Introduction to 2 Integers and Algebraic...

22Introduction toIntegers andAlgebraicExpressions

2.1 Integers and the Number Line

2.2 Addition of Integers

2.3 Subtraction of Integers

2.4 Multiplication of Integers

2.5 Division of Integers and Order of Operations

2.6 Introduction to Algebra and Expressions

2.7 Like Terms and Perimeter

2.8 Solving Equations

Real-World ApplicationSurface temperatures on Mars vary from �128Cduring polar night to 27C at the equator duringmidday at the closest point in orbit to the sun. Findthe difference between the highest value and thelowest value in this temperature range.

Source : Mars Institute

This problem appearsas Exercise 73 inSection 2.3.

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Prealgebra, Fifth Edition, by Marvin L. Bittinger, David J. Ellenbogen, and Barbara L. Johnson. Published by Addison-Wesley. Copyright © 2008 by Pearson Education, Inc.

In this section, we extend the set of whole numbers to form the set ofintegers. You have probably already used negative numbers. For example,the outside temperature could drop to negative five degrees and a creditcard statement could indicate activity of negative forty-eight dollars.

To create the set of integers, we begin with the set of whole numbers, 0, 1,2, 3, and so on. For each number 1, 2, 3, and so on, we obtain a new numberthe same number of units to the left of zero on a number line.

For the number 1, there is the opposite number �1 (negative 1).

For the number 2, there is the opposite number �2 (negative 2).

For the number 3, there is the opposite number �3 (negative 3), andso on.

The integers consist of the whole numbers and these new numbers. Weillustrate them on a number line as follows.

The integers to the left of zero on the number line are called negative inte-gers and those to the right of zero are called positive integers. Zero is neitherpositive nor negative and serves as its own opposite.

INTEGERS

The integers:

Integers and the Real World

Integers correspond to many real-world problems and situations. The following examples will help you get ready to translate problem situations to mathematical language.

EXAMPLE 1 Tell which integercorresponds to this situation:Researcher Robert Ballard discoveredthe wreck of the Titanic 12,500 ftbelow sea level.Source: Office of Naval Research

12,500 ft below sea levelcorresponds to the integer �12,500.

. . . , �5, �4, �3, �2, �1, 0, 1, 2, 3, 4, 5, . . .

�6 60�5 �4 �3 �2 �1 54321

Negative integers Positive integers

Opposites

Integers

0, neither positive nor negative

2.12.1 INTEGERS AND THE NUMBER LINEObjectivesState the integer thatcorresponds to a real-world situation.

Form a true sentence using� or �.

Find the absolute value ofany integer.

Find the opposite of anyinteger.

Study Tips

LEARN DEFINITIONS

Take time to learn thedefinitions in each section. Try to go beyond memorizingthe words of a definition tounderstanding the meaning.Asking yourself questionsabout a definition can aid inunderstanding. For example,for the word integers, firstmemorize the list of integers,and then see if you can thinkof some numbers that are not integers.

96

CHAPTER 2: Introduction to Integers and Algebraic Expressions

To the student:

In the Preface, at the front ofthe text, you will find a StudentOrganizer card. This pulloutcard will help you keep track ofimportant dates and usefulcontact information. You canalso use it to plan time for class,study, work, and relaxation. Bymanaging your time wisely, youwill provide yourself the bestpossible opportunity to besuccessful in this course.

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EXAMPLE 2 Tell which integers correspond to this situation: Elaine reversedthe disc in her DVD player 17 min and then advanced it 25 min.

The integers �17 and 25 correspond to the situation. The integer �17 cor-responds to the reversing and 25 corresponds to the advancing.

Do Exercises 1–5.

Order on the Number Line

Numbers are written in order on the number line, increasing as we move to theright. For any two numbers on the line, the one to the left is less than the one tothe right.

Since the symbol � means “is less than,” the sentence is read “�5is less than 9.” The symbol � means “is greater than,” so the sentence is read “�4 is greater than �8.”

EXAMPLES Use either � or � for to form a true sentence.

3. �9 2 Since �9 is to the left of 2, we have .

4. 7 �13 Since 7 is to the right of �13, we have .

5. �19 �6 Since �19 is to the left of �6, we have .

Do Exercises 6–9 on the next page.

Absolute Value

From the number line, we see that some integers, like 5 and �5, are the samedistance from zero.

How far is 5 from 0? How far is �5 from 0? Since distance is never negative (itis “nonnegative,” that is, either positive or zero), it follows that both 5 and �5are 5 units from 0.

ABSOLUTE VALUE

The absolute value of a number is its distance from zero on a number line. We use the symbol to represent the absolute value of a number x.

Like distance, the absolute value of a number is never negative; it is alwayseither positive or zero.

x

�6 �5 �4 �3 �2 �1 1 2 3 4 5 60

5 units5 units

�19 � �6

7 � �13

�9 � 2

�6 �5 �4 �3 �2 �1 1 2 3 4 5 60�7 7 8 9�8�9

�4 � �8�5 � 9

Tell which integers correspond toeach situation.

1. The halfback gained 8 yd onfirst down. The quarterback wassacked for a 5-yd loss onsecond down.

2. Temperature High and Low.The highest recorded tempera-ture in Nevada is 125F (degreesFahrenheit) on June 29, 1994, inLaughlin. The lowest recordedtemperature in Nevada is 50Fbelow zero on January 8, 1937,in San Jacinto.Sources: National Climatic Data Center,Asheville, NC, and Storm Phillips,STORMFAX, INC.

3. Stock Decrease. The stock ofWendy’s decreased from $41 pershare to $38 per share over arecent period.Source: The New York Stock Exchange

4. At 10 sec (seconds) before liftoff,ignition occurs. At 148 sec afterliftoff, the first stage is detachedfrom the rocket.

5. Jacob owes $137 to thebookstore. Fortunately, he has$289 in a savings account.

Answers on page A-4

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EXAMPLES Find the absolute value of each number.

6. The distance of �3 from 0 is 3, so .

7. The distance of 25 from 0 is 25, so .

8. The distance of 0 from 0 is 0, so .

To find a number’s absolute value:

1. If a number is positive or zero, use the number itself.2. If a number is negative, make the number positive.

Do Exercises 10–13.

Opposites

Recall that the set of integers can be represented on a number line. Given anumber on one side of 0, we can get a number on the other side by reflectingthe number across zero. For example, the reflection of 2 is �2. We can read �2as “negative 2” or “the opposite of 2.”

NOTATION FOR OPPOSITES

The opposite of a number x is written �x (read “the opposite of x”).

EXAMPLE 9 If x is �3, find �x.

To find the opposite of x when x is �3, we reflect �3 to the other sideof 0.

When We substitute �3 for x. We have . Theopposite of �3 is 3.

EXAMPLE 10 Find �x when x is 0.

When we try to reflect 0 “to the other side of 0,” we go nowhere:

when x is 0. The opposite of 0 is 0.

In Examples 9 and 10, the variable was replaced with a number. When thisoccurs, we say that we are evaluating the expression.

EXAMPLE 11 Evaluate �x when x is 4.

To find the opposite of x when x is 4, we reflect 4 to the other side of 0.

We have . The opposite of 4 is �4.��4� � �4

�6 �5 �4 �3 �2 �1 1 2 3 4 5 60

�x � 0

��3� � 3�x � ��3�.x � �3,

�6 �5 �4 �3 �2 �1 1 2 3 4 5 60

�6 �5 �4 �3 �2 �1 1 2 3 4 5 60

0 � 0025 � 2525�3 � 3�3

Use either � or � for to form atrue sentence.

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CALCULATORCORNER

Entering NegativeNumbers To enter a nega-tive number on many calcula-tors, we use the key.This key gives the opposite ofwhatever number is currentlydisplayed. Thus, to enter �27,we press . Othercalculators have a key. Toenter �27 on such a calculator,we simply press .Be careful not to confuse the

or key with thekey which is used for

subtraction.

Exercises: Press theappropriate keys so that yourcalculator displays each of thefollowing numbers.

1. �63

2. �419

3. �2004

4. List the keystrokesneeded to compute

.

Keystrokes will vary by calculator.

��7 � 4�

�

(�)��

72(�)

(�)

��72

��

6. 13 7 7. 12 �3

8. �13 �3 9. �4 �20

Find the absolute value.

10. 11. �918

12. 13. 52�29

Answers on page A-4

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A negative number is sometimes said to have a negative sign. A positivenumber is said to have a positive sign, even though it rarely is written in.

EXAMPLES Determine the sign of each number.

12. �7 Negative 13. 23 Positive

Replacing a number with its opposite, or additive inverse, is sometimescalled changing the sign.

EXAMPLES Change the sign (find the opposite, or additive inverse) of eachnumber.

14. �6 15. �10

16. 0 17. 14


EXAMPLE 18 If x is 2, find .

We replace x with 2:

Read “the opposite of the opposite of x”

We copy the expression, replacing x with 2

The opposite of the opposite of 2 is 2, or .

EXAMPLE 19 Evaluate for .

We replace x with �4:

Using an extra set of parentheses to avoid notation like ��4

Changing the sign of �4

Changing the sign of 4

Thus, .

When we change a number’s sign twice, we return to the original number.


It is important not to confuse parentheses with absolute-value symbols.

EXAMPLE 20 Evaluate for .

We replace x with 2:

Replacing x with 2

The absolute value of �2 is 2.

Thus,

Note that , whereas .

Do Exercises 25 and 26.

��2 � �2��2� � 2

��2 � �2.

� �2.

� ��2 ��x

x � 2��x

��4�� 4

� �4.

� �� 4 �

� ��4�� x�

x � �4��x�

��2� � 2

��2�.�

��x�

��x�

��14� � �14��0� � 0

��10� � 10��6� � 6

In each case draw a number line, if necessary.

14. Find �x when x is 1.

15. Find �x when x is �2.

16. Evaluate �x when x is 0.

Change the sign. (Find the opposite,or additive inverse.)

17. �4 18. �13

19. 39 20. 0

21. If x is 7, find .

22. If x is 1, find .

23. Evaluate for .

24. Evaluate for .

25. Find

26. Find

Answers on page A-4

��39.

��7.

x � �2��x�

x � �6��x�

��x�

��x�

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100


Tell which integers correspond to each situation.

2.12.1 Student’sSolutionsManual

VideoLectures on CD Disc 1

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MyMathLabMathXLEXERCISE SET For Extra Help

1. Pollution Fine. In 2003, The Colonial PipelineCompany was fined a record $34 million for pollution.Source: Green Consumer Guide.com

�34,000,000

2. Highest and Lowest Temperatures. The highesttemperature ever created on earth was 950,000,000F.The lowest temperature ever created was approximately460F below zero. 950,000,000; �460Source: The Guinness Book of Records, 2004

3. The recycling program for Colchester once received $40 for a ton of office paper. More recently, they’ve had to pay $15 to get rid of a ton of office paper.

40; �15

4. The space shuttle stood ready, 3 sec before liftoff. Solidfuel rockets were released 128 sec after liftoff.

�3; 128

5. At tax time, Janine received an $820 refund while Davidowed $541.

820; �541

6. Oceanography. At a depth of 2438 meters researchersfound the first hydrothermal vent ever seen by humans.This depth is approximately 8000 ft below sea level.Source: Office of Naval Research

�2438; �8000

7. Geography. Death Valley, California, is 280 ft below sealevel. Mt. Whitney, the highest point in California, hasan elevation of 14,491 ft. �280; 14,491

8. Geography. The Dead Sea, between Jordan and Israel,is 1286 ft below sea level; Mt. Rainier in WashingtonState is 14,410 ft above sea level. �1286; 14,410

Dea

d S

ea

JORDAN

ISRAEL

EGYPT

Mediterr

anean

Sea

Use either � or � for to form a true sentence.

9. �8 0 � 10. 7 0 � 11. 9 0 � 12. �7 0 � 13. 8 �8 �

14. 6 �6 � 15. �6 �4 � 16. �1 �7 � 17. �8 �5 � 18. �5 �3 �

19. �13 �9 � 20. �5 �11 � 21. �3 �4 � 22. �6 �5 �

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101

Exercise Set 2.1

Find the absolute value.

23. 57 24. 11 25. 0 26. 4 27. 24�24�401157

28. 36 29. 53 30. 54 31. 8 32. 79�79�85453�36

Find �x when x is each of the following.

33. �7 7 34. �6 6 35. 7 �7 36. 6 �6 37. 0 0

38. �1 1 39. �19 19 40. 50 �50 41. 42 �42 42. �73 73

Change the sign. (Find the opposite, or additive inverse.)

43. �8 8 44. �7 7 45. 7 �7 46. 10 �10 47. �29 29

48. �14 14 49. �22 22 50. 0 0 51. 1 �1 52. �53 53

Evaluate when x is each of the following.��x�

53. 7 7 54. �8 �8 55. �9 �9 56. 3 3 57. �17 �17 58. �19 �19

59. 23 23 60. 0 0 61. �1 �1 62. 73 73 63. 85 85 64. �37 �37

Evaluate when x is each of the following.��x

65. 47 �47 66. 92 �92 67. 345 �345 68. 729 �729

69. 0 0 70. 1 �1 71. �8 �8 72. �3 �3

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Use either �, �, or � for to write a true sentence.

87. � 88. � 89. �8�8�74�2�5

SKILL MAINTENANCE

SYNTHESIS

75. Add: . [1.2b] 825 76. Evaluate: . [1.9b] 12553327 � 498

83. If is true, does it follow that is alsotrue? Why or why not?

84. Does always represent a positive number? Whyor why not?

xDW�b � �aa � bDW

77. Multiply: . [1.5a] 7106 78. Solve: . [1.7b] 4300 � x � 1200209 � 34

79. Evaluate: . [1.9b] 81 80. Multiply: . [1.5a] 155031 � 5092

81. Simplify: . [1.9c] 10 82. Simplify: . [1.9c] 427�9 � 3�5�8 � 6�

85. On your calculator list the sequence of keystrokesneeded to find the opposite of the sum of 549 and 387.

. Answers may vary.

86. On your calculator list the sequence of keystrokesneeded to find the opposite of the product of 438 and 97.

. Answers may vary.��79�834��783�945

Simplify.

90. �3 91. �8 92. �2 93. �7�7��2��8�3

Solve. Consider only integer replacements.

94. �7, 7 95. �1, 0, 1x � 2x � 7

96. Simplify , , and .

x, �x, x

97. List these integers in order from least to greatest.

, �5, , 4, , �100, 0, , ,

�100, �5, 0, , 4, , , , , 2102710272�63

10272273�6210

��x��x��x�

73. Does �x always represent a negative number? Whyor why not?

74. Explain in your own words why .��x� � xDWDW

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Add, using a number line.

1.

2.

3.

4.

For each illustration, write acorresponding addition sentence.

5.

6.

7.

Answers on page A-4

�4 �3 �2 �1 1 2 3 4 5 60

�6 �5 �4 �3 �2 �1 1 2 3 40

�5 �4 �3 �2 �1 1 2 3 4 50

�5 � 5

�3 � 7

�3 � ��5�

3 � ��4�

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Addition

To explain addition of integers, we can use the number line. Once our under-standing is developed, we will streamline our approach.

