a check for $70.What is the corrected balance?

23. Business and Finance Ylena’s checking account shows a balance of $947. Shehas discovered that a check for $86 was recorded as a deposit of $86. What is thecorrected balance?

24. Technology How long ago was the year 1250 B.C.E.? What year was 3,300 yearsago? Make a number line and locate the following events, cultures, and objects onit. How long ago was each item in the list? Which two events are the closest to eachother? You may want to learn more about some of the cultures in the list and themathematics and science developed by each culture.

Inca culture in Peru—1400 A.D. The Ahmes Papyrus, a mathematical text from Egypt—1650 B.C.E.Babylonian arithmetic develops the use of a zero symbol—300 B.C.E.First Olympic Games—776 B.C.E.Pythagoras of Greece dies—500 B.C.E.Mayans in Central America independently develop use of zero—500 A.D. The Chou Pei, a mathematics classic from China—1000 B.C.E.The Aryabhatiya, a mathematics work from India—499 A.D. Trigonometry arrives in Europe via the Arabs and India—1464 A.D. Arabs receive algebra from Greek, Hindu, and Babylonian sources and develop

it into a new systematic form—850 A.D. Development of calculus in Europe—1670 A.D. Rise of abstract algebra—1860 A.D. Growing importance of probability and development of statistics—1902 A.D.

2




© 2

010

McG

raw

-Hill

Co

mp

anie

s

144 SECTION 2.3

25. Complete the following statement: “3 � (�7) is the same as ____ because . . .”Write a problem that might be answered by doing this subtraction.

26. Explain the difference between the two phrases: “a number subtracted from 5”and “a number less than 5.” Use algebra and English to explain the meaning ofthese phrases. Write other ways to express subtraction in English. Which ones areconfusing?

27. Science and Medicine The greatest one-day temperature drop in the UnitedStates happened on Christmas Eve, 1924, in Montana. The temperature went from63�F during the day to �21�F at night. What was the total temperature drop?

28. Science and Medicine A similar one-day temperature drop happened in Alaska.The temperature went from 47�F during the day to �29�F at night. What was thetotal temperature drop?

29. Science and Medicine The tide at the mouth of the Delaware River tends to varybetween a maximum of �10 ft and a minimum of �5 ft. What is the difference infeet between the high tide and the low tide?

30. Science and Medicine The tide at the mouth of the Sacramento River tends tovary between a maximum of �7 ft and a minimum of �2 ft. What is the differencein feet between the high tide and the low tide?

Calculator Exercises

Use your calculator to perform the following operations.

31. �789 � (�128) 32. �910 � (�567)

33. �349 � (�431) 34. �412 � (�367)

35. 47 � (�25) 36. 123 � (�219)

37. 234 � (�456) 38. 412 � (�123)

Answers1. 8 3. 37 5. �2 7. �21 9. �8 11. �23 13. 1615. 19 17. �12 19. 8 21. 82° � 12° � 70�F23. 947 � 86 � (�86); $775 is the balance 25. 27. 84°F

29. 15 31. �917 33. �780 35. 72 37. 690

2

2

2

2

ANSWERS

25.

26.

27. 84°F

28. 76°F

29. 15

30. 9

31. �917

32. �1,477

33. �780

34. �779

35. 72

36. 342

37. 690

38. 535


To use this rule in multiplying two integers with different signs, multiply their absolutevalues and attach a negative sign.

Multiply.

(a) � �30

The product is negative.

(b) � �100

(c) � �96(8)(�12)

(�10)(10)

(5)(�6)

Example 1 Multiplying Integers

© 2

010

McG

raw

-Hill

Co

mp

anie

s

2.4 Multiplication of Integers

1. Find the product of two or more integers2. Use the order of operations with integers

When you first considered multiplication in arithmetic, it was thought of as repeatedaddition. Now we look at what our work with the addition of integers can tell us about mul-tiplication when integers are involved. For example,

3 � 4 � 4 � 4 � 4 � 12

We interpret multiplication as repeatedaddition to find the product, 12.

Now, consider the product (3)(�4):

(3)(�4) � (�4) � (�4) � (�4) � �12

Looking at this product suggests the first portion of our rule for multiplying integers.The product of a positive integer and a negative integer is negative.

�

2.4 OBJECTIVES

The product of two integers with different signs is negative.

Property: Multiplying Integers Case 1: Different Signs

OBJECTIVE 1

Multiply.

(a) (�7)(5) (b) (�12)(9) (c) (�15)(8)


145



The product of two negative integers is harder to visualize. The following pattern mayhelp you see how we can determine the sign of the product.

(3)(�2) � �6

(2)(�2) � �4

(1)(�2) � �2 Do you see that the product isincreasing by 2 each time as you go down?

