1.3 Adding and Subtracting Signed Numbers · Adding and Subtracting Signed Numbers 1.3 ... mum and maximum. ... The median is one-half the sum of the middle two numbers. 2 3 2 1 2

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Adding and Subtracting Signed Numbers

1.3

1.3 OBJECTIVES

1. Find the median of a set of signed numbers2. Find the difference of two signed numbers3. Find the range of a set of signed numbers

In Section 0.4 we introduced the idea of signed numbers. Now we will examine the fourarithmetic operations (addition, subtraction, multiplication, and division) and see howthose operations are performed when signed numbers are involved. We start by consideringaddition.

An application may help. As before, let’s represent a gain of money as a positive num-ber and a loss as a negative number.

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

3 � 4 � 7

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

�3 � (�4) � �7

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

3 � (�4) � �1

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

�3 � 4 � 1

The number line can be used to illustrate the addition of signed numbers. Starting at theorigin, we move to the right for positive numbers and to the left for negative numbers.

71

Example 1

Adding Signed Numbers

(a) Add (�3) � (�4).

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

(�3) � (�4) � �7

(b) Add

0�2 �1�23

�23�

21

��3

2� � ��1

2�.

0�7 �3

�4 �3

72 CHAPTER 1 THE LANGUAGE OF ALGEBRA

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You have probably noticed some helpful patterns in the previous examples. These pat-terns will allow you to do the work mentally without having to use the number line. Lookat the following rule.

As before, we start at the origin. From that point move units left. Then move another unit left to find the sum. In this case

��3

2� � ��1

2� � �2

1

2

3

2

Add.

(a) (�4) � (�5) (b) (�3) � (�7)

(c) (�5) � (�15) (d) ��5

2� � ��3

2�

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

Let’s again use the number line to illustrate the addition of two numbers. This time thenumbers will have different signs.

Example 2

If two numbers have the same sign, add their absolute values. Give the sum thesign of the original numbers.

NOTE This means that the sumof two positive numbers ispositive and the sum of twonegative numbers is negative.We first encountered absolutevalues in Section 0.4.


(a) Add 3 � (�6).

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

3 � (�6) � �3

(b) Add �4 � 7.

�4

�7

4 30�

3

0 3�3

�6

Rules and Properties: Adding Signed Numbers Case 1: Same Sign

ADDING AND SUBTRACTING SIGNED NUMBERS SECTION 1.3 73©

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(a) � �12

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

(b) � �3

Subtract the absolute values . The sum has the sign (�) of the number

with the larger absolute value,

(c) � �3.7

Subtract the absolute values (8.2 � 4.5 � 3.7). The sum has the sign (�) of the numberwith the larger absolute value, �8.2.

�8.2 � 4.5

�13

2.

�13

2�

7

2�

6

2� 3�

�13

2�

7

2

7 � (�19)

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

Add.

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

3�

16

3


Example 3

NOTE Remember, signednumbers can be fractions anddecimals as well as integers.

NOTE Again, we firstencountered absolute values in Section 0.4.

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

Rules and Properties: Adding Signed Numbers Case 2: Different Signs

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

�4 � 7 � 3


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Add.

(a) � 9

(b)

(c) � �25(�25) � 0

� �5

40 � ��

5

4�9 � 0

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

For any number a,

a � 0 � 0 � a � a

Rules and Properties: Additive Identity Property

NOTE No number loses itsidentity after addition with 0.Zero is called the additiveidentity.

Example 4

Add.

(a) 8 � 0 (b) (c) (�36) � 00 � ��8

3�


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

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

Our second property states that the sum of any number and its opposite is 0.

�3

3

0 3

3

NOTE 3 and �3 are opposites.

Add mentally.

(a) 5 � (�14) (b) �7 � (�8) (c) �8 � 15

(d) 7 � (�8) (e) (f) 5.3 � (�2.3)�2

3� ��

7

3�


In Section 1.2 we discussed the commutative, associative, and distributive properties.There are two other properties of addition that we should mention. First, the sum of anynumber and 0 is always that number. In symbols,


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

a � (�a) � (�a) � a � 0

Rules and Properties: Additive Inverse Property

NOTE Here �a represents theopposite of the number a. Thesum of any number and itsopposite, or additive inverse, is 0.