ADDING INTEGERS

To perform the addition we start at a, and then moveaccording to b.

a) If b is positive, we move to the right.b) If b is negative, we move to the left.c) If b is 0, we stay at a.

EXAMPLE 1 Add:

EXAMPLE 2 Add:

EXAMPLE 3 Add:

Do Exercises 1–7.

You may have noticed a pattern in Example 2 and Margin Exercises 2 and6. When two negative integers are added, the result is negative.

ADDING NEGATIVE INTEGERS

To add two negative integers, add their absolute values and changethe sign (making the answer negative).

�4 � 9 � 5�6 �5 �4 �3 �2 �1 1 2 3 4 5 60 7 8 9

Start at �4. Move9 units to the right.

�4 � 9.

�1 � ��3� � �4�6 �5 �4 �3 �2 �1 1 2 30�7�8�9

Start at �1.Move3 units to the left.

�1 � ��3�.

2 � ��5� � �3�6 �5 �4 �3 �2 �1 1 2 3 4 5 60

Start at 2.Move5 units to the left.

2 � ��5�.

a � b,

2.22.2 ADDITION OF INTEGERSObjectiveAdd integers without using a number line.

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EXAMPLES Add.

4. Think : Add the absolute values: Make the answer negative, �12.

5. We can visualize the number line without actuallydrawing it.


Note that the sum of two positive integers is positive, and the sum of twonegative integers is negative.

When the number 0 is added to any number, that number remains un-changed. For this reason, the number 0 is referred to as the additive identity.

EXAMPLES Add.

6. 7. 8.


When we add a positive integer and a negative integer, as in Examples 1and 3, the sign of the number with the greater absolute value is the sign ofthe answer.

ADDING POSITIVE AND NEGATIVE INTEGERS

To add a positive integer and a negative integer, find the difference oftheir absolute values.

a) If the negative integer has the greater absolute value, the answer isnegative.

b) If the positive integer has the greater absolute value, the answer ispositive.

EXAMPLES Add.

9. Think : The absolute values are 3 and 5. The difference is 2. Since the negative number has the larger absolutevalue, the answer is negative, �2.

10. Think : The absolute values are 11 and 8. The differenceis 3. The positive number has the larger absolute value,so the answer is positive, 3.

11. 12.

13. 14.


Sometimes �a is referred to as the additive inverse of a. This terminologyis used because adding any number to its additive inverse always results inthe additive identity, 0.

�8 � 8 � 0, 14 � ��14� � 0, and 0 � 0 � 0.

�6 � 10 � 47 � ��3� � 4

�7 � 4 � �31 � ��6� � �5

11 � ��8� � 3

3 � ��5� � �2

17 � 0 � 170 � ��9� � �9�4 � 0 � �4

�8 � ��2� � �10

5 � 7 � 12.�5 � ��7� � �12

Add. Do not use a number lineexcept as a check.

8.

9.

10.

11.

Add.

12.

13.

14.

Add, using a number line only asa check.

15.

16.

17.

18.

Answers on page A-4

10 � ��7�

5 � ��7�

�7 � 3

�4 � 6

�56 � 0

49 � 0

0 � ��17�

�11 � ��11�

�20 � ��14�

�9 � ��3�

�5 � ��6�

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ADDING OPPOSITES

For any integer a,

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


Suppose we wish to add several numbers, positive and negative:

Because of the commutative and associative laws for addition, we can groupthe positive numbers together and the negative numbers together and addthem separately. Then we add the two results.

EXAMPLE 15 Add:

First add the positive numbers:

Then add the negative numbers:

Finally, add the results:

We can also add in any other order we wish, say, from left to right:


� 17.

� 29 � ��12� � 34 � ��5� � ��12� � 20 � 14 � ��5� � ��12�

15 � ��2� � 7 � 14 � ��5� � ��12� � 13 � 7 � 14 � ��5� � ��12�

36 � ��19� � 17.

�2 � ��5� � ��12� � �19.

15 � 7 � 14 � 36.

15 � ��2� � 7 � 14 � ��5� � ��12�.

15 � ��2� � 7 � 14 � ��5� � ��12�.

a � ��a� � �a � a � 0.

Add, using a number line only asa check.

19.

20.

21.

22.

Add.

23.

24.

25.

Answers on page A-4

31 � ��12��35 � 17 � 14 � ��27� �

18 � ��31�42 � ��81� � ��28� � 24 �

��59� � ��14��15� � ��37� � 25 � 42 �

89 � ��89�

�10 � 10

�6 � 6

5 � ��5�

Study Tips HELP SESSIONS

Often students find that a tutoring session would behelpful. The following comments may help you tomake the most of such sessions.

� Work on the topics before you go to the help ortutoring session. Do not regard yourself as anempty cup that the tutor will fill with knowledge.The primary source of your ability to learn iswithin you. When students go to help or tutoringsessions unprepared, they waste time and, inmany cases, money. Go to class, study thetextbook, work exercises, and mark trouble spots.Then use the help and tutoring sessions to workon the trouble spots.

� Do not be afraid to ask questions in thesesessions! The more you talk to your tutor, themore the tutor can help you.

� Try being a “tutor” yourself. Explaining a topicto someone else—a classmate, your instructor—is often the best way to master it.

Make the most oftutoring sessions

by doing what youcan ahead of time

and knowing thetopics with which

you need help.

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106


Add, using a number line.



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6. 0 7. 0 8. 9. 10. �11�2 � ��9��4�3 � ��1��13�8 � ��5��7 � 79 � ��9�

11. 12. 9 13. 5 14. �1�3 � 2�7 � 12�4 � 13�54 � ��9�

Add. Use a number line only as a check.

15. 16. 17. 18. �24�10 � ��14��11�6 � ��5��10�3 � ��7��12�3 � ��9�

19. 0 20. 0 21. 0 22. 0�3 � 3�2 � 210 � ��10�5 � ��5�

23. 6 24. 7 25. 0 26. 0�17 � 1713 � ��13�7 � 00 � 6

27. 28. 29. 30. �190 � ��19��270 � ��27��43�43 � 0�25�25 � 0

31. 0 32. 0 33. 34. �11�11 � 0�8�8 � 012 � ��12��31 � 31

35. 5 36. 1 37. 38. �30 � ��3��9�4 � ��5��7 � 89 � ��4�

39. 40. 41. 9 42. 11�3 � 1414 � ��5��210 � ��12��50 � ��5�

1. 2. 3. 4. 5 5. 6�3 � 98 � ��3��4�9 � 5�41 � ��5��5�7 � 2

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107

Exercise Set 2.2

43. 44. 45. 0 46. �7�10 � 3�19 � 19�340 � ��34��3�11 � 8

47. 48. 49. 50. �20�15 � ��5��24�17 � ��7��10�15 � 5�10�16 � 6

51. 52. 6 53. 54. 0�8 � 8�21�15 � ��6��8 � 14�511 � ��16�

55. 2 56. 57. 6 58. 019 � ��19��11 � 17�33�14 � ��19�11 � ��9�

59. 60. 22 61. 25 62. 3740 � ��8� � 530 � ��10� � 523 � ��5� � 4�21�15 � ��7� � 1

63. 64. �9�25 � 25 � ��9��17�23 � ��9� � 15

65. 6 66. 5763 � ��18� � 1240 � ��40� � 6

67. 68. �63�35 � ��63� � 35�6512 � ��65� � ��12�

69. 70. 3975 � ��14� � ��17� � ��5��160�24 � ��37� � ��19� � ��45� � ��35�

71. 72. 5227 � ��54� � ��32� � 65 � 46�6228 � ��44� � 17 � 31 � ��94�

73. 74. 2935 � ��51� � 29 � 51 � ��35��23�19 � 73 � ��23� � 19 � ��73�

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108


SKILL MAINTENANCE

SYNTHESIS

75. Explain in your own words why the sum of twonegative numbers is always negative.

76. A student states “�45 is bigger than �21.” Whatmistake do you think the student is making?DWDW

Subtract. [1.3d]

77.

324

78.

3625

79.

1484

80.

23,337

� 1 9 , 8 7 64 3 , 2 1 3

� 1 4 0 72 8 9 1

� 2 6 8 96 3 1 4

� 2 1 95 4 3

81. Write in expanded notation: 39,417. [1.1b]

3 ten thousands 9 thousands 4 hundreds 1 ten

82. Round to the nearest hundred: 746. [1.4a] 700

� 7 ones��

83. Round to the nearest thousand: 32,831. [1.4a] 84. Multiply: [1.5a] 235242 � 56.33,000

85. Divide: [1.6c] 32 86. Round to the nearest ten: 3496. [1.4a] 3500288 9.

87. Without using the words “absolute value,” explainhow to find the sum of a positive number and anegative number.

88. Why is it important to understand the associativeand commutative laws when adding more than twointegers at a time?

DWDW

Add.

89. 90. 17�32 � ��15��40�27 � ��13�

91. �6483 92. �2531497 � ��3028��3496 � ��2987�

93. �1868 94. �2784�7623 � 4839�7846 � 5978

95. For what numbers x is positive? All negative 96. For what numbers x is negative? All positive�x�x

Tell whether each sum is positive, negative, or zero.

97. If n is positive and m is negative, then is.

98. If and n is negative, then is.positive

�n � ��m�n � mnegative

�n � m

99. If n is negative and m is less than n, then is.

100. If n is positive and m is greater than n, then is.positive

n � mnegative

n � m

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Subtract.

1.

Think: What number can beadded to 4 to get �6?

2.

Think: What number can beadded to �10 to get �7?

3.

Think: What number can beadded to �2 to get �7?

Complete the addition and comparewith the subtraction.

4.

5.

6.

7.

Answers on page A-4

�5 � 3 ��5 � ��3� � �2;

�5 � 9 ��5 � ��9� � 4;

�3 � ��8� ��3 � 8 � �11;

4 � ��6� �4 � 6 � �2;

�7 � ��2�

�7 � ��10�

�6 � 4

109


Subtraction

We now consider subtraction of integers. To find the difference we lookfor a number to add to b that gives us a.

THE DIFFERENCE

The difference is the number that when added to b gives a.

For example, because Let’s consider anexample in which the answer is a negative number.

EXAMPLE 1 Subtract:

Think: is the number that when added to 8 gives 5. What numbercan we add to 8 to get 5? The number must be negative. The number is �3:

That is, because

Do Exercises 1–3.

The definition of above does not always provide the most efficientway to subtract. To understand a faster way to subtract, consider findingusing a number line. We start at 5. Then we move 8 units to the left to do thesubtracting. Note that this is the same as adding the opposite of 8, or �8, to 5.

Look for a pattern in the following table.

Do Exercises 4–7.

Perhaps you have noticed that we can subtract by adding the opposite ofthe number being subtracted. This can always be done.

5 � 8 � �3�6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6

Move 8 unitsto the left. Start at 5.

5 � 8a � b

8 � ��3� � 5.5 � 8 � �3

5 � 8 � �3.

5 � 8

5 � 8.

28 � 17 � 45.45 � 17 � 28

a � b

b�a

a � b,

2.32.3 SUBTRACTION OF INTEGERSObjectivesSubtract integers andsimplify combinations ofadditions and subtractions.

Solve applied problemsinvolving addition andsubtraction of integers.

�7 � 2 � �5 �7 � ��2� � �5

�7 � 10 � 3 �7 � ��10� � 3

�6 � ��4� � �10 �6 � 4 � �10

5 � ��8� � �3 5 � 8 � �3

SUBTRACTIONS ADDING AN OPPOSITE

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SUBTRACTING BY ADDING THE OPPOSITE

To subtract, add the opposite, or additive inverse, of the numberbeing subtracted:

This is the method generally used for quick subtraction of integers.

EXAMPLES Equate each subtraction with a corresponding addition. Thenwrite the equation in words.

2. ;

Adding the opposite of 30

Negative twelve minus thirty is negative twelve plus negative thirty.

3. ;

Adding the opposite of �17

Negative twenty minus negative seventeen is negative twenty plus seventeen.


Once the subtraction has been rewritten as addition, we add as in Section 2.2.

EXAMPLES Subtract.

4. The opposite of 6 is �6. We change the subtraction to addition and add the opposite. Instead of subtracting 6, we add �6.

5. The opposite of �9 is 9. We change the subtraction to addition and add the opposite. Instead of subtracting �9, we add 9.

6. We change the subtraction to addition and add the opposite. Instead of subtracting 8, we add �8.

7. We change the subtraction to addition and add the opposite.Instead of subtracting 7, we add �7.

8. Instead of subtracting �9, we add 9.

To check, note that .

9. Instead of subtracting �3, we add 3.

Check: .


�4 � ��3� � �7 � �4

�7 � ��3� � �7 � 3

5 � ��9� � �4 � 5

�4 � ��9� � �4 � 9

� 3

10 � 7 � 10 � ��7�

� �12

�4 � 8 � �4 � ��8�

� 13

4 � ��9� � 4 � 9

� �4

2 � 6 � 2 � ��6�

�20 � ��17� � �20 � 17

�20 � ��17�

�12 � 30 � �12 � ��30��12 � 30

a � b � a � ��b�.

Equate each subtraction with acorresponding addition. Then writethe equation in words.

8.

9.

10.

11.

12.

Subtract.

13.

14.

15.

16.

17.

18.

Answers on page A-4

5 � ��8�

�8 � ��2�

�7 � ��9�

13 � 8

�6 � 10

7 � 11

�14 � ��14�

�12 � 10

�12 � ��9�

13 � 5

3 � 10

110


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When several additions and subtractions occur together, we can makethem all additions. The commutative law for addition can then be used.

EXAMPLE 10 Simplify: .

Adding opposites

Using acommutative law


Applications and Problem Solving

We need addition and subtraction of integers to solve a variety of appliedproblems.

EXAMPLE 11 Toll Roads. The E-Z Pass program allows drivers in theNortheast to travel certain toll roads without having to stop to pay. Instead,a transponder attached to the vehicle is scanned as the vehicle rolls througha toll booth. Recently the Ramones began a trip to New York City with a bal-ance of $12 in their E-Z Pass account. Their trip accumulated $15 in tolls,and because they overspent their balance, the Ramones had to pay $80 infines and administrative fees. By how much were the Ramones in debt as aresult of their travel on the toll roads?Source: State of New Jersey

We solve by first subtracting the cost of the tolls from the original bal-ance in the account. Then we subtract the cost of the fees and fines fromthe new balance in the account:


,

and


.

The Ramones were $83 in debt as a result of their travel on toll roads.


� �83

�3 � 80 � �3 � ��80�

� �3

12 � 15 � 12 � ��15�

� 3

� �12 � 15

� �3 � ��9� � 5 � 4 � 6

�3 � ��5� � 9 � 4 � ��6� � �3 � 5 � ��9� � 4 � 6

�3 � ��5� � 9 � 4 � ��6�

Simplify.

19.

20.

21. E-Z Pass. (See Example 11.)Suppose the Ramones had abalance of $11 in their account,accumulated $25 in tolls, andhad to pay $85 in fines andadministrative fees. By howmuch would the Ramones be in debt?