(0)(�2) � 0

(�1)(�2) � 2

What should the product (�2)(�2) be? Continuing the pattern shown, we see that

(�2)(�2) � 4

This suggests that the product of two negative integers is positive, which is the case. We canextend our multiplication rule.

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Multiply.

(a) 10 � 12 (b) (�8)(�9)


The multiplicative identity property and multiplicative property of zero studied in Sec-tion 1.5 can be applied to integers, as illustrated in Example 3.

Find each product.

(a) � �7

(b) � 15

(c) � 0

(d) � 00 # 12

(�7)(0)

(15)(1)

(1)(�7)

Example 3 Multiplying Integers by One and Zero

NOTE This number is decreasingby 1.

NOTE (�1)(�2) is the oppositeof �2.

The product of two integers with the same sign is positive.

Property: Multiplying Integers Case 2: Same Sign

Multiply.

(a) � 63 The product of two positive numbers(same sign, �) is positive.

(b) � 40 The product of two negative numbers(same sign, �) is positive.

(�8)(�5)

9 # 7



MULTIPLICATION OF INTEGERS SECTION 2.4 147

We can now extend the rules for the order of operations learned in Section 1.8 tosimplify expressions containing integers. First, we will work with integers raised to apower.

Evaluate each expression.

(a) (�3)2 � (�3)(�3) � 9

(b) (�3)3 � (�3)(�3)(�3) � �27

(c) �32 � �(3 � 3) � �9 Note that the negative is not squared.

Example 4 Integers with Exponents

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Multiply.

(a) (�10)(1) (b) (0)(�17)



(a) (�4)2 (b) (�4)3 (c) �42


In Example 5 we will apply the order of operations.


(a) 7(�9 � 12) Evaluate inside the parentheses first.

� 7(3) � 21

(b) (�8)(�7) � 40 Multiply first, then subtract.

� 56 � 40

� 16

(c) (�5)2 � 3 Evaluate the power first.

� (�5)(�5) � 3

� 25 � 3

� 22

(d) �52 � 3 Note that �52 � �25. The power applies only to the 5.

� �25 � 3

� �28

Note that (�5)2 � (�5)(�5)� 25

Example 5 Using Order of Operations with Integers

NOTE In part (b) of Example 4,we have a negative integerraised to a power.

In part (c), only the 3 israised to a power. We havethe opposite of 3 squared.

OBJECTIVE 2



© 2

010

McG

raw

-Hill

Co

mp

anie

s

1. (a) �35; (b) �108; (c) �120 2. (a) 120; (b) 72 3. (a) �10; (b) 0

4. (a) 16; (b) �64; (c) �16 5. (a) �16; (b) 22; (c) 20; (d) �12


READING YOUR TEXT


Section 2.4

(a) The product of two integers with different signs is .

(b) The product of two integers with the same sign is .

(c) Given the expression �32, the is not squared.

(d) The rules for order of operations were learned in Section .


(a) 8(�9 � 7) (b) (�3)(�5) � 7

(c) (�4)2 � (�4) (d) �42 � (�4)



© 2

010

McG

raw

-Hill

Co

mp

anie

s

Name

Section Date

ANSWERS

1. 40

2. 42

3. �60

4. �20

5. �72

6. �36

7. 56

8. 72

9. 60

10. 21

11. 0

12. 0

13. 0 14. 0

15. 25 16. 21

17. �6 18. �12

19. �6 20. �18

21. 21 22. 20

23. �11 24. �75

25. �33 26. 16

27. �2 28. �4

29. �1 30. �4

31. �22 32. �11

33. �1 34. �5

35. 32 36. 16

SECTION 2.4 149

2.4 Exercises

[Objective 1]

Multiply.

1. 4 � 10 2. 3 � 14

3. (5)(�12) 4. (10)(�2)

5. (�8)(9) 6. (�12)(3)

7. (�8)(�7) 8. (�9)(�8)

9. (�5)(�12) 10. (�7)(�3)

11. (0)(�18) 12. (�17)(0)

13. (15)(0) 14. (0)(25)

[Objective 2]

Do the indicated operations. Remember the rules for the order of operations.

15. 5(7 � 2) 16. 7(8 � 5)

17. 2(5 � 8) 18. 6(14 � 16)

19. �3(9 � 7) 20. �6(12 � 9)

21. �3(�2 � 5) 22. �2(�7 � 3)

23. (�2)(3) � 5 24. (�6)(8) � 27

25. 4(�7) � 5 26. (�3)(�9) � 11

27. (�5)(�2) � 12 28. (�7)(�3) � 25

29. (3)(�7) � 20 30. (2)(�6) � 8

31. �4 � (�3)(6) 32. �5 � (�2)(3)

33. 7 � (�4)(�2) 34. 9 � (�2)(�7)

35. (�7)2 � 17 36. (�6)2 � 20




© 2

010

McG

raw

-Hill

Co

mp

anie

s

150 SECTION 2.4

37. (�5)2 � 18 38. (�2)2 � 10

39. �62 � 4 40. �52 � 3

41. (�4)2 � (�2)(�5) 42. (�3)3 � (�8)(�2)

43. (�8)2 � 52 44. (�6)2 � 42

45. (�6)2 � (�3)2 46. (�8)2 � (�4)2

47. �82 � 52 48. �62 � 32

49. �82 � (�5)2 50. �92 � (�6)2

51. Business and Finance Stores occasionally sell products at a loss in order to drawin customers or to reward good customers. The theory is that customers will buyother products along with the discounted item and the store will ultimately profit.