Example 5


(a) � 0

(b) � 0

(c) � 0

(d) � 04

5� ��

4

5�(�2.3) � 2.3

�15 � 15

9 � (�9)

In Section 0.4 we saw that the least and greatest elements of a set were called the mini-mum and maximum. The middle value of an ordered set is called the median. The medianis sometimes used to represent an average of the set of numbers.

Add.

(a) (�17) � 17 (b) 12 � (�12)

(c) (d) (�1.6) � 1.6 1

3� ��

1

3�


Example 6

Finding the Median

Find the median for each set of numbers.

(a) 9, �5, �8, 3, 7

First, rewrite the set in ascending order.

�8, �5, 3, 7, 9

The median is then the element that has just as many numbers to its right as it has to its left.In this set, 3 is the median, because there are two numbers that are larger (7 and 9) and twonumbers that are smaller (�8 and �5).

(b) 3, �2, 18, �20, �13


�20, �13, �2, 3, 18

The median is then the element that is exactly in the middle. The median for this set is �2.


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Add the middle two numbers (�1 and 3), then divide their sum by 2.

The median is 1.

(b) 8, 3, �2, 4, �5, �7

Rewrite the set in ascending order.

�7, �5, �2, 3, 4, 8

The median is one-half the sum of the middle two numbers.

�2 � 3

2�

1

2� 0.5

(�1) � (3)

2�

2

2� 1


(a) 2, �5, 15, 8, 3, �4

(b) 8, 3, 6, �8, 9, �7


In the previous example, each set had an odd number of elements. If we had an evennumber of elements, there would be no single middle number.

To find the median from a set with an even number of elements, add the two middlenumbers and divide their sum by 2.

Example 7

Finding the Median


(a) �3, 3, �8, 4, �1, �7, 5, 9


�8, �7, �3, �1, 3, 4, 5, 9


(a) �3, 2, 7, �6, �1 (b) 5, 1, �10, 2, �20


To begin our discussion of subtraction when signed numbers are involved, we can lookback at a problem using natural numbers. Of course, we know that

8 � 5 � 3 (1)


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1. Rewrite the subtraction problem as an addition problem by

a. Changing the minus sign to a plus sign.b. Replacing the number being subtracted with its opposite.

2. Add the resulting signed numbers as before.In symbols,

a � b � a � (�b)

Rules and Properties: Subtracting Signed Numbers

Example 8 illustrates the use of this definition while subtracting.

NOTE This is the definition ofsubtraction.

Subtracting Signed Numbers

Subtraction Addition

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

(a) � 15 � (�7)

Replace 7 with its opposite, �7.

� 8

(b) � 9 � (�12) � �3

(c) � �6 � (�7) � �13

(d)

(e) � 2.1 � (�3.4) � �1.3

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

�2 � 5 � �2 � (�5) � �7

2.1 � 3.4

� �3

5� ��

7

5� � �10

5� �2�

3

5�

7

5

�6 � 7

9 � 12

15 � 7

Example 8

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

8 � (�5) � 3 (2)

Comparing equations (1) and (2), we see that the results are the same. This leads us toan important pattern. Any subtraction problem can be written as a problem in addition.Subtracting 5 is the same as adding the opposite of 5, or �5. We can write this fact asfollows:

8 � 5 � 8 � (�5) � 3

This leads us to the following rule for subtracting signed numbers.


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Given a set of numbers, the range is the difference between the maximum and theminimum.

Subtract.

(a) 8 � (�2) (b) 3 � (�10) (c) �7 � (�2)

(d) �9.8 � (�5.8) (e) 7 � (�7)


Example 9


Subtraction Addition

Change the subtractionto an addition.

(a) � 5 � (�2) � 5 � 2 � 7

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

(b) � 7 � (�8) � 7 � 8 � 15

(c) � �9 � 5 � �4

(d) � �12.7 � 3.7 � �9

(e)

(f) Subtract �4 from �5. We write

�5 � (�4) � �5 � 4 � �1

� �3

4� ��

7

4� �4

4� 1�

3

4� ��

7

4��12.7 � (�3.7)

�9 � (�5)

7 � (�8)

5 � (�2)

The subtraction rule is used in the same way when the number being subtracted is neg-ative. Change the subtraction to addition. Replace the negative number being subtractedwith its opposite, which is positive. Example 9 will illustrate this principle.

Subtract.

(a) 18 � 7 (b) 5 � 13 (c) �7 � 9

(d) (e) �2 � 7 (f) 5.6 � 7.8�5

6�

7

6



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Example 10

Find the range for each set of numbers.