22. Temperature Extremes. InChurchill, Manitoba, Canada,the average daily lowtemperature in January is �31C (degrees Celsius). Theaverage daily low temperature in Key West, Florida, is 50warmer. What is the averagedaily low temperature in Key West, Florida?

Answers on page A-4

9 � ��6� � 7 � 9 � 8 � ��20�

�6 � ��2� � ��4� � 12 � 3

Study Tips USING THE ANSWER SECTION

When using the answers listed in the back of thisbook, try not to “work backward” from the answer. If you find that you frequently require two or moreattempts to answer an exercise correctly, you probably need to work more carefully and/or rereadthe section preceding the exercise set. Remember thaton quizzes and tests you have only one attempt toanswer each question.

It is easy to becomeoverdependent on

the answer section.

111


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112


Subtract.



Math TutorCenter

InterActMath


1. 2. 3. 4. �90 � 9�80 � 8�53 � 8�52 � 7

5. 6. 2 7. 0 8. 0�6 � ��6��11 � ��11��6 � ��8��3�7 � ��4�

9. 10. 11. 12. 2630 � 4�720 � 27�514 � 19�413 � 17

13. 14. 2 15. 0 16. 0�9 � ��9��40 � ��40��7 � ��9��5�9 � ��4�

17. 0 18. 0 19. 14 20. 84 � ��4�7 � ��7�9 � 97 � 7

21. 11 22. 23. 24. 166 � ��10��14�6 � 8�11�7 � 48 � ��3�

25. 6 26. 27. 28. �62 � 8�81 � 9�16�14 � 2�3 � ��9�

29. 30. 31. 18 32. 115 � ��6�8 � ��10��1�4 � ��3��1�6 � ��5�

33. 34. 35. 36. �2�3 � ��1��3�5 � ��2��230 � 23�100 � 10

37. 38. 39. 5 40. 10 � ��1�0 � ��5��25�9 � 16�21�7 � 14

41. 42. 43. 12 44. 117 � ��4�7 � ��5��9�9 � 0�8�8 � 0

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113

Exercise Set 2.3

45. 46. 47. 48. �81�18 � 63�68�42 � 26�4518 � 63�196 � 25

49. 50. 51. 116 52. 12148 � ��73�24 � ��92��52�49 � 3�81�72 � 9

53. 0 54. 0 55. 55 56. �10�25 � ��15��30 � ��85��70 � ��70��50 � ��50�

Simplify.

57. 19 58. 13 59. �62�31 � ��28� � ��14� � 17�5 � ��8� � 3 � ��7�7 � ��5� � 4 � ��3�

60. 22 61. 62. 539 � ��88� � 29 � ��83��139�34 � 28 � ��33� � 44�43 � ��19� � ��21� � 25

63.

6

64.

4

65.

107

�5 � ��30� � 30 � 40 � ��12�84 � ��99� � 44 � ��18� � 43�93 � ��84� � 41 � ��56�

66.

116

67.

219

68.

190

81 � ��20� � 14 � ��50� � 53132 � ��21� � 45 � ��21�14 � ��50� � 20 � ��32�

Solve.

69. Reading. Before falling asleep, Alicia read from thetop of page 37 to the top of page 62 of her book. Howmany pages did she read? 25 pages

70. Writing. During a weekend retreat, James wrote fromthe bottom of page 29 to the bottom of page 37 of hismemoirs. How many pages did he write? 8 pages

71. Through exercise, Rod went from 8 lb above his “ideal”body weight to 9 lb below it. How many pounds didRod lose? 17 lb

72. Laura has a charge of $476.89 on her credit card, butshe then returns a sweater that cost $128.95. Howmuch does she now owe on her credit card? $347.94

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73. Surface Temperatures on Mars. Surface temperatureson Mars vary from �128C during polar night to 27Cat the equator during midday at the closest point inorbit to the sun. Find the difference between thehighest value and the lowest value in this temperature range. 155CSource: Mars Institute

74. Carla is completing the production work on a trackthat is to appear on her band’s upcoming CD. In doing so, she resets the digital recorder to 0, advancesthe recording 16 sec, and then reverses the recording25 sec. What reading will the recorder then display?

�9

75. While recording a 60-minute television show, thereading on Kate’s VCR changes from �21 min to 29 min. How many minutes have been recorded?Has she recorded the entire show? 50 min; No

76. As a result of coaching, Cedric’s average golf scoreimproved from 3 over par to 2 under. By how manystrokes did his score change? 5 strokes

77. Temperature Changes. One day the temperature inLawrence, Kansas, is 32 at 6:00 A.M. It rises 15 bynoon, but falls 50 by midnight when a cold frontmoves in. What is the final temperature? �3

78. Midway through a movie, Lisa resets the counter onher DVD player to 0. She then reverses the disc 8 min,and then advances the movie 11 min. What does thecounter now read? 3

79. Profit. Teapots and Treasures lost $5000 in 2004. In2005, the store made a profit of $8000. How muchmore did the store make in 2005 than in 2004?

$13,000

80. Tallest Mountain. The tallest mountain in the world,when measured from base to peak, is Mauna Kea(White Mountain) in Hawaii. From its base 19,684 ftbelow sea level in the Hawaiian Trough, it rises 33,480 ft. What is the elevation of the peak?Source: The Guinness Book of Records

13,796 ft above sea level

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115

Exercise Set 2.3

SKILL MAINTENANCE

83. Toll Roads. The Murrays began a trip with $13 in theirE-Z Pass account (see Example 11). They accumulated$20 in tolls and had to pay $80 in fines and administra-tive fees. By how much were the Murrays in debt as aresult of their travel on toll roads?

�87, or $87 in debt

84. Toll Roads. Suppose the Murrays (see Exercise 83)incurred $25 in tolls and $85 in fines and administrativefees. By how much would the Murrays be in debt?

�97, or $97 in debt

85. Write a subtraction problem for a classmate tosolve. Design the problem so that the solution is “Clara ends up $15 in debt.”

86. If a negative number is subtracted from a positivenumber, will the result always be positive? Why or why not?

DWDW

81. Offshore Oil. In 1998, the elevation of the world’sdeepwater drilling record was �7718 ft. In 2005, thedeepwater drilling record was 2293 ft deeper. What wasthe elevation of the deepwater drilling record in 2005?Source: www.deepwater.com/FactsandFirsts.cfm

�10,011 ft

82. Oceanography. The deepest point in the PacificOcean is the Marianas Trench, with a depth of 11,033 m. The deepest point in the Atlantic Ocean is the Puerto Rico Trench, with a depth of 8648 m.What is the difference in the elevation of the two trenches? 2385 m

– 11,033 m – 8648 m

MarianasTrench

Pacific OceanAtlantic Ocean

Puerto RicoTrench

1998 2005

– 7718 ft

?

Evaluate.

87. [1.9b] 64 88. [1.5a] 4896 89. [1.9b] 1 90. [1.5a] 4147143 � 291768 � 7243

91. How many 12-oz cans of soda can be filled with 96 ozof soda? [1.8a] 8 cans

92. A case of soda contains 24 bottles. If each bottlecontains 12 oz, how many ounces of soda are in the case? [1.8a] 288 oz

Simplify.

93. [1.9c] 35 94. [1.9c] 345 �22 � 11�5 � 42 � 2 � 7

95. [1.9c] 32 96. [1.9c] 165�13 � 2� �13 � 2��9 � 7� �9 � 7�

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SYNTHESIS

97. Explain why the commutative law was used inExample 10.

98. Is subtraction of integers associative? Why or why not?DWDW

Subtract.

99. �309,882 100. 83,44323,011 � ��60,432�123,907 � 433,789

For Exercises 101–106, tell whether each statement is true or false for all integers a and b. If false, show why.

101. False; 102. False; 0 � 3 � 30 � a � a3 � 0 � 0 � 3a � 0 � 0 � a

103. If , then . True 104. If , then . Truea � b � 0a � �ba � b � 0a � b

105. If , then a and b are opposites. True 106. If , then .

False; , but .3 � �33 � 3 � 0

a � �ba � b � 0a � b � 0

107. If is find the value of a. 17 108. If is find the value of x. 33�15,x � 48�37,a � 54

109. Doreen is a stockbroker. She kept track of the weekly changes in the stock market over a period of 5 weeks. By how manypoints (pts) had the market risen or fallen over this time? Up 15 points

110. Blackjack Counting System. The casino game ofblackjack makes use of many card-counting systems to give players an advantage if the count becomesnegative. One such system is called High–Low, firstdeveloped by Harvey Dubner in 1963. Each cardcounts as �1, 0, or 1 as follows:

2, 3, 4, 5, 6 count as �1;

7, 8, 9 count as 0;

10, J, Q, K, A count as �1.

Source: Patterson, Jerry L. Casino Gambling. New York:Perigee, 1982

a) Find the total count on the sequence of cards �2

K, A, 2, 4, 5, 10, J, 8, Q, K, 5.

b) Does the player have a winning edge? Yes

Down 13 pts Down 16 pts Up 36 pts Down 11 pts Up 19 pts

WEEK 1 WEEK 2 WEEK 3 WEEK 4 WEEK 5

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1. Complete, as in the example.

Multiply.

2.

3.

4.

5. Complete, as in the example.

Answers on page A-5

3 � (�10) �2 � (�10) �1 � (�10) �0 � (�10) �

�1 � (�10) ��2 � (�10) ��3 � (�10) �

�30�20

9��1�

20 � ��5�

�3 � 6

4 � 10 �3 � 10 �2 � 10 �1 � 10 �0 � 10 �

�1 � 10 ��2 � 10 ��3 � 10 �

4030

117


Multiplication

Multiplication of integers is like multiplication of whole numbers. The differ-ence is that we must determine whether the answer is positive or negative.

MULTIPLICATION OF A POSITIVE INTEGER AND A NEGATIVE INTEGERTo see how to multiply a positive integer and a negative integer, consider thefollowing pattern.

Do Exercise 1.

According to this pattern, it looks as though the product of a negative in-teger and a positive integer is negative. To confirm this, use repeated addition:

MULTIPLYING A POSITIVE AND A NEGATIVE INTEGER

To multiply a positive integer and a negative integer, multiply theirabsolute values and make the answer negative.

EXAMPLES Multiply.

1. 2. 3.

Do Exercises 2–4.

MULTIPLICATION OF TWO NEGATIVE INTEGERSHow do we multiply two negative integers? Again we look for a pattern.

Do Exercise 5.

4 � (�5) �3 � (�5) �2 � (�5) �1 � (�5) �0 � (�5) �

�1 � (�5) ��2 � (�5) ��3 � (�5) �

�20�15�10

�505

1015

This number decreasesby 1 each time.

This number increasesby 5 each time.

�7 � 6 � �4250��1� � �508��5� � �40

�3 � 5 � 5 � ��3� � �3 � ��3� � ��3� � ��3� � ��3� � �15

�2 � 5 � 5 � ��2� � �2 � ��2� � ��2� � ��2� � ��2� � �10

�1 � 5 � 5 � ��1� � �1 � ��1� � ��1� � ��1� � ��1� � �5

4 � 5 �3 � 5 �2 � 5 �1 � 5 �0 � 5 �

�1 � 5 ��2 � 5 ��3 � 5 �

201510

50

�5�10�15



2.42.4 MULTIPLICATION OF INTEGERSObjectivesMultiply integers.

Find products of three ormore integers and simplifypowers of integers.

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According to the pattern, the product of two negative integers is positive.This leads to the second part of the rule for multiplying integers.

MULTIPLYING TWO NEGATIVE INTEGERS

To multiply two negative integers, multiply their absolute values. Theanswer is positive.

EXAMPLES Multiply.

4.

5.

6.

Do Exercises 6–8.

The following is another way to state the rules for multiplication.

To multiply two integers:

a) Multiply the absolute values.b) If the signs are the same, the answer is positive.c) If the signs are different, the answer is negative.

MULTIPLICATION BY ZERONo matter how many times 0 is added to itself, the answer is 0. This leads tothe following result.

For any integer a,

(The product of 0 and any integer is 0.)

EXAMPLES Multiply.

7.

8.


Multiplication of More Than Two Integers

Because of the commutative and the associative laws, to multiply three ormore integers, we can group as we please.

EXAMPLES Multiply.

9. a) Multiplying the first two numbers

Multiplying the results

b) Multiplying the negatives

The result is the same as above. � 48

�8 � 2��3� � 24 � 2

� 48

�8 � 2��3� � �16��3�

0��7� � 0

�19 � 0 � 0

a � 0 � 0.

��9� ��1� � 9

��10� ��7� � 70

��2� ��4� � 8

Multiply.

6.

7.

8.

Multiply.

9.

10.

Answers on page A-5

�23 � 0

0 � ��5�

��1� ��6�

�9��5�

��3� ��4�

Study Tips

SLEEP WELL

Being well rested, alert, andfocused is very importantwhen studying math. Often,problems that may seemconfusing to a sleepy personare easily understood after agood night’s sleep. Using yourtime efficiently is alwaysimportant, so you should beaware that an alert, wide-awake student can oftenaccomplish more in10 minutes than a sleepystudent can accomplish in30 minutes.

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10. Multiplying the first two numbers and thelast two numbers

11. a) Each pair of negatives gives apositive product.

b) Making use ofExample 11(a)

We can see the following pattern in the results of Examples 9–11.

The product of an even number of negative integers is positive.The product of an odd number of negative integers is negative.


POWERS OF INTEGERSA positive number raised to any power is positive. When a negative number israised to a power, the sign of the result depends upon whether the exponentis even or odd.

EXAMPLES Simplify.

12. The result is positive.

13.

The result is negative.

14.

The result is positive.

15.

The result is negative.

Perhaps you noted the following.

When a negative number is raised to an even exponent, the result ispositive.

When a negative number is raised to an odd exponent, the result isnegative.


When an integer is multiplied by �1, the result is the opposite of thatinteger.

For any integer a,

�1 � a � �a

� �32

� 16��2� � 4 � 4 � ��2�

��2�5 � ��2� ��2� ��2� ��2� ��2�

� 81

� 9 � 9

��3�4 � ��3� ��3� ��3� ��3�

� �64

� 16��4� ��4�3 � ��4� ��4� ��4�

��7�2 � ��7� ��7� � 49

� �180

�5 � ��2� � ��3� � ��6� � ��1� � 10 � 18 � ��1� � 180

�5 � ��2� � ��3� � ��6� � 10 � 18

� �56

7��1� ��4� ��2� � ��7�8 Multiply.

11.

12.

13.

Simplify.

14.

15.

16.

17.

Answers on page A-5

25

��1�9

��9�2

��2�3

��1� ��1� ��2� ��3� ��1� ��1�

��4� ��5� ��2� ��3� ��1�

�2 � ��5� � ��4� � ��3�

119


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EXAMPLE 16 Simplify:

Since lacks parentheses, the base is 7, not �7. Thus we regard as

The rules for order of operations tell us to square first.