Beguhn Industries sells five different products. The company makes $18 oneach product-A item sold, loses $4 on product-B items, earns $11 on product C,makes $38 on product D, and loses $15 on product E.

One month, Beguhn Industries sold 127 units of product A, 273 units of prod-uct B, 201 units of product C, 377 units of product D, and 43 units of product E.What was their profit or loss that month?

52. Statistics In Atlantic City, Nick played the slot machines for 12 h. He lost $45 anhour. Use integers to represent the change in Nick’s financial status at the end ofthe 12 h.

53. Science and Medicine The temperature is �6�F at 5:00 in the evening. If the tem-perature drops 2 degrees every hour, what is the temperature at 1:00 A.M.?

Answers1. 40 3. �60 5. �72 7. 56 9. 60 11. 0 13. 015. 25 17. �6 19. �6 21. 21 23. �11 25. �3327. �2 29. �1 31. �22 33. �1 35. 32 37. 4339. �40 41. 6 43. 39 45. 27 47. �89 49. �8951. �$17,086 53. �22�F

2

ANSWERS

37. 43

38. 14

39. �40

40. �28

41. 6

42. �43

43. 39

44. 20

45. 27

46. 48

47. �89

48. �45

49. �89

50. �117

51. �$17,086

52. �$540

53. �22�F


151

Again, the rule is easy to use. To divide two integers, divide their absolute values. Thenattach the proper sign according to the rule.

Divide.

(a)Positive

� 4 PositivePositive

(b)Negative

� 9 PositiveNegative

(c)Negative

� �6 NegativePositive

(d)Positive

� �25 NegativeNegative

75

�3

�42

7

�36

�4

28

7

Example 1 Dividing Integers

© 2

010

McG

raw

-Hill

Co

mp

anie

s

2.5 Division of Integers

1. Find the quotient of two integers2. Use the order of operations with integers

You know from your work in arithmetic that multiplication and division are related opera-tions. We can use that fact, and our work of Section 2.4, to determine rules for the divisionof integers. Every division problem can be stated as an equivalent multiplication problem.For instance,

because

because

because

These examples illustrate that because the two operations are related, the rule of signsthat we stated in Section 2.4 for multiplication is also true for division.

�30 � (�5)(6)�30

�5� 6

�24 � (6)(�4)�24

6� �4

15 � 5 # 315

5� 3

2.5 OBJECTIVES

1. The quotient of two integers with different signs is negative.

2. The quotient of two integers with the same sign is positive.

Property: Dividing Integers

OBJECTIVE 1



As discussed in Section 1.6, we must be very careful when 0 is involved in a divisionproblem. Remember that 0 divided by any nonzero number is just 0. This rule can beextended to include integers, so that

because

However, if zero is the divisor, we have a special problem. Consider

This means that �9 � 0 � ?.Can 0 times a number ever be �9? No, so there is no solution.

Because cannot be replaced by any number, we agree that division by 0 is not

allowed. We say that division by 0 is undefined.

Divide, if possible.

(a) is undefined.

(b) is undefined.

(c) � 0

(d) � 0

Note: The expression is called an indeterminate form. You will learn more about this

in later mathematics classes.

0

0

0

�8

0

5

�9

0

7

0


�9

0

�9

0� ?

0 � (�7)(0)0

�7� 0

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Divide.

(a) (b)

(c) (d)144

�12

�48

�8

80

20

�55

11



DIVISION OF INTEGERS SECTION 2.5 153

The fraction bar, like parentheses and the absolute value bars, serves as a grouping sym-bol. This means that all operations in the numerator and denominator should be performedseparately. Then the division is done as the last step. Example 3 illustrates this property.


(a)

(b)

(c)

Divide as the last step.��16

�8� 2

Multiply in the numerator. Thenadd in the numerator andsubtract in the denominator.

��4 � (�12)

�6 � 2

�4 � (2)(�6)

�6 � 2

Add in the numerator, thendivide.�

�9

3� �3

3 � (�12)

3

Multiply in the numerator, thendivide.�

42

3� 14

(�6)(�7)

3

Example 3 Using Order of Operations

© 2

010

McG

raw

-Hill

Co

mp

anie

s

Divide, if possible.