(a) 2, �4, 7, �3, �1 (b) �3, 4, �7, 5, 9, �4

C H E C K Y O U R S E L F 1 0

Example 11

NOTE Some graphingcalculators have a negative sign

that acts to change thesign of a number.(�)

NOTE If you have a graphingcalculator, the key sequencewill be

12.43 3.516 ENTER�(�)

Your scientific calculator can be used to do arithmetic with signed numbers. Before welook at an example, there are some keys on your calculator with which you should becomefamiliar.

There are two similar keys you must find on the calculator. The first is used for subtrac-tion ( ) and is usually found in the right column of calculator keys. The second will“change the sign” of a number. It is usually a and is found on the bottom row.

We will use these keys in our next example.��

�

Finding the Range

Find the range for each set of numbers.

(a) 5, �2, �7, 9, 3

Rewrite the set in ascending order. The maximum is 9, the minimum is �7. The range is thedifference.

9 � (�7) � 9 � 7 � 16

The range is 16.

(b) 3, 8, �17, 12, �2

Rewrite the set in ascending order. The maximum is 12. The minimum is �17. The range is12 � (�17) � 29.


Using your calculator, find the difference.

(a) �12.43 � 3.516

Enter the 12.43 and push the to make it negative. Then push 3.516 . The resultshould be �15.946.

(b) 23.56 � (�4.7)

��

The key sequence is

23.56 4.7

The answer should be 28.26.

��


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Use your calculator to find the difference.

(a) �13.46 � 5.71 (b) �3.575 � (�6.825)

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

1. (a) �9; (b) �10; (c) �20; (d) �4 2. (a) 2; (b) �4; (c) 5; (d) �4

3. (a) �9; (b) �15; (c) 7; (d) �1; (e) �3; (f ) 3 4. (a) 8; (b) ; (c) �36

5. (a) 0; (b) 0; (c) 0; (d) 0 6. (a) �1; (b) 1 7. (a) 2.5; (b) 4.5

8. (a) 11; (b) �8; (c) �16; (d) �2; (e) �9; (f ) �2.2

9. (a) 10; (b) 13; (c) �5; (d) �4; (e) 14 10. (a) 7 � (�4) � 11;(b) 9 � (�7) � 16 11. (a) �19.17; (b) 3.25

�8

3

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

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Exercises

Add.

1. 3 � 6 2. 5 � 9

3. 11 � 5 4. 8 � 7

5. 6.

7. 8.

9. (�2) � (�3) 10. (�1) � (�9)

11. 12.

13. 14.

15. (�1.6) � (�2.3) 16. (�3.5) � (�2.6)

17. 9 � (�3) 18. 10 � (�4)

19. 20.

21. 22. ��11

6 � �5

12��4

5� �9

20

2

3� ��

1

6�3

4� ��

1

2�

��4

7� � ��3

14��1

2� � ��3

8�

��3

5� �12

5��3

5� � ��7

5�

2

3�

5

9

1

2�

4

5

7

3�

8

3

3

4�

5

4

1.3

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

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23. �11.4 � 13.4 24. �5.2 � 9.2

25. �3.6 � 7.6 26. �2.6 � 4.9

27. �9 � 0 28. �15 � 0

29. 7 � (�7) 30. 12 � (�12)

31. �4.5 � 4.5 32.

33. 7 � (�9) � (�5) � 6 34. (�4) � 6 � (�3) � 0

35. 7 � (�3) � 5 � (�11) 36.

37. 38.

39. 2.3 � (�5.4) � (�2.9) 40. �5.4 � (�2.1) � (�3.5)

Subtract.

41. 21 � 13 42. 36 � 22

43. 82 � 45 44. 103 � 56

45. 46.

47. 7.9 � 5.4 48. 11.7 � 4.5

49. 8 � 10 50. 14 � 19

17

8�

9

8

15

7�

8

7

1

3� ��

5

6� � ��1

2��3

2� ��

7

4� �1

4

�6

5� ��

13

5 � �4

5

��2

3� �2

3

ANSWERS

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

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51. 24 � 45 52. 136 � 352

53. 54.

55. 7.8 � 11.6 56. 14.3 � 25.5

57. �5 � 3 58. �15 � 8

59. �9 � 14 60. �8 � 12

61. 62.