Compare Examples 12 and 16 and note that In fact, theexpressions and are not even read the same way: is read“negative seven squared,” whereas is read “the opposite of sevensquared.”


�72��7�2�72��7�2

��7�2 � �72.

� �49.

� �1 � 49

� �1 � 7 � 7

�72 � �1 � 72

�1 � 72:�72�72

�72.18. Simplify:

19. Simplify: .

20. Write and in words.

Answers on page A-5

�82��8�2

��5�2

�52.

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CALCULATOR CORNER

Exponential Notation When using a calculator to calculateexpressions like or it is important to use the correctsequence of keystrokes.Calculators with key: On some calculators, a key must bepressed after a number is entered to make the number negative. For thesecalculators, appropriate keystrokes for are

.

To calculate we must first raise 39 to the power 4. Then the sign ofthe result must be changed. This can be done with the keystrokes

or by multiplying by �1:

.

Calculators with key: On some calculators, the key is pressedbefore a number to indicate that the number is negative. This is similar tothe way the expression is written on paper. For these calculators, isfound by pressing

and � is found by pressing

.

You can either experiment or consult a user’s manual if you are unsure ofthe proper keystrokes for your calculator.

Exercises: Use a calculator to determine each of the following.

1. 148,035,889

2. �1,419,857

3. �1,124,864

4. 1,048,576��4�10

��104�3

��17�5

��23�6

ENTER �4�93(�)

394

ENTER �4�)93(�)(

��39�4

(�)(�)

��1��4xy93

394

��4xy93

�394,

�4xy��93

��39�4

��

�394,��39�4

5. �531,441

6. �117,649

7. �7776

8. �19,683�39

�65

�76

�96

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Multiply.

121

Exercise Set 2.4

EXERCISE SET For Extra Help2.42.4 Student’sSolutionsManual


Math TutorCenter

InterActMath

MyMathLabMathXL

1. �16 2. �21 3. �18 4. �49�7 � 7�9 � 2�7 � 3�2 � 8

5. �48 6. �24 7. �30 8. �72�9 � 8�10 � 38 � ��3�8 � ��6�

9. 15 10. 16 11. 18 12. 72��8� ��9��9 � ��2��8 � ��2��3��5�

13. 42 14. 24 15. 20 16. 72�9��8��10��2��8 � ��3��6� ��7�

17. �120 18. �120 19. 300 20. 200�25��8��6��50�15��8�12��10�

21. 72 22. �123 23. �340 24. �43��1�43��20�1741��3��72� ��1�

25. 0 26. 0 27. 0 28. 00��38�0��14��17 � 0�47 � 0

Multiply.

29. 24 30. �28 31. 4207��4� ��3�5��7� � ��4� � ��1�3 � ��8� � ��1�

32. 756 33. �70 34. 150��2� ��5� ��3� ��5��2��5� ��7�9��2� ��6�7

35. 30 36. �270 37. 0��15� ��29�0 � 8�6��5� ��9��5� ��2� ��3� ��1�

38. 0 39. �294 40. 600��5�6��4�5��7� ��1� �7� ��6�19��7� ��8�0 � 6

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Simplify.

41. 36 42. 64 43. �125 44. 16��2�4��5�3��8�2��6�2

45. 10,000 46. �1 47. �16 48. 64��2�6�24��1�5��10�4

49. �243 50. �10,000 51. 1 52. �1��1�13��1�12�104��3�5

53. �729 54. �64 55. �64 56. �32�25�43�26�36

Write each of the following expressions in words.

57.

The opposite of eight tothe fourth power

58.

Negative six to theeighth power

59.

Negative nine to thetenth power

60.

The opposite of five tothe fourth power

�54��9�10��6�8�84

61. Explain in your own words why is positive. 62. Explain in your own words why is negative.�910DW��9�10DW

SKILL MAINTENANCE

SYNTHESIS

63. Round 532,451 to the nearest hundred. [1.4a]532,500

64. Write standard notation for sixty million. [1.1c]60,000,000

65. Divide: [1.6c] 80 66. Multiply: [1.5a] 255075 � 34.2880 36.

67. Simplify: . [1.9c] 5 68. Simplify: . [1.9c] 482 � 52 � 3 � 23 �3 � 32�10 � 23 � 6 2

69. A rectangular rug measures 5 ft by 8 ft. What is the areaof the rug? [1.5c], [1.8a] 40 sq ft

70. How many 12-egg cartons can be filled with 2880 eggs?[1.8a] 240 cartons

71. A ferry can accommodate 12 cars and 53 cars arewaiting. How many trips will be required to ferrythem all? [1.8a] 5 trips

72. An elevator can hold 16 people and 50 people arewaiting to go up. How many trips will be required totransport all of them? [1.8a] 4 trips

73. Which number is larger, or Why? 74. Describe all conditions for which is negative.axDW��5�79?��3�79DWSimplify.

75. 243 76. �8 77. 0 78. 25�52��1�29�94 � ��9�4��2�3 � ��1�2946��3�5��1�379

79. 7 80. 294�12��3�2 � 53 � 62 � ��5�2��2�5 � 32 � �3 � 7�2

81. �2209 82. �2809 83. 130,321 84. 279,841��23�4��19�4�532�472

85. �2197 86. 87.116,875

88.�85,525,504

��17�4�129 � 133�5�935�238 � 243�3�3375��49 � 34�3�73 � 86�3

89. Jo wrote seven checks for $13 each. If she had a balanceof $68 in her account, what was her balance afterwriting the checks? �$23

90. After diving 95 m below the surface, a diver rises at arate of 7 meters per minute for 9 min. What is the diver’snew elevation? �32 m

91. What must be true of m and n if is to be(a) negative? (b) positive?(a) Both m and n must be odd. (b) At least one of m and nmust be even.

92. What must be true of m and n if is to be(a) positive? (b) zero? (c) negative?

(a) m and n must have different signs. (b) At least one of mand n must be zero. (c) m and n must have the same sign.

�mn��5�mn

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Divide.

1.

Think : What numbermultiplied by �2 gives 6?

2.

Think : What numbermultiplied by �3 gives �15?

3.

Think : What numbermultiplied by 8 gives �24?

4.

5.

6.

Answers on page A-5

�459

30�5

0�4

�24 8

�15�3

6 ��2�

123


We now consider division of integers. Because of the way in which divi-sion is defined, its rules are similar to those for multiplication.

Division of Integers

THE QUOTIENT

The quotient (or , or ) is the number, if there is one, that

when multiplied by b gives a.

Let’s use the definition to divide integers.

EXAMPLES Divide, if possible. Check each answer.

1. Think: What number multiplied by �7 gives 14? The number is �2. Check: .

2. Think: What number multiplied by �4 gives �32? The number is 8. Check: .

3. Think: What number multiplied by 7 gives �21? The number is �3. Check: .

4. Think: What number multiplied by �5 gives 0? The number is 0. Check: .

5. is not defined.Think : What number multiplied by 0 gives �5? There is no such number because the product of 0 and any number is 0.

The rules for determining the sign of a quotient are the same as for determining the sign of a product. We state them together.

To multiply or divide two integers:

a) Multiply or divide the absolute values.b) If the signs are the same, the answer is positive.c) If the signs are different, the answer is negative.

Do Exercises 1–6.

Recall that, in general, and are different numbers. In Exam-ple 4, we divided into 0. In Example 5, we attempted to divide by 0. Since anynumber times 0 gives 0, not �5, we say that is not defined or is unde-fined. Also, since any number times 0 gives 0, is also not defined.0 0

�5 0

b aa b

�50

0��5� � 00

�5� 0

��3� � 7 � �21�21 7 � �3

8��4� � �32�32�4

� 8

��2� ��7� � 1414 ��7� � �2

a�ba bab

b�a

2.52.5 DIVISION OF INTEGERS AND ORDER OF OPERATIONS

ObjectivesDivide integers.

Use the rules for order ofoperations with integers.

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EXCLUDING DIVISION BY 0

Division by 0 is not defined:

is undefined for all real numbers a.

DIVIDING 0 BY OTHER NUMBERSNote that

DIVIDENDS OF 0

Zero divided by any nonzero real number is 0:

EXAMPLES Divide.

6. 7. 8. is undefined.

Do Exercises 7–9.

Rules for Order of Operations

When more than one operation appears in a calculation or problem, we applythe same rules that were used in Section 1.9. We repeat them here for review,now including absolute-value symbols.

RULES FOR ORDER OF OPERATIONS

1. Do all calculations within parentheses, brackets, braces, absolute-value symbols, numerators, or denominators.

2. Evaluate all exponential expressions.3. Do all multiplications and divisions in order from left to right.4. Do all additions and subtractions in order from left to right.

EXAMPLES Simplify.

9.

With no grouping symbols or exponents, we begin with the third rule.

Carrying out all multiplications anddivisions in order from left to right

10.

We first simplify within the absolute-value symbols.

Dividing

Multiplying

Subtracting � 12

� 2 � ��10��2 � 2 � 2 � 5��2�

� �2 � 5��2��2� ��2� ��2� � �8 ��2�3 4 � 5��2� � �8 4 � 5��2�

��2�3 4 � 5��2�

� �3

� 17 � 20

17 � 10 2 � 4 � 17 � 5 � 4

17 � 10 2 � 4

�30

012

� 00 ��6� � 0

0a

� 0, a � 0.

0 8 � 0 because 0 � 0 � 8.

a 0, or a0

,

Divide, if possible.

7.

8.

9.

Answers on page A-5

�52 0

0 ��4�

34 0

Study Tips

ORGANIZATION

When doing homework,consider using a spiral orthree-ring binder. You want tobe able to go over yourhomework when studying fora test. Therefore, you need tobe able to easily access anyproblem. Write legibly, labeleach section and each exerciseclearly, and show all steps.Writing clearly will also beappreciated by your instructorif homework is collected. Mosttutors and instructors can bemore helpful if they can seeand understand all the stepsin your work.

When you are finishedwith your homework, checkthe answers to the odd-numbered exercises at theback of the book or in theStudent’s Solutions Manualand make corrections. If youdo not understand why ananswer is wrong, draw a starby it so you can ask questionsin class or during yourinstructor’s office hours.

124


⎫⎬⎭

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EXAMPLE 11 Simplify:

Carrying out all operations insideparentheses first, multiplying 23 by 2,following the rules for order ofoperations within the parentheses

Completing the addition insideparentheses

Evaluating exponential expressions

Doing all multiplications

Doing all additions and subtractions in order from left to right

Always regard a fraction bar as a grouping symbol. It separates any calcu-lations in the numerator from those in the denominator.

EXAMPLE 12 Simplify: .

Calculating within the numerator:and

Dividing


� 2

5 � 9 � �4��3�2 � ��3� ��3� � 9 �

�4�2

5 � ��3�2

�2�

5 � 9�2

5 � ��3�2

�2

� 137

� 220 � 83

� 16 � 204 � 83

� 16 � 51 � 4 � 83

� 24 � 51 � 4 � 83

� 24 � 51 � 4 � �37 � 46�24 � 51 � 4 � �37 � 23 � 2�

24 � 51 � 4 � �37 � 23 � 2�. Simplify.

10.

11.

12.

13.

Answers on page A-5

��5� ��9�1 � 2 � 2

52 � 5 � 53 � �42 � 48 4�

��2� � 3 � 22 � 5

5 � ��7� ��3�2

125


⎫⎪⎬⎪⎭

CALCULATOR CORNER

Grouping Symbols Most calculators now provide grouping symbols.Such keys may appear as and or and . Grouping symbolscan be useful when we are simplifying expressions written in fractionform. For example, to simplify

,

we press .Failure to include grouping symbols in the above keystrokes would meanthat we are simplifying a different expression:

.

Exercises: Use a calculator with grouping symbols to simplify each ofthe following.

1. �4 2. �2 3.

787

785 �285 � 54

17 � 3 � 51311 � 172

2 � 1338 � 178

5 � 30

38 �1422

� 47

�)74�2()241�83(

38 � 1422 � 47

...)][(...)(

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Divide, if possible, and check each answer by multiplying. If an answer is undefined, state so.



Math TutorCenter

InterActMath


1. �7 2. �5 3. �14 4. �226 ��13�28�2

35�7

28 ��4�

5. �9 6. 11 7. 4 8. 7�63 ��9��48�12

�22 ��2�18�2

9. �9 10. �2 11. 2 12. �50�400

8�100 ��50�

�5025

�728

13. �43 14. �16 15. �8 16. 21�651 ��31�200�25

�1288

�344 8

17. Undefined 18. 0 19. �8 20. 29�145

�588

�110

�5�56

0

21. �23 22. �31 23. 0 24. Undefined�13

00

�2�

2177

�27612

25. �19 26. �17 27. �41 28. �2323 ��1��41 1�17

119�1

Simplify, if possible. If an answer is undefined, state so.

29. �7 30. 6 31. 19 32. �36�8 � 2� �3 � 9�9 � 2�3 � 8�5 � �2 � 3 � 7�5 � 2 � 3 � 6

33. 34. 35. 23 36. 1024 � 22 � 1040 � 32 � 23�16010 � 20 � 15 � 24�33416 � ��24� � 50

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127

Exercise Set 2.5

57. �7988 58. 12 59. 60. �549�7�34� � 18�30002 � 103 � 500020 � 43 ��8�12 � 203

61. 60 62. 63.

1

64.

264

256 � ��32� ��4��1000 ��100� 10�1448��6 � 13� � 116�9 � �3 � 4�

65. 2 66. 67. 7 68. 19 � 5 � 739 � 7 � 32178 � 7 � 9 � 2 � 18 � 7 � 9 � 3

69.

Undefined

70.

2

71.

3

72.

40

5 � 62 �22 � 5� � 72

32 � 42 � ��2�3 � 22 � 32 �32 � �2 � 1��

52 � 62 � 22��3�4 2 � 42 � 3 � 2

�7 � 4�3 � 2 � 5 � 463 � 7 � 34 � 25 � 9

�1 � 23�3 � 73

73.

2

74.

�10

75.

0

76.

Undefined

�16 � 28 22

5 � 25 � 532 � 43 � 4 � 32

193 � 174�3 � 5�2 � �7 � 13�

�2 � 5�3 � 2 � 4��5�3 � 17

10�2 � 6� � 2�5 � 2�

77. Explain in your own words why isundefined.

78. Without performing any calculations, Stefanreports that is negative. How do you think he reached this conclusion?

�192 � 172��162 � 182�DW17 0DW

45. �9 46. 47.

18

48. 09 � 0 5 � 418 � 0�32 � 52 � 7 � 4��3514 2��6� � 720 5��3� � 3

49. 10 50. �1 51. 52. �63 � 32�25�3 � 8�2 ��1��2 � 5�2 ��9�4 � 52 10

53. �983 54. �95 55. 82 56. 535 � 62 3 � 222 � 102 5 � 2230 � ��5�317 � 103

41. �10 42. 4 43. �86 44. 5010��5� 1��1�8��7� � 6��5�100 � 62

��5�2 � 3292 � 11 � 32

37.

8

38.

305

39.

12

40.