(a) (b) (c) (d)0

�9

�7

0

5

0

0

3



(a) (b) (c)(�2)(�4) � (�6)(�5)

(�2)(11)

3 � (2)(�6)

�5

�4 � (�8)

6


OBJECTIVE 2

Using Your Calculator toMultiply and Divide Integers

Finding the product of two integers using a calculator is relatively straightforward.

Find the product. (457) (�734)

457 734

Your display should read �335,438.

��




Finding the quotient of integers is also straightforward.

Find the quotient.

384 16


��

�384

16


© 2

010

McG

raw

-Hill

Co

mp

anie

s

Find the quotient.

(�7,865) � (�242)


We can also use the calculator to raise an integer to a power.

Evaluate.

(�3)6

3 6

or, on some calculators

3 6

Either way, your display should read 729.

Enter^)(�)(

�yx)��(

Example 6 Raising a Number to a Power

Evaluate.

(�2)9


NOTE The parentheses ensurethat the negative is attached tothe 3 before it is raised to apower.

Find the products.

(a) (36) (�91) (b) (�12) (�284)



DIVISION OF INTEGERS SECTION 2.5 155©

201

0 M

cGra

w-H

ill C

om

pan

ies

READING YOUR TEXT


Section 2.5

(a) The quotient of two integers with different signs is .

(b) The quotient of two integers with the same sign is .

(c) Division by is not allowed.

(d) The fraction bar serves as a symbol.

1. (a) �5; (b) 4; (c) 6; (d) �12 2. (a) 0; (b) undefined; (c) undefined; (d) 0

3. (a) �2; (b) �3; (c) 1 4. (a) �3,276; (b) 3,408 5. 32.5 6. �512



© 2

010

McG

raw

-Hill

Co

mp

anie

s

156 SECTION 2.5

Name

Section Date

ANSWERS

1. 5

2. 5

3. 8

4. �3

5. �10

6. 4

7. �13

8. �8

9. 25

10. �4

11. 0

12. 5 13. 9

14. Undefined

15. 12 16. �10

17. Undefined

18. 0

19. �17 20. 27

21. 9 22. �25

23. 9 24. 15

25. 4 26. 4

27. �2 28. �9

29. 8 30. 3

31. �2 32. 3

33. Undefined

34. Undefined

2.5 Exercises

[Objective 1]

Divide.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

16. 17. 18.

19. 20. 21.

22.

[Objective 2]

Perform the indicated operations.

23. 24. 25.

26. 27. 28.

29. 30. 31.

32. 33. 34.10 � 6

4 � 4

7 � 5

2 � 2

�11 � 7

�14 � 8

55 � 19

�12 � 6

�14 � 4

�6

�12 � 12

�3

36

�7 � 3

24

�4 � 8

(7)(�8)

�14

(�8)(2)

�4

(�9)(5)

�3

(�6)(�3)

2

�150

6

�144

�16

�27

�1

�17

1

0

8

18

0

�20

2

�96

�8

�10

0

�9

�1

�125

�25

0

�8

�60

15

�75

�3

56

�7

�52

4

�32

�8

50

�5

�24

8

48

6

70

14

�20

�4




© 2

010

McG

raw

-Hill

Co

mp

anie

sANSWERS

35. 125 � 9 � 9 � $44

36. � 14 weeks

37. �$1,256

38. �675

39. �936

40. �1,736

41. 952

42. 1,349

43. 2

44. 625

45. �1,024

20,000 � 16,232——

3

�42—�3

SECTION 2.5 157

For exercises 35 to 37, use integers to write an expression that represents the situation. Thenanswer the question.

35. Business and Finance Patrick worked all day mowing lawns and was paid$9 per hour. If he had $125 at the end of a 9-h day, how much did he have beforehe started working?

36. Social Science A woman lost 42 lb. If she lost 3 lb each week, how long has shebeen dieting?

37. Business and Finance Suppose that you and your two brothers bought equalshares of an investment for a total of $20,000 and sold it later for $16,232. Howmuch did each person lose?

Calculator Exercises

Use your calculator to multiply and divide.

38. (15) (�45) 39. (78) (�12)

40. (�56) (31) 41. (�34) (�28)

42. (�71) (�19) 43.

44. (�5)4 45. (�4)5

Answers1. 5 3. 8 5. �10 7. �13 9. 25 11. 0 13. 915. 12 17. Undefined 19. �17 21. 9 23. 9 25. 427. �2 29. 8 31. �2 33. Undefined 35. 125 � 9 � 9 � $44

37. $1,256 39. �936 41. 952 43. 2

45. �1,024

20,000 � 16,232

3�

�28

�14


INTEGERS AND INTRODUCTION TO ALGEBRA

Documents

Transcript of INTEGERS AND INTRODUCTION TO ALGEBRA