63. �3.4 � 4.7 64. �8.1 � 7.6

65. 5 � (�11) 66. 7 � (�5)

67. 7 � (�12) 68. 3 � (�10)

69. 70.

71. 72.

73. 8.3 � (�5.7) 74. 6.5 � (�4.3)

75. 8.9 � (�11.7) 76. 14.5 � (�24.6)

77. �36 � (�24) 78. �28 � (�11)

11

16� ��

7

8�6

7� ��

5

14�

5

6� ��

7

6�3

4� ��

3

2�

�5

9�

7

18�

2

5�

7

10

5

9�

32

9

7

6�

19

6

ANSWERS

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

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79. �19 � (�27) 80. �11 � (�16)

81. 82.

83. �12.7 � (�5.7) 84. �5.6 � (�2.6)

85. �6.9 � (�10.1) 86. �3.4 � (�7.6)

Use your calculator to evaluate each expression.

87. �4.1967 � 5.2943 88. 5.3297 � (�4.1897)

89. �4.1623 � (�3.1468) 90. (�3.6829) � 4.5687

91. �6.3267 � 8.6789 92. �6.6712 � 5.3245

Find the median for each of the following sets.

93. 1, 3, 5, 7, 9 94. 2, 4, 6, 8, 10

95. 8, 7, 2, 25, 5, 13, 3 96. 53, 23, 34, 21, 32, 30, 32

Determine the range for each of the following sets.

97. 2, 7, 9, 15, 24 98. 4, 8, 11, 15, 27

99. �4, �3, 2, 7, 9 100. �7, �2, 1, 8, 11

101. 102.

103. 3, 2, �5, 6, �3 104. 1, �9, 7, �2, 3

Solve the following problems.

105. Checking account. Amir has $100 in his checking account. He writes a check for$23 and makes a deposit of $51. What is his new balance?

106. Checking account. Olga has $250 in her checking account. She deposits $52 andthen writes a check for $77. What is her new balance?

Bal: Dep:CK # 1111:

3, 5

6, �7, �

1

3,

2

3

7

8, 2, �

1

2, �8,

3

4

�1

2� ��

5

8��3

4� � ��11

4 �

ANSWERS

79.

80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104.

105.

106.

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107. Football yardage. On four consecutive running plays, Ricky Watters of the SeattleSeahawks gained 23 yards, lost 5 yards, gained 15 yards, and lost 10 yards. Whatwas his net yardage change for the series of plays?

108. VISA balance. Ramon owes $780 on his VISA account. He returns three itemscosting $43.10, $36.80, and $125.00 and receives credit on his account. Next, hemakes a payment of $400. He then makes a purchase of $82.75. How much doesTom still owe?

109. Temperature. The temperature at noon on a June day was 82. It fell by 12 in thenext 4 hours. What was the temperature at 4:00 P.M.?

110. Mountain climbing. Chia is standing at a point 6000 feet (ft) above sea level. Shedescends to a point 725 ft lower. What is her distance above sea level?

111. Checking account. Omar’s checking account was overdrawn by $72. He wroteanother check for $23.50. How much was his checking account overdrawn afterwriting the check?

112. Personal finance. Angelo owed his sister $15. He later borrowed another $10.What positive or negative number represents his current financial condition?

113. Education. A local community college had a decrease in enrollment of 750students in the fall of 1999. In the spring of 2000, there was another decrease of425 students. What was the total decrease in enrollment for both semesters?

114. Temperature. At 7 A.M., the temperature was �15F. By 1 P.M., the temperaturehad increased by 18F. What was the temperature at 1 P.M.?

115. Education. Ezra’s scores on five tests taken in a mathematics class were 87, 71, 95,81, and 90. What was the range of his scores?