126

53 � 4 � 9 � �8 � 9 � 3�4 � 5 � 2 � 6 � 443 � 10 � 20 � 82 � 234 � �6 � 8��4 � 3�

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SKILL MAINTENANCE

SYNTHESIS

79. Fabrikant Fine Diamonds ran a 4-in. by 7-in.advertisement in The New York Times. Find the area of the ad. [1.5c], [1.8a] 28 sq in.

80. A classroom contains 7 rows of chairs with 6 chairs in each row. How many chairs are there in theclassroom? [1.8a] 42 chairs

81. Cindi’s Ford Focus gets 32 mpg (miles per gallon). Howmany gallons will it take to travel 384 mi? [1.8a]

12 gal

82. Craig’s Chevy Blazer gets 14 mpg. How many gallonswill it take to travel 378 mi? [1.8a]

27 gal

83. A 7-oz bag of tortilla chips contains 1050 calories. Howmany calories are in a 1-oz serving? [1.8a]

150 cal

84. A 7-oz bag of tortilla chips contains 8 g (grams) of fatper ounce. How many grams of fat are in a cartoncontaining 12 bags of chips? [1.8a] 672 g

85. There are 18 sticks in a large pack of Trident gum. If 4 people share the pack equally, how many wholepieces will each person receive? How many extrapieces will remain? [1.8a] 4 pieces; 2 pieces

86. A bag of Ricola throat lozenges contains 24 coughdrops. If 5 people share the bag equally, how manylozenges will each person receive? How many extralozenges will remain? [1.8a] 4 lozenges; 4 lozenges

87. Ty claims that is �2. What mistake doyou think he is making?

88. Bryn contends that is �1. Whatmistake do you think she is making?

13 � 10�2 � 5DW8 � 32 � 1DW

Simplify, if possible.

89. 0 90. Undefined73 � 9 � 62 � 8 � 43 � 6

52 � 259 � 32

2 � 42 � 52 � 9 � 82 � 7

95. 992 96. 48693 � 363�122 � 92282 � 362�42 � 172

91.

0

92.

0

93.

�2

94.

�159

195 � ��15�3

195 � 7 � 5219 � 172

132 � 34�7 � 8�37

72 � 82 � �98 � 72 � 2��25 � 42�3

172 � 162 � ��6�2 � 62�

97. Write down the keystrokes needed to calculate .

�)x24�x23()3yx5�x251(

152 � 53

32 � 42

98. Write down the keystrokes needed to calculate .

�)x25�4�3()32�42�x261(

162 � 24 � 233 � 4 � 52

99. Evaluate the expression for which the keystrokes are as follows: . 56�201�4

Determine the sign of each expression if m is negative and n is positive.

101.

Positive

102.

Negative

103.

Negative

104.

Negative

105.

Positive

��n�m �� n

�m ��nm �n

�m�nm

100. Evaluate the expression for which the keystrokes are . 2)6�2(61�4

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1. Evaluate for and

2. Evaluate for and

3. Evaluate 5t for

Answers on page A-5

t � �14.

y � 29.x � 57x � y

b � 26.a � 38a � b

In this section, we will learn to write equivalent expressions by making useof the distributive law, both of which are very important concepts.

Algebraic Expressions

In arithmetic, we work with expressions such as

In algebra, we use both numbers and letters and work with algebraicexpressions such as

We have already worked with expressions like these.When a letter can stand for various numbers, we call the letter a variable.

A number or a letter that stands for just one number is called a constant. Letthe speed of light. Then c is a constant. Let the speed of an Amtrak

metroliner. Then a is a variable since the value of a can vary.An algebraic expression consists of variables, constants, numerals, and

operation signs. When we replace a variable with a number, we say that we aresubstituting for the variable. This process is called evaluating the expression.

EXAMPLE 1 Evaluate for and

We substitute 37 for x and 29 for y and carry out the addition:

The number 66 is called the value of the expression.

Algebraic expressions involving multiplication, like “8 times a,” can bewritten as or simply 8a. Two letters written together withoutan operation symbol, such as ab, also indicate multiplication.

EXAMPLE 2 Evaluate 3y for

Parentheses are required here.

Do Exercises 1–3.

Algebraic expressions involving division can also be written several ways.

For example, “8 divided by t” can be written as or

EXAMPLE 3 Evaluate and for and

We substitute 35 for a and 7 for b:

ab

�357

� 5;�a�b

��35�7

� 5.

b � 7.a � 35�a�b

ab

8t

.8�t,8 t,

3y � 3��14� � �42

y � �14.

8�a�,8 � a,8 � a,

x � y � 37 � 29 � 66.

y � 29.x � 37x � y

a �c �

x � 86, 7 � t, 19 � y, andab

.

37 � 86, 7 � 8, 19 � 7, and38

.

2.62.6 INTRODUCTION TO ALGEBRA AND EXPRESSIONS

ObjectivesEvaluate an algebraicexpression by substitution.

Use the distributive law tofind equivalent expressions.

Study Tips

READING A MATH TEXT

Do not expect a math text toread like a magazine or novel.On one hand, most assignedreadings in a math text consistof only a few pages. On theother hand, every sentenceand word is important andshould make sense. If theydon’t, seek help as soon aspossible.

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We substitute 15 for a and 3 for b:

Examples 3 and 4 illustrate the following.

and represent the same number.

and all represent the same number.

Do Exercises 4–7.

EXAMPLE 5 Evaluate for

This expression can be used to find the Fahrenheit temperature that cor-responds to 20 degrees Celsius:

Do Exercise 8.

EXAMPLE 6 Evaluate for and

The rules for order of operations specify that the replacement for x besquared. That result is then multiplied by 5:

Example 6 illustrates that when opposites are raised to an even power, theresults are the same.


EXAMPLE 7 Evaluate and for

When we evaluate for we have

Substitute 7 for x. Then evaluatethe power.

To evaluate we again substitute 7 for x. We must recall that taking the op-posite of a number is the same as multiplying that number by �1.

The opposite of a number is the same asmultiplying by �1.

Substituting 7 for x

Using the rules for order of operations;calculating the power before multiplying

Example 7 shows that


��x�2 � �x2.

� �1 � 49 � �49.

�72 � �1 � 72

�x2 � �1 � x2

�x2,

��x�2 � ��7�2 � ��7� ��7� � 49.

x � 7,��x�2

x � 7.�x2��x�2

5x2 � 5��3�2 � 5�9� � 45.

5x2 � 5�3�2 � 5�9� � 45;

x � �3.x � 35x2

9C5

� 32 �9 � 20

5� 32 �

1805

� 32 � 36 � 32 � 68.

C � 20.9C5

� 32

a�b

�ab

,�

ab

,

ab

�a�b

�ab

� �153

� �5;�ab

��15

3� �5;

a�b

�15�3

� �5.

b � 3.a � 15a

�b�ab

,�ab

,For each number, find two equivalentexpressions with negative signs indifferent places.

4.

5.

6.

7. Evaluate and for

and

8. Find the Fahrenheit temperaturethat corresponds to 10 degreesCelsius (see Example 5).

9. Evaluate for and

10. Evaluate for and

11. Evaluate and for

12. Evaluate and for


Answers on page A-5

x � �2.x � 2x5

x � 2.�x2��x�2

x � 3.�x2��x�2

a � �3.a � 3a4

x � �4.x � 43x2

b � 4.a � 28

�ab

�ab

,a

�b,

r�4

�mn

�6x

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Equivalent Expressions and the Distributive Law

It is useful to know when two algebraic expressions will represent the samenumber. In many situations, this will help with problem solving.

EXAMPLE 8 Evaluate and for and and compare theresults.

We substitute 3 for x in and again in 2x :

Next we repeat the procedure, substituting �5 for x :

The results can be shown in a table. Itappears that and 2x represent thesame number.


Example 8 suggests that and 2x represent the same number for anyreplacement of x. When this is known to be the case, we can say that and2x are equivalent expressions.

EQUIVALENT EXPRESSIONS

Two expressions that have the same value for all allowablereplacements are called equivalent.

In Examples 3 and 7 we saw that the expressions and are equivalent

but that the expressions and are not equivalent.An important concept, known as the distributive law, is useful for finding

equivalent algebraic expressions. The distributive law involves two opera-tions: multiplication and either addition or subtraction.

To review how the distributive law works, consider the following:

This is This is

This is the sum

To carry out the multiplication, we actually added two products. That is,

The distributive law says that if we want to multiply a sum of severalnumbers by a number, we can either add within the grouping symbols andthen multiply, or multiply each of the terms separately and then add.

7 � 45 � 7�40 � 5� � 7 � 40 � 7 � 5.

7 � 40 � 7 � 5. 3 1 5

7 � 40. 2 8 07 � 5. 3 5

� 7 4 5

�x2��x�2

ab

�a�b

x � xx � x

x � x

x � x � �5 � ��5� � �10; 2x � 2��5� � �10.

x � x � 3 � 3 � 6; 2x � 2 � 3 � 6.

x � x

x � �5x � 32xx � x

Complete each table by evaluatingeach expression for the given values.

14.

15.

Answers on page A-5

131


x � �5

x � 3 6 6

�10 �10

x � x 2x

CALCULATORCORNER

Evaluating Powers Toevaluate an expression like

for with acalculator, it is imperative thatwe keep in mind the rules fororder of operations. On somecalculators, this expression isevaluated by pressing

. Other calculators usethe keystrokes

.Consult your owner’s manual,an instructor, or simplyexperiment if your calculatorbehaves differently.

Exercises: Evaluate.

1. 243

2. 1024

3. �32

4. �3125�x5 for x � 5

�x5 for x � 2

�x5 for x � �4

�a5 for a � �3

ENTER �3�)4

1(�)((�)

��

�3xy��41

x � �14�x 3

x � 0

x � �2

x � 4 20 20

�10 �10

0 0

3x � 2x 5x

x � 0

x � �2

x � 2 6 6

�6 �6

0 0

4x � x 3x

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THE DISTRIBUTIVE LAW

For any numbers a, b, and c,


We have

and

It is impossible to overemphasize the importance of the parentheses inthe statement of the distributive law. Were we to omit the parentheses, wewould have To see that note that but

EXAMPLE 10 Use the distributive law to write an expression equivalent to

Note that the � sign between l and w nowappears between and

Try to go directly to this step.


Since subtraction can be regarded as addition of the opposite, it followsthat the distributive law holds in cases involving subtraction.

EXAMPLE 11 Use the distributive law to write an expression equivalent toeach of the following:

a) b) c) d)

a)


b)

Again, try to go directly to this step.

c) Using a commutative law

Using the distributive law

Using a commutative law to write baalphabetically and with the constant first

d) Using thedistributive law

Using an associativelaw (twice)



� �4x � 8y � 12z

� �4x � ��8y� � ��12z�

� �4x � ��4 � 2�y � ��4 � 3�z

�4�x � 2y � 3z� � �4 � x � ��4� �2y� � ��4� �3z�

b � 7 � ab � 7b

� b � a � b � 7

�a � 7�b � b�a � 7�

� 9x � 45

9�x � 5� � 9x � 9�5�

� 7a � 7b

7�a � b� � 7 � a � 7 � b

�4�x � 2y � 3z��a � 7�b;9�x � 5�;7�a � b�;

� 2l � 2w.

2 � w.2 � l 2�l � w� � 2 � l � 2 � w

2�l � w�.

3 � 4 � 2 � 14.3�4 � 2� � 18,a�b � c� � ab � c,ab � c.

ab � ac � 3 � 4 � 3 � 2 � 12 � 6 � 18.

a�b � c� � 3�4 � 2� � 3 � 6 � 18

c � 2.b � 4,a � 3,ab � aca�b � c�

a�b � c� � ab � ac.

Use the distributive law to write anequivalent expression.

16.

17.

Use the distributive law to write anequivalent expression.

18.

19.

20.

21.

Answers on page A-5

�8�2a � b � 3c�

�m � 4�6

3�a � b � c�

4�x � y�

6�x � y � z�

5�a � b�

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Evaluate.

133

Exercise Set 2.6



Math TutorCenter

InterActMath

MyMathLabMathXL

1. 12n, for

(The cost, in cents, of sending 2 text messages)

24

2. 39n, for

(The cost, in cents, of sending 2 letters)

78¢

n � 2

¢

n � 2

3. for and �2 4. 9mn

, for m � 18 and n � 2y � �3x � 6xy

,

5. 1 6. �35yz

, for y � 15 and z � �252qp

, for p � 6 and q � 3

7.

(The approximate doubling time, in years, for aninvestment earning 4% interest per year)

18 yr

8.

(The approximate doubling time, in years, for aninvestment earning 2% interest per year)

36 yr

72i

, for i � 272r

, for r � 4

9. 13 10. 39 � 2 � x, for x � 33 � 5 � x, for x � 2

11.

(The perimeter, in feet, of a 3-ft by 4-ft rectangle)

14 ft

12.

18

3�a � b�, for a � 2 and b � 42l � 2w, for l � 3 and w � 4

13.

(The perimeter, in feet, of a 3-ft by 4-ft rectangle)

14 ft

14.

18

3a � 3b, for a � 2 and b � 42�l � w�, for l � 3 and w � 4

15. 21 16. 204x � 4y, for x � 6 and y � 17a � 7b, for a � 5 and b � 2

17. 21 18. 204�x � y�, for x � 6 and y � 17�a � b�, for a � 5 and b � 2

19.

(The distance, in feet, that an object falls in 5 sec)

400 ft

20.

(The distance, in meters, that an object falls in 10 sec)

490 m

49t 2

10, for t � 1016t 2, for t � 5

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21. 10 22. 25�x � y�2, for x � 2 and y � 3a � �b � a�2, for a � 6 and b � 4

23. 0 24. 08x � 8y, for x � 17 and y � �179a � 9b, for a � 13 and b � �13

25.

(For determining the number of handshakes possibleamong 9 people) 36

26.

(For converting 50 degrees Fahrenheit to degreesCelsius) 10

5�F � 32�9

, for F � 50n2 � n

2, for n � 9

27. 100 28. 66a6 � a, for a � �2m3 � m2, for m � 5

For each expression, write two equivalent expressions with negative signs in different places.

29. 30. 31. 32.�3

r;

3�r

�3r

n�b

; �nb

�nb

�7x

; �7x

7�x

�5t

; 5

�t�

5t

33. 34. 35. 36.23

�m; �

23m

�23m

14�w

; �14w

�14w

u�5

; �u5

�u5

�9p

; �9p

9�p

Evaluate and for the given values.�ab

a�b

,�ab

,

37. �5; �5; �5 38. �8; �8; �8a � 40, b � 5a � 45, b � 9

39. �27; �27; �27 40. �8; �8; �8a � 56, b � 7a � 81, b � 3

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135

Exercise Set 2.6

Evaluate.

41. 36; �12 42. 36; �18��2x�2 and �2x2, for x � 3��3x�2 and �3x2, for x � 2

43. 45; 45 44. 50; 502x2, for x � 5 and x � �55x2, for x � 3 and x � �3

45. 216; �216 46. 64; 64x6, for x � 2 and x � �2x3, for x � 6 and x � �6

47. 1; 1 48. 243; �243x5, for x � 3 and x � �3x8, for x � 1 and x � �1

49. 32; �32 50. 1; �1a7, for a � 1 and a � �1a5, for a � 2 and a � �2

Use the distributive law to write an equivalent expression.

51. 52. 53. 4x � 44�x � 1�7x � 7y7�x � y�5a � 5b5�a � b�

54. 55. 56. 3x � 183�x � 6�2b � 102�b � 5�6a � 66�a � 1�

57. 58. 59. 30x � 126�5x � 2�4 � 4y4�1 � y�7 � 7t7�1 � t�

60. 61. 62. 20x � 32 � 12p4�5x � 8 � 3p�8x � 56 � 48y8�x � 7 � 6y�54m � 639�6m � 7�

63. 64. 65. 3x � 6�x � 2�3�9y � 63�9� y � 7��7y � 14�7� y � 2�

66. 67. 68. 16x � 40y � 64z8�2x � 5y � 8z��4x � 12y � 8z�4�x � 3y � 2z�2x � 8�x � 4�2

69. 70. 71. 4x � 12y � 28z4�x � 3y � 7z��6a � 12b � 6c�6�a � 2b � c�8a � 24b � 8c8�a � 3b � c�

72. 73. 74.

63a � 28b � 21c � 7d

�9a � 4b � 3c � d�7

20a � 25b � 5c � 10d

�4a � 5b � c � 2d�5

45x � 5y � 40z

5�9x � y � 8z�

75. Does always represent a negative number?

Why or why not?

76. Is always negative? Why or why not?�x2DW�xy

DW

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SKILL MAINTENANCE

SYNTHESIS

77. Write a word name for 23,043,921. [1.1c]Twenty-three million, forty-three thousand, nine hundredtwenty-one

78. Multiply: [1.5a]

901

17 � 53.

79. Estimate by rounding to the nearest ten. Show yourwork. [1.4b]

80. Divide: [1.6c]

994

2982 3.

5280 � 2480 � 2800

� 2 4 7 5 5 2 8 3

81. On January 6, it snowed 9 in., and on January 7, itsnowed 8 in. How much did it snow altogether? [1.8a]

17 in.

82. On March 9, it snowed 12 in., but on March 10, the sunmelted 7 in. How much snow remained? [1.8a]

5 in.

83. For Brett’s party, his wife ordered two cheese pizzas at$11 apiece and two pepperoni pizzas for $13 apiece.How much did she pay for the pizza? [1.8a]

$48

84. For Tania’s graduation party, her husband orderedthree buckets of chicken wings at $12 apiece and3 trays of nachos at $9 a tray. How much did he pay forthe wings and nachos? [1.8a]

$63

85. Under what condition(s) will the expression be nonnegative? Explain.

86. Ted evaluates for and gets 100 as theresult. What mistake did he probably make?

a � 5a � a2DWax2DW

87. A car’s catalytic converter works most efficiently after it is heated to about 370°C. To what Fahrenheit temperature doesthis correspond? (Hint : see Example 5.) 698F

Evaluate.

88. �2016x2 � xy2 2 � y, for x � 24 and y � 6

89. for and 4438 90. for and �8454y � �21x � 18x2 � 23y � y3,b � �16a � 19a � b3 � 17a,

91. 279 92. for and 2496b � �6a � �8a3b � a2b2 � ab3,r 3 � r 2t � rt 2, for r � �9 and t � 7

93. 2 94. 2x1492 � x1493, for x � �1a1996 � a1997, for a � �1

95. 2,560,000 96. 205a3a�4, for a � 2�m3 � mn�m, for m � 4 and n � 6

Replace the blanks with , , , or to make each statement true.��

97. 98. 59 17 59

59 17 59 8 � 1475��

8 � 1475

29� � �1888��88��32

29� � �1888�88�32

Classify each statement as true or false. If false, write an example showing why.

99. For any choice of x,

True


False; 23 � 8; but �23 � �8

x3 � �x3.x2 � ��x�2.


True


True

��3x�2 � 9x2.x6 � x4 � ��x�6 � ��x�4.

103. If the Fahrenheit temperature is doubled, does it follow that the corresponding Celsius temperature is also doubled?(Hint: see Example 5.)DW IS

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0-558-68755-5


What are the terms of eachexpression?

1.

2.

Identify the like terms.

3.

4.

Answers on page A-6

3xy � 5x2 � y2 � 4xy � y

9a3 � 4ab � a3 � 3ab � 7

�4y � 2x �xy

5x � 4y � 3

137


One common way in which equivalent expressions are formed is by com-bining like terms. In this section we learn how this is accomplished and applythe concept to geometry.

Combining Like Terms

A term is a number, a variable, a product of numbers and/or variables, or aquotient of numbers and/or variables. Terms are separated by addition signs.If there are subtraction signs, we can find an equivalent expression that usesaddition signs.

EXAMPLE 1 What are the terms of ?

Separating parts with � signs

The terms are 3xy, �4y, and .


Terms in which the variable factors are exactly the same, such as 9x and�4x, are called like, or similar, terms. For example, and are like terms,whereas 5x and are not. Constants, like 7 and 3, are also like terms.

EXAMPLES Identify the like terms.

2.

7x and 2x are like terms; 8 and 1 are like terms.

3.

5ab and �2ab are like terms; and are like terms.


When an algebraic expression contains like terms, an equivalent ex-pression can be formed by combining, or collecting, like terms. To com-bine like terms, we rely on the distributive law.

EXAMPLE 4 Combine like terms to form an equivalent expression.

a) b)

c) d)

a) Using the distributive law (in “reverse”)

We usually go directly to this step.

b) Try to do this mentally.

, or simply �mn

c) Rewriting as addition

Using a commutative law

Try to go directly to this step. � 4y � 3

� 7y � ��3y� � ��2� � 5

7y � 2 � 3y � 5 � 7y � ��2� � ��3y� � 5

� �1mn

6mn � 7mn � �6 � 7�mn

� 7x

4x � 3x � �4 � 3�x

2a5 � 9ab � 3 � a5 � 7 � 4ab7y � 2 � 3y � 5

6mn � 7mn4x � 3x

7a3a3

5ab � a3 � a2b � 2ab � 7a3

7x � 5x2 � 2x � 8 � 5x3 � 1

6x27y23y2

2z

3xy � 4y �2z

� 3xy � ��4y� �2z

3xy � 4y �2z

2.72.7 LIKE TERMS AND PERIMETERObjectivesCombine like terms.

Determine the perimeter of apolygon.

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d)

Rearranging terms

Think of as ;

Do Exercises 5–7.

Perimeter

PERIMETER OF A POLYGON

A polygon is a closed geometric figure with three or more sides. Theperimeter of a polygon is the distance around it, or the sum of thelengths of its sides.

EXAMPLE 5 Find the perimeter of this polygon.

We add the lengths of all sides. Since all the units are the same, we areeffectively combining like terms.

Using the distributive law


Caution!

When units of measurement are given in the statement of a problem, as inExample 5, the solution should also contain units of measurement.


A rectangle is a polygon with four sides and four 90 angles. Oppositesides of a rectangle have the same measure. The symbol CR or CR indicatesa 90 angle. A 90 angle is often referred to as a right angle.

EXAMPLE 6 Find the perimeter of a rectangle that is 3 cm by 4 cm.

Do Exercise 10.

� 14 cm

� �3 � 3 � 4 � 4� cm

Perimeter � 3 cm � 3 cm � 4 cm � 4 cm

3 cm

4 cm

90

� 29 m

� �6 � 5 � 4 � 5 � 9� m Perimeter � 6 m � 5 m � 4 m � 5 m � 9 m

9 m

5 m

5 m

6 m

4 m

� 3a5 � 5ab � 4

2a5 � a5 � 3a51a5a5� 3a5 � 5ab � ��4�� 2a5 � a5 � 9ab � ��4ab� � 3 � ��7�� 2a5 � 9ab � 3 � a5 � ��7� � ��4ab�

2a5 � 9ab � 3 � a5 � 7 � 4abCombine like terms to form anequivalent expression.

5.

6.

7.

Find the perimeter of each polygon.

8.

9.

10. Find the perimeter of a rectangle that is 2 cm by 4 cm.

Answers on page A-6

2 cm

4 cm

7 mm

7 mm7 mm

7 mm 7 mm

7 cm

11 cm 4 cm

2 cm2 cm

4m � 2n2 � 5 � n2 � m � 9

5x2 � 9 � 2x2 � 3

2a � 7a

138


A polygon with five sidesis called a pentagon.

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The perimeter of the rectangle in Example 6 is , orequivalently . This can be generalized, as follows.

PERIMETER OF A RECTANGLE

The perimeter P of a rectangle of length l and width w is given by

, or .

EXAMPLE 7 A common door size is 3 ft by 7 ft. Find the perimeter of sucha door.

We could also use .

Try to do this mentally.

Combining like terms

The perimeter of the door is 20 ft.

Do Exercise 11.

A square is a rectangle in which all sides have the same length.

EXAMPLE 8 Find the perimeter of a square with sides of length 9 mm.

Note that.

Do Exercise 12.

PERIMETER OF A SQUARE

The perimeter P of a square is four times s, the length of a side:

.

EXAMPLE 9 Find the perimeter of a square garden with sides of length 12 ft.

The perimeter of the garden is 48 ft.

Do Exercise 13.

� 48 ft

� 4 � 12 ft

P � 4s

s s

s

s

� 4s

P � s � s � s � s

� 36 mm 9 � 9 � 9 � 9 � 4 � 9 � �9 � 9 � 9 � 9� mm

P � 9 mm � 9 mm � 9 mm � 9 mm

9 mm

9 mm

� 20 ft

� 14 ft � 6 ft

� �2 � 7� ft � �2 � 3� ft � 2 � 7 ft � 2 � 3 ft

P � 2�l � w� P � 2l � 2w

ww

l

l

P � 2 � �l � w�P � 2l � 2w

2�3 cm � 4 cm�2 � 4 cm2 � 3 cm � 11. Find the perimeter of a 4-ft by

8-ft sheet of plywood.

12. Find the perimeter of a squarewith sides of length 10 km.

13. Find the perimeter of a squaresandbox with sides of length 6 ft.

Answers on page A-6

10 km

10 km

Study Tips

UNDERSTAND YOURMISTAKES

When you receive a gradedquiz, test, or assignment backfrom your instructor, it isimportant to review andunderstand what yourmistakes were. Too oftenstudents simply file away oldpapers without first making aneffort to learn from theirmistakes.

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The goal of these matchingquestions is to practice step (2),Translate, of the five-step problem-solving process. Translate eachword problem to an equation andselect a correct translation fromequations A–O.

A.

B.

C.

D.

E.

F.

G.

H.

I.

J.

K.

L.

M.

N.

O.

Answers on page A-6

75 � 120 � x

75 � 150 � x

75 � 120 � x

15 � ��10� � x

750 15 � x

8 � x � 120

�10 � ��15� � x

2 � 150 � 2 � 75 � x

75 � ��150� � x

15 � 750 � x

150 � 75 � x

8 � 120 � x

�10 � 15 � x

15 750 � x

75 � x � 120

6. Account Balance. Lorenzo had$75 in his checking account. Hethen wrote a check for $150.What was the balance in hisaccount?

7. Laptop Computers. GreatGraphics purchased a laptopcomputer for each of its 15 employees. If each laptopcosts $750, how much did thecomputers cost?

8. Basketball. A basketball teamscored 75 points in one game. Inthe next game, the team scoreda record 120 points. How manypoints did the team score in thetwo games?

9. Pizza Sales. A youth club sold120 pizzas for a fundraiser. Ifeach pizza sold for $8, howmuch money was taken in?

10. Temperature. The temperaturein Fairbanks was �10 at 6:00 P.M.and fell another 15 by midnight.What was the temperature atmidnight?

1. Yarn Cost. It costs $8 for theyarn for each scarf that Annetteknits. If she used $120 worth ofyarn, how many scarves did she knit?

2. Elevation. Genine startedhiking a 10-mi trail at anelevation that was 150 ft belowsea level. At the end of the trail,she was 75 ft above sea level.How many feet higher was sheat the end of the trail than at the beginning?

3. Community Service. In order to fulfill the requirements for asociology class, Glen must log120 hr of community service. So far, he has spent 75 hrvolunteering at a youth center.How many more hours must he serve?

4. Disaster Relief. Each packagethat is prepared for a disasterrelief effort contains 15 mealbars. How many packages canbe filled from a donation of 750 bars?

5. Perimeter. A rectangularbuilding lot is 75 ft wide and 150 ft long. What is theperimeter of the lot?

Translatingfor Success

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List the terms of each expression.

141

Exercise Set 2.7



Math TutorCenter

InterActMath

MyMathLabMathXL

1. 2a, 5b, �7c 2. 4x, �6y, 7z 3. 9mn, �6n, 89mn � 6n � 84x � 6y � 7z2a � 5b � 7c

4. 7rs, 4s, �5 5. , , 6. , , �9b3ab24a3b4a3b � ab2 � 9b3�2z3�4y23x2y3x2y � 4y2 � 2z37rs � 4s � 5

Combine like terms to form an equivalent expression.

7. 14x 8. 16a9a � 7a5x � 9x

9. �3a 10. �15x�16x � x10a � 13a

11. 12. 10a � 5b3a � 5b � 7a11x � 6z2x � 6z � 9x

13. 14. 38x � 442x � 6 � 4x � 2�13a � 6227a � 70 � 40a � 8

15. 16. �7 � 3a � 6b8 � 4a � 9b � 7a � 3b � 15�4 � 4t � 6y9 � 5t � 7y � t � y � 13

17. 18. 4x � 5y � 5x � 7y � 5 � 2y � 3x6a � 4b � 2a � 3b � 5a � 2 � b

19. 20. 21.

7x2 � 3y

8x2 � 3y � x2

3x � 2 � 2y

8x � 5x � 6 � 3y � y � 4

�1 � 17a � 12b

�8 � 11a � 5b � 6a � 7b � 7

22. 23. 24.

11a5 � 5b4

13a5 � 9b4 � 2a5 � 4b4

7x4 � y3

11x4 � 2y3 � 4x4 � y3

12y3 � 3z

8y3 � 3z � 4y3

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25. 26. 27.

3x3 � 8x2 � 4

x3 � 5x2 � 2x3 � 3x2 � 4

2a2 � 8a3 � 5

3a2 � 7a3 � a2 � 5 � a3

6a2 � 3a

9a2 � 4a � a � 3a2

28. 29. 30.

4a2b � 3ab2 � 2ab

8a2b � 3ab2 � 4a2b � 2ab

7a3 � 3ab � 3

7a3 � 4ab � 5 � 7ab � 8

7xy � 6y2 � 1

9xy � 4y2 � 2xy � 2y2 � 1

31. 32. 33.

�4a6 � 11b4 � 2a6b4

3a6 � 9b4 � 2a6b4 � 7a6 � 2b4

�4x4 � 6y4 � 8x4y4

3x4 � 2y4 � 8x4y4 � 7x4 � 8y4

9x3y � 2xy3 � 3xy

9x3y � 4xy3 � 6xy3 � 3xy

34. x6 � y5 � 3x6y9x6 � 5y5 � 3x6y � 8x6 � 4y5

Find the perimeter of each polygon.