= ____1 + 5

Name:___________

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____4 x 3

= ____3 + 6

= ____9 + 4

= ____3 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____8 x 3

= ____3 + 6

= ____6 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____9 + 4

= ____4 x 3 = ____

1 + 5 = ____

2 x 5 = ____

4 + 5 = ____

15 - 2 = ____

4 x 3 = ____

3 + 6 = ____

9 + 4 = ____

3 + 9 = ____

1 x 2 = ____

13 - 4 = ____

5 + 6

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____8 x 3

= ____3 + 6

= ____6 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____9 + 4

= ____4 x 3

= ____1 + 5

= ____2 x 5

= ____4 + 5

= ____

15 - 2

= ____4 x 3

= ____3 + 6

= ____9 + 4

= ____3 + 9

= ____1 x 2

= ____

13 - 4

= ____5 + 6

= ____2 x 5

= ____4 + 5

= ____

15 - 2

= ____8 x 3

= ____3 + 6

= ____6 + 9

= ____1 x 2

= ____

13 - 4

= ____5 + 6

= ____9 + 4

= ____4 x 3

= ____1 + 5

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____4 x 3

= ____3 + 6

= ____9 + 4

= ____3 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____8 x 3

= ____3 + 6

= ____6 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____9 + 4

= ____4 x 3

= ____1 + 5

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____4 x 3

= ____3 + 6

= ____9 + 4

= ____3 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____2 x 5

= ____4 + 5

= ____15 - 2

= ____8 x 3

= ____3 + 6

= ____6 + 9

= ____1 x 2

= ____13 - 4

= ____5 + 6

= ____9 + 4

= ____4 x 3

Name:___________

Name:___________

Name:___________

Name:___________

ANSWERS

107.

108.

109.

110.

111.

112.

113.

114.

115.

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116. The bar chart shown represents the total yards gained by the leading passer in theNFL from 1993 to 1997. Use a signed number to represent the change in totalpassing yards from one year to the next.

(a) from 1993 to 1994 (b) from 1994 to 1995

(c) from 1995 to 1996 (d) from 1996 to 1997

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

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

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

(c) Think of other prefixes that negate or change the meaning of a word to itsopposite. Make a list of words formed with these prefixes, and write a sentencewith three of the words you found. Make a sentence with two words andphrases from each of the lists above. Look up the meaning of the wordirregardless.

What is the value of �[�(�5)]? What is the value of �(�6)? Howdoes this relate to the above examples? Write a short description about thisrelationship.

118. The temperature on the plains of North Dakota can change rapidly, falling or risingmany degrees in the course of an hour. Here are some temperature changes duringeach day over a week.

0

1000

2000

3000

4000

5000

1993 1994 1995 1996 1997

Yar

ds

Year

40304453

2575

3328 3281

ANSWERS

116.

117.

118.

86

Day Mon. Tues. Wed. Thurs. Fri. Sat. Sun.

Temp.Changefrom 10 A.M.to 3 P.M. �13 �20 �18 �10 �25 �5 �15

Write a short speech for the TV weather reporter that summarizes the dailytemperature change. Use the median as you characterize the average daily middaychange.

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119. Science. The daily average temperatures in degrees Fahrenheit for a week inFebruary were �1, 3, 5, �2, 4, 12, and 10. What was the range of temperatures forthat week?

120. How long ago was the year 1250 B.C.E.? What year was 3300 years ago? Make anumber line and locate the following events, cultures, and objects on it. How longago was each item in the list? Which two events are the closest to each other? Youmay want to learn more about some of the cultures in the list and the mathematicsand science developed by that culture.

Inca culture in Peru—1400 A.D.The Ahmes Papyrus, a mathematical text from Egypt—1650 B.C.E.Babylonian arithmetic develops the use of a zero symbol—300 B.C.E.First Olympic Games—776 B.C.E.Pythagoras of Greece dies—580 B.C.E.Mayans in Central America independently develop use of zero—500 A.D.The Chou Pei, a mathematics classic from China—1000 B.C.E.The Aryabhatiya, a mathematics work from India—499 A.D.Trigonometry arrives in Europe via the Arabs and India—1464 A.D.Arabs receive algebra from Greek, Hindu, and Babylonian sources and develop it

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

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

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

ANSWERS

119.

120.

121.

122.

a.

b.

c.

d.

e.

f .

87

Add.

(a) (�1) � (�1) � (�1) � (�1)

(b) 3 � 3 � 3 � 3 � 3

(c) 9 � 9 � 9

(d) (�10) � (�10) � (�10)

(e) (�5) � (�5) � (�5) � (�5) � (�5)

(f) (�8) � (�8) � (�8) � (�8)

Getting Ready for Section 1.4 [Section 0.3]

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Answers

1. 9 3. 16 5. 2 7. 9. �5 11. �2 13.

15. �3.9 17. 6 19. 21. 23. 2 25. 4 27. �9

29. 0 31. 0 33. �1 35. �2 37. �3 39. �6 41. 843. 37 45. 1 47. 2.5 49. �2 51. �21 53. �2 55. �3.8

57. �8 59. �23