35.

10 ft

36.

20 in.

37.

42 km

Each side7 km5 in.

5 in.

2 ft

3 ft

38.

17 mm

39.

8 m

40.

18 m

4 m

4 m4 m

5 m

1 m

3 m

3 m

1 m 1 m4 mm 6 mm

7 mm

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143

Exercise Set 2.7

Tennis Court. A tennis court contains many rectangles.Use the diagram of a regulation tennis court to calculate theperimeters in Exercises 41–44.

41. The perimeter of a singles court 210 ft

42. The perimeter of a doubles court 228 ft

43. The perimeter of the rectangle formed by the servicelines and the singles sidelines 138 ft

44. The perimeter of the rectangle formed by a service line,a baseline, and the singles sidelines 90 ft

Center service line

Singles sideline

Singles sideline

Doubles sideline

78 ft

18 ft 42 ft

Doubles sideline

27 ftSingles

36 ftDoubles

Serv

ice

line

Bas

elin

e

Serv

ice

line

45. Find the perimeter of a rectangular 8-ft by 10-ft bedroom. 36 ft

46. Find the perimeter of a rectangular 3-ft by 4-ft doghouse. 14 ft

47. Find the perimeter of a checkerboard that is 14 in. oneach side. 56 in.

48. Find the perimeter of a square skylight that is 2 m oneach side. 8 m

49. Find the perimeter of a square frame that is 65 cm oneach side. 260 cm

50. Find the perimeter of a square garden that is 12 yd oneach side. 48 yd

51. Find the perimeter of a 12-ft by 20-ft rectangular deck.

64 ft

52. Find the perimeter of a 40-ft by 35-ft rectangularbackyard. 150 ft

53. Explain in your own words what it means for twoalgebraic expressions to be equivalent.

54. Can the formula for the perimeter of a rectangle beused to find the perimeter of a square? Why or why not?DWDW

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SKILL MAINTENANCE

SYNTHESIS

55. A box of Shaw’s Corn Flakes contains 510 grams (g) ofcorn flakes. A serving of corn flakes weighs 30 g. Howmany servings are in one box? [1.8a] 17 servings

56. Estimate the difference by rounding to the nearest ten.[1.4b] 210

� 4 8 67 0 4

Simplify. [1.9c]

57. 29 58. 7 59. 812 3 � 2�9 � 7�4 � 325 � 3 � 23

60. 27 61. 16 62. 2630 � 42 8 � 215 � 3 � 2 � 727 3�2 � 1�

Solve. [1.7b]

63. 16 64. 1319 � x � 625 � t � 9

65. 15 66. 2550 � 2t45 � 3x

67. Does doubling the length of a square’s side doublethe perimeter of the original square? Why or why not?

68. Why was it necessary to introduce the distributivelaw before discussing how to combine like terms?DWDW

Simplify. (Multiply and then combine like terms.)

69. 70. 71. �29 � 3a2�3 � 4a� � 5�a � 7�10a � 73�a � 7� � 7�a � 4�7x � 15�x � 3� � 2�x � 7�

72. 73. 74.

17 � 39x � 15y

3�4 � 2x� � 5�9x � 3y � 1�

�10 � x � 27y

�5�2 � 3x � 4y� � 7�2x � y��10 � 32x7�2 � 5x� � 3�x � 8�

75. In order to save energy, Andrea plans to run a bead ofcaulk sealant around 3 exterior doors and 13 windows.Each window measures 3 ft by 4 ft, each door measures3 ft by 7 ft, and there is no need to caulk the bottom ofeach door. If each cartridge of caulk seals 56 ft and costs$5.95, how much will it cost Andrea to seal the windowsand doors? $29.75

76. Eric is attaching lace trim to small tablecloths that are 5 ft by 5 ft, and to large tablecloths that are 7 ft by 7 ft. If the lace costs $1.95 per yard, how much will the trimcost for 6 small tablecloths and 6 large tablecloths?

$187.20

77. A square wooden rack is used to store the 15 numbered balls as well as the cue ball in pool. If a pool ball has a diameter of 57 mm, find the inside perimeter of the storage rack. 912 mm

78. A rectangular box is used to store six Christmasornaments. Find the perimeter of such a box if eachornament has a diameter of 72 mm. 720 mm

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Classify each pair as equivalentexpressions or equivalent equations.

1. ;

2. ;

Answers on page A-6

7a � 9a � 9 � 6a

a � 2a � 5 � �3

145


In Section 1.7, we learned to solve certain equations by writing a “relatedequation.” We now extend this approach to include negative integers, as wellas equations that involve both addition and multiplication.

The Addition Principle

In Section 1.7, we learned to solve an equation like by writingthe related subtraction, , or . Note that is an equa-tion, not a solution. Of course, the solution of the equation is obvi-ously 15. The solution of is also 15. Because their solutions areidentical, and are said to be equivalent equations.

EQUIVALENT EQUATIONS

Equations with the same solutions are called equivalent equations.

It is important to be able to distinguish between equivalent expressionsand equivalent equations.

• 6a and are equivalent expressions because, for any replacementof a, both expressions represent the same number.

• and are equivalent equations because any solution ofone equation is also a solution of the other equation.

EXAMPLE 1 Classify each pair as either equivalent equations or equivalentexpressions:

a) ; b) ; .

a) First note that these are expressions, not equations. To see if they areequivalent, we combine like terms in the second expression:

Regrouping and using the distributive law

.

We see that and are equivalent expressions.

b) First note that both and are equations. The solution ofis �7. We substitute to see if �7 is also the solution of

TRUE

Since and have the same solution, they are equivalentequations.


There are principles that enable us to begin with one equation and createan equivalent equation similar to , for which the solution is obvious.One such principle, the addition principle, is stated on the next page.

Suppose that a and b stand for the same number and some number c isadded to a. We get the same result if we add c to b, because a and b are equal.

x � 15

x � 2 � �5x � �7

�7 � 2 � �5

x � 2 � �5

x � 2 � �5:x � �7x � 2 � �5x � �7

5x � 12x � 4 � 3x � 5

� 5x � 1

2x � 4 � 3x � 5 � �2 � 3�x � ��4 � 5�

x � 2 � �5x � �72x � 4 � 3x � 55x � 1

4x � 203x � 15

4a � 2a

x � 12 � 27x � 15x � 12 � 27

x � 15x � 15x � 15x � 27 � 12

x � 12 � 27

2.82.8 SOLVING EQUATIONSObjectivesUse the addition principle tosolve equations.

Use the division principle tosolve equations.

Decide which principleshould be used to solve an equation.

Solve equations that requireuse of both the additionprinciple and the divisionprinciple.

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THE ADDITION PRINCIPLE

For any numbers a, b, and c,

is equivalent to .

EXAMPLE 2 Solve: .

We have

Using the addition principle: adding 7 to both sides

Adding 7 “undoes” the subtraction of 7.

.

The solution appears to be 5. To check, we use the original equation.

Check:

TRUE

The solution is 5.


Recall from Section 2.3 that we can subtract by adding the opposite ofthe number being subtracted. Because of this, the addition principle allowsus to subtract the same number from both sides of an equation.

EXAMPLE 3 Solve: .

We have

Using the addition principle to add �7 or to subtract 7 on both sides

Subtracting 7 “undoes” the addition of 7.

.

The solution is 16. The check is left to the student.

To visualize the addition principle, think of a jeweler’s balance. Whenboth sides of the balance hold equal amounts of weight, the balance islevel. If weight is added or removed, equally, on both sides, the balance remains level.


16 � t

16 � t � 0

23 � 7 � t � 7 � 7

23 � t � 7

23 � t � 7

�2 5 � 7 ? �2

x � 7 � �2

x � 5

x � 0 � 5

x � 7 � 7 � �2 � 7

x � 7 � �2

x � 7 � �2

a � c � b � ca � b

Solve.

3.

4.

Solve.

5.

6.

Answers on page A-6

a � 8 � �6

42 � x � 17

x � 9 � �12

x � 5 � 19

Study Tips

USE A PENCIL

It is no coincidence that thestudents who experience thegreatest success in this coursework in pencil. We all makemistakes and by using penciland eraser we are more willing to admit to ourselves thatsomething needs to berewritten. Please work with apencil and eraser if you aren’tdoing so already.

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The Division Principle

In Section 1.7, we solved by dividing both sides by 8:

Dividing both sides by 8

. 8 times n, divided by 8, is n. is 12.

You can check that and are equivalent. We can divide bothsides of an equation by any nonzero number in order to find an equivalentequation.

THE DIVISION PRINCIPLE

For any numbers a, b, and c ,

is equivalent to .

In Chapter 3, after we have discussed multiplication of fractions, we willuse an equivalent form of this principle: the multiplication principle.

EXAMPLE 4 Solve: .

We have

.

Check:

TRUE

The solution is 7.


EXAMPLE 5 Solve: .

It is important to distinguish between an opposite, as we have in �8n,and subtraction, as we had in (margin exercise 3). To undo multi-plication by �8, we use the division principle:

Dividing both sides by �8

.

Check:

TRUE

The solution is �6.

48 48 ? �8��6�

48 � �8n

�6 � n

48�8

��8n�8

48 � �8n

x � 5 � 19

48 � �8n

63 9 � 7 ? 63

9x � 63

x � 7

Using the division principle todivide both sides by 9

9x9

�639

9x � 63

9x � 63

ac

�bc

a � b

�c � 0�

n � 128 � n � 96

96 8 n � 12

8 � n

8�

968

8 � n � 96

8n � 96

Solve.

7.

8.

Solve.

9.

10.

Answers on page A-6

�6x � 72

63 � �7n

�24 � 3t

7x � 42

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Be sure that you understand why the addition principle is used in Example 2and the division principle is used in Example 5.

Do Exercises 9 and 10 on the previous page.

Equations like or often give students difficulty. Oneway to handle problems of this sort is to multiply both sides of the equa-tion by �1.

EXAMPLE 6 Solve: .

To solve an equation like , remember that when an expression ismultiplied or divided by �1, its sign is changed. Here we multiply on bothsides by �1 to change the sign of �x :

Multiplying both sides by �1

Note that is the same as .

Check:

TRUE


Another way to solve Example 6 is to note that Then we candivide both sides by �1:


Selecting the Correct Approach

It is important for you to be able to determine which principle should be usedto solve a particular equation.

EXAMPLES Solve.

7.

Note that �3 is added to t. To undo addition of �3, we subtract �3 or sim-ply add 3 on both sides:

Using the addition principle

.

Check:

TRUE

The solution is 42.

39 39 ? �3 � 42

39 � �3 � t

42 � t

42 � 0 � t

3 � 39 � 3 � ��3� � t

39 � �3 � t

x � �7.

�1 � x

�1�

7�1

�1 � x � 7

�x � 7

�x � �1 � x.

7 ��7� ? 7

�x � 7

��1� ��1�x��1� ��x� x � �7.

��1� ��x� � ��1� � 7

�x � 7

�x � 7

�x � 7

�t � �3�x � 7

Solve.

11.

12.

Answers on page A-6

�t � �3

�x � 23

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8.

To undo multiplication by �3, we divide by �3 on both sides:

Using the division principle

.

Check:

TRUE



Using the Principles Together

Suppose we want to determine whether 7 is the solution of . Tocheck, we replace x with 7 and simplify.

Check:

TRUE

This shows that 7 is the solution.


In the check above, note that the rules for order of operations require thatwe multiply before we subtract (or add).

The rules for order of operations dictate that unless grouping symbols in-dicate otherwise, multiplication and division are performed before any addi-tion or subtraction. Thus, to evaluate ,

we select a value: x

then multiply by 5: 5x

and then subtract 8: .

In Example 9, which follows, these steps are reversed to solve for x:

We will add 8: ,

then divide by 5:

and isolate x: x.

In general, the last step performed when calculating is the first step to be re-versed when finding a solution.

5x5

5x � 8 � 8

5x � 8

5x � 8

27 35 � 8

5 � 7 � 8 ? 27

5x � 8 � 27

5x � 8 � 27

39 39 ? �3��13�

39 � �3t

�13 � t

39�3

��3t�3

39 � �3t

39 � �3t Solve.

13.

14.

15.

16. Determine whether �9 is thesolution of .

17. Determine whether �6 is thesolution of .

Answers on page A-6

4x � 3 � �25

7x � 8 � �55

x � 7 � �28

�2 � x � �52

�2x � �52

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EXAMPLE 9 Solve: .

We first note that the term containing x is 5x. To isolate 5x, we add 8 onboth sides:

Using the addition principle

Try to do this step mentally.

.

Next, we isolate x by dividing by 5 on both sides:

Using the division principle


.

The check was performed on the previous page. The solution is 7.

Do Exercise 18.

EXAMPLE 10 Solve: .

We first isolate �9x by subtracting 2 on both sides:

Subtracting 2 (or adding �2) from both sides


Now that we have isolated �9x on one side of the equation, we can divide by�9 to isolate x:

Dividing both sides by �9

Simplifying

Check:

TRUE


Do Exercise 19.

38 36 � 2 38 ? �9 � ��4� � 2

38 � �9x � 2

�4 � x.

36�9

��9x�9

36 � �9x

36 � �9x.

36 � �9x � 0

38 � 2 � �9x � 2 � 2

38 � �9x � 2

38 � �9x � 2

x � 7

1x � 7

5x5

�355

5x � 35

5x � 35

5x � 0 � 35

5x � 8 � 8 � 27 � 8

5x � 8 � 27

5x � 8 � 2718. Solve:

19. Solve:

Answers on page A-6

�3x � 2 � 47

2x � 9 � 43

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Classify each pair as either equivalent expressions or equivalent equations.

151

Exercise Set 2.8



Math TutorCenter

InterActMath

MyMathLabMathXL

1. ;

Equivalent equations

2. ;

Equivalent expressions

3. ;


4a � 3 � 3a7a � 36 � 4x � 54x � 15x � 252x � 10

4. ;


5. ;


6. ;


r � 7 � r2r � 78 � 4r � 54r � 34t � 87t � 14

7. ;


8. ;


9. ;


5 � 3t � 13�t � 2�t � 6 � 9t � 4 � 19x � 3 � 20x � 9 � 8

10. ;


11. ;


12. ;


3x � 28 � x4�x � 7�2x � �24x � 4 � �8x � 2 � �92x � �14

Solve.

13. �3 14. �2 15. �8 16. 12x � 7 � 5x � 4 � �12x � 5 � �7x � 6 � �9

17. 18 18. �12 19. �14 20. 8t � 5 � 13x � 8 � �6x � 9 � �3a � 7 � 25

21. 32 22. �12 23. �17 24. 2317 � n � 6�12 � x � 5�9 � x � 324 � t � 8

25. 17 26. �14 27. 0 28. 0�7 � t � �7�8 � �8 � t3 � 17 � x�5 � a � 12

Solve.

29. �4 30. �5 31. �14 32. 83x � 24�3t � 42�8t � 406x � �24

33. 5 34. �32 35. 0 36. 13�5n � �650 � 6x64 � �2t�7n � �35

37. �11 38. �83 39. �56 40. 0�2x � 0�x � 56�x � 8355 � �5t

41. 12 42. 475 43. 390 44. 7n��6� � �42�x � �390�x � �475n��4� � �48

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Solve.

45. 4 46. �15 47. �9 48. �24x � 9 � �156x � �543t � �45t � 6 � �2

49. �15 50. �9 51. �26 52. 42�42 � �x�21 � x � 5�13 � x � 415 � �x

Solve.

61. 7 62. 4 63. 3 64. 73t � 5 � 264t � 2 � 147x � 3 � 255x � 1 � 34

65. �3 66. �5 67. �7 68. �103x � 5 � �352x � 9 � �238a � 3 � �376a � 1 � �17

69. �8 70. 5 71. 8 72. 6�7x � 4 � �46�8t � 3 � �67�4t � 3 � �17�2x � 1 � 17

73. 24 74. �14 75. 6 76. 49 � 4x � 77 � 2x � 5�x � 6 � 8�x � 9 � �15

77. 5 78. �7 79. �8 80. �512 � 7 � x13 � 5 � x33 � 5 � 4x13 � 3 � 2x

81. To solve , Eva decides to add 5 to bothsides of the equation. Is there anything wrong with her doing this? Why or why not?

82. Gary decides to solve by adding 5 toboth sides of the equation. Is there anything wrongwith his doing this? Why or why not?

x � 9 � �5DW�5x � 13DW

53. �5 54. �25 55. �4 56. �44�34 � x � 10�17x � 687 � t � �1835 � �7t

57. 58. 4 59. �50 60. 15�135 � �9t�27 � x � 23�48 � t��12��17818 � t � �160

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153

Exercise Set 2.8

SKILL MAINTENANCE

SYNTHESIS

83. A(n) is a closed geometric figure. [2.7b]

84. Terms are if they have the same variablefactor(s). [2.7a]

85. Numbers we multiply together are called .[1.5a]

86. Equations are if they have the samesolutions. [2.8a]

87. The result of an addition is a(n) . [1.2a]

88. A(n) is a letter that can stand for variousnumbers. [2.6a]

89. The of a number is its distance from zeroon a number line. [2.1c]

90. We for a variable when we replace it witha number. [2.6a]

substitute

absolute value

variable

sum

equivalent

factors

similar

polygon absolute value

opposite

constant

variable

factors

addends

evaluate

substitute

polygon

perimeter

equivalent

similar

sum

product

i VOCABULARY REINFORCEMENT

In each of Exercises 83–90, fill in the blank with the correct term from the given list. Some of the choices may notbe used and some may be used more than once.

91. Explain how equivalent expressions can be usedto write equivalent equations.

92. To solve , Wilma divides both sides by 2. Can this first step lead to a solution? Why or why not?

2x � 8 � 24DWDW

Solve.

93. 8 94. 19 95. 2917 � 32 � 4 � t � 529 � x � 5 � 232x � 7x � �40

96. 3 97. �20 98. �378��42�3 � 142t��7�2 � 5 � t � 43��9�2 � 23t � �3 � 6 � 1�t

99. 1027 100. 45 101. �343353 � �125t232 � x � 222x � �19�3 � �183

102. �7 103. 17529 � 143x � �1902248 � 24 � 32x

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IMPORTANT PROPERTIES AND FORMULAS

For any integers a, b, and c :

Perimeter of a Rectangle : or

Perimeter of a Square :

The Addition Principle : is equivalent to .

The Division Principle : For is equivalent to .ac

�bc

a � bc � 0,

a � c � b � ca � b

P � 4s

P � 2�l � w�P � 2l � 2w,

a�b � c� � ab � aca � 0 � 0;a � b � a � ��b�;a � ��a� � 0;

Review Exercises

1. Tell which integers correspond to this situation: [2.1a]

Bonnie has $527 in her campus account and Roger is$53 in debt.

527; �53

Use either � or � for to form a true statement. [2.1b]

2. 0 �5

�

3. �7 6

�

4. �4 �19

�

The review that follows is meant to prepare you for a chapter exam. It consists of three parts. The first part, ConceptReinforcement, is designed to increase understanding of the concepts through true/false exercises. The second part isa list of important properties and formulas. The third part is the Review Exercises. These provide practice exercises forthe exam, together with references to section objectives so you can go back and review. Before beginning, stop andlook back over the skills you have obtained. What skills in mathematics do you have now that you did not have beforestudying this chapter?

154


Summary and Review22

i CONCEPT REINFORCEMENT

Determine whether the statement is true or false. Answers are given at the back of the book.

1. The absolute value of a number is always nonnegative. True

2. The opposite of the opposite of a number is the original number. True

3. The product of an even number of negative numbers is positive. True

4. The expression is equivalent to the expression False

5. and are equivalent equations. False

6. Collecting like terms is based on the distributive law. True

5x � �33 � x � 4x

2 � x � 3.2�x � 3�

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155

Summary and Review: Chapter 2

Find the absolute value. [2.1c]

5. 39 6. 23 7. 0023�39

8. Find when . [2.1d] 72x � �72�x

9. Find when [2.1d] 59x � 59.��x�

Compute and simplify. [2.2a], [2.3a], [2.4a, b], [2.5a, b]

10. �9 11. �11�5 � ��6��14 � 5

12. 6 13. �240 � ��24�14 � ��8�

14. �12 15. 239 � ��14�17 � 29

16. �1 17. 7�3 � ��10��8 � ��7�

18. �4 19. 128 � ��9� � 7 � 2�3 � 7 � ��8�

20. 92 21. �847��12��23 � ��4�

22. �40 23. �315 ��5�2��4� ��5� ��1�

24. �5 25. 007

�5511

26. �25 27. �20��3� 4 � 32 � 57 12 � ��3� � 4

28. Evaluate for and [2.6a] 7b � �5.a � 43a � b

29. Evaluate and for and

[2.6a] �6, �6, �6

y � 5.x � 30�xy

x�y

,�xy

,

Use the distributive law to write an equivalent expression.[2.6b]

30. 31.

6a � 12b � 15

3�2a � 4b � 5�

20x � 36

4�5x � 9�

Combine like terms. [2.7a]

32. 17a 33. 6x�7x � 13x5a � 12a

34. �3m � 69m � 14 � 12m � 8

35. Find the perimeter of a rectangular frame that is 8 in. by10 in. [2.7b] 36 in.

36. Find the perimeter of a square pane of glass that is25 cm on each side. [2.7b] 100 cm

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Solve. [2.8a, b, c, d]

37. �8x � 9 � �17

38. �9�4t � 36

39. �1313 � �x

40. 1156 � 6x � 10

41. 15�x � 3 � �12

42. �718 � 4 � 2x

43. Explain the difference between equivalentexpressions and equivalent equations. [2.8a]

Equivalent expressions are expressions that have the samevalue when evaluated for various replacements of thevariable(s). Equivalent equations are equations that havethe same solution(s).

DW

44. Is a number’s absolute value ever less than thenumber itself? Why or why not? [2.1c]A number’s absolute value is the number itself if thenumber is nonnegative, and the opposite of the number ifthe number is negative. In neither case is the result lessthan the number itself, so “no,”a number’s absolute value isnever less than the number itself.

DW

45. A classmate insists on reading as “negative x.”When asked why, the response is “because isnegative.” What mistake is this student making? [2.1d]

The notation “�x” means “the opposite of x.” If x is anegative number, then �x is a positive number. Forexample, if then �x � 2.x � �2,

�x�xDW

46. Are and equivalent for all choicesof a and b? Why or why not? Experiment with differentreplacements for a and b. [2.6a]

The expressions and are equivalent for all choices of a and b because and areopposites. When opposites are raised to an even power, the results are the same.

b � aa � b�b � a�2�a � b�2

�b � a�2�a � b�2DW

Simplify. [2.5b]

47. 662,58287 3 � 293 � ��6�6 � 1957

48. �88,1741969 � ��8�5 � 17 � 153

49. �240113 � 173

15 � 83 � 507

50. For what values of x will be negative? [2.6a]

x � �2

8 � x3

51. For what values of x is [2.1b, c] x � 0x � x?

SYNTHESIS

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1. Tell which integers correspond to this situation: The Tee Shop sold 542 fewer muscle shirts than expected inJanuary and 307 more than expected in February.

2. Use either � or � for to form a true statement.

�14 �21

157

Test: Chapter 2

Chapter Test22 Work It Out!Chapter Test Video

on CD

For Extra Help

3. Find the absolute value: . 4. Find when �x � �19.��x��739

Compute and simplify.

5. 6. 7. �8 � 17�9 � ��12�6 � ��17�

8. 9. 10. �5 � 197 � 220 � 12

11. 12. 13. ��4�331 � ��3� � 5 � 9�8 � ��27�

14. 15. 16. �72 ��9��9 � 027��10�

17. 18. 19. 29 � �3 � 5�28 2 � 2 � 32�567

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20. Antarctica Highs and Lows. The continent ofAntarctica, which lies in the southern hemisphere,experiences winter in July. The average hightemperature is �67F and the average low temperatureis �81F. How much higher is the average high than theaverage low?

Source: National Climatic Data Center

Antarctica

Africa

Australia

SouthAmerica

Pacific

O

cean

IndianO

cean

Atlantic Ocean

21. Jeannie rewound a tape in her video camera from the8 minute mark to the �15 minute mark. How manyminutes of tape were rewound?

22. Evaluate for and b � 10.a � �8a � b

6

23. Use the distributive law to write an equivalentexpression.

24. Combine like terms. 9x � 14 � 5x � 3

7�2x � 3y � 1�

Solve.

25. 26. �a � 9 � �3�7x � �35

SYNTHESIS

27. Monty plans to attach trim around the doorway and along the base of the walls in a 12-ft by 14-ft room. If the doorway is3 ft by 7 ft, how many feet of trim are needed? (Only three sides of a doorway get trim.)

Simplify.

28. 29. 15x � 3�2x � 7� � 9�4 � 5x�9 � 5�x � 2�3 � 4x� � 14

30. 31. 3487 � 16 4 � 4 28 � 14449 � 143 74 � 19262 62

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1. Write standard notation for the number written inwords in the following sentence: In 2003 there wereabout one hundred eighty-one million, five hundredninety-nine thousand, nine hundred telephone lines inuse in the United States. [1.1c] 181,599,900

2. Write a word name for 5,380,001,437.

[1.1c] Five billion, three hundred eighty million, onethousand, four hundred thirty-seven

159

Cumulative Review: Chapters 1–2

Cumulative ReviewCHAPTERS

1–21–2

Add.

3.

[1.2b] 18,827

4.

[1.2b] 8857

� 7 9 7 8 9

7 9 8 9 � 2 , 9 3 5

1 5 , 8 9 2

Subtract.

5.

[1.3d] 7846

6.

[1.3d] 2428

� 5 7 8 3 0 0 6

� 4 3 0 8 2 7 6

Multiply.

7.

[1.5a] 16,767

8.

[1.5a] 8,266,500

9.

[2.4a] �344

10.

[2.4a] 72

�12��6�43 � ��8� � 3 3 0 0

2 5 0 5 � 2 7

6 2 1

Divide.

11.

[1.6c] 104

12.

[1.6c] 62

13.

[2.5a] 0

14.

[2.5a] �5

60 ��12�0 ��67�6 2 � 3 8 4 46 3 � 6 5 5 2

15. Round 427,931 to the nearest thousand.

[1.4a] 428,000

16. Round 5309 to the nearest hundred.

[1.4a] 5300

Estimate each sum or product by rounding to the nearest hundred. Show your work.

17.

[1.4b]

18.

[1.5b] 700 � 500 � 350,000

� 5 3 1 7 4 9

749,600 � 301,400 � 1,051,000

� 3 0 1 , 3 6 2 7 4 9 , 5 5 9

19. Use � or � for to form a true sentence:

�26 19.

[2.1b] �

20. Find the absolute value:

[2.1c] 279

�279.

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Simplify.

21.

[1.9c], [2.5b] 36

22.

[1.9d], [2.5b] 2

�17 � �8 � �5 � 2 � 2�� 3 � 12 6�35 � 25 5 � 2 � 3

23.

[1.9c], [2.5b] �86

24.

[1.9b] 125

5310 1��5� � 62


[2.6a] 3

26. Evaluate for

[2.6a] 28

x � �2.7x2y � 4.x � 11x � y

5

Use the distributive law to write an equivalent expression.

27.

[2.6b]

28.

[2.6b] 18x � 12y � 24

6�3x � 2y � 4�

�2x � 10

�2�x � 5�

Simplify.

29.

[2.2a] �26

30.

[2.4a] 30

31.

[2.3a] �15

32.

[2.5a] �32

64 ��2�23 � 38��3� ��10��12 � ��14�

33.

[2.3a] 13

34.

[2.4a] �30

35.

[2.3a] 16

36.

[2.5b] �57

16 2��8� � 73 � ��8� � 2 � ��3��2� ��3� ��5��12 � ��25�

Solve.

37.

[1.7b], [2.8a] 27

38.

[2.8b] �3

39.

[2.8a] 15

40.

[2.8d] �8

�39 � 4x � 76 � x � �9�12t � 36x � 8 � 35

Solve.

41. In the movie Little Big Man, Dustin Hoffman plays acharacter who ages from 17 to 121. This represents thegreatest age range depicted by one actor in one film.How many years did Hoffman’s character age?Source: Guinness Book of World Records

[1.8a] 104 yr

42. The ten largest hotels in the United States are in LasVegas. Of these, the four largest are the MGM Grand, theLuxor, the Excalibur, and the Circus Circus. These have5034 rooms, 4408 rooms, 4008 rooms, and 3770 rooms,respectively. What is the total number of rooms in thesefour hotels?Source: http://govegas.about.com/cs/hotels/tp/largesthotels.htm

[1.8a] 17,220 rooms

43. Amanda is offered a part-time job paying $4940 a year.How much is each weekly paycheck?

[1.8a] $95

44. Eastside Appliance sells a refrigerator for $600 and $30tax with no delivery charge. Westside Appliance sells thesame model for $560 and $28 tax plus a $25 deliverycharge. Which is the better buy?

[1.8a] Westside Appliance

45. Write an equivalent expression by combining like terms: .

[2.7a] 10x � 14

7x � 9 � 3x � 5

SYNTHESIS

46. A soft-drink distributor has 166 loose cans of cola. Thedistributor wishes to form as many 24-can cases aspossible and then, with any remaining cans, as manysix-packs as possible. How many cases will be filled?How many six-packs? How many loose cans willremain? [1.8a] Cases: 6; six-packs: 3; loose cans: 4

47. Simplify: [2.5b] 4a

48. Simplify: .[2.5b] �1071

49. Find two solutions of [2.8d] �35x � 2 � 13.

37 � 64 42 � 2 � �73 � ��4�5�

a � �3a � �4a � �2a � 4a��.

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Introduction to 2 Integers and Algebraic...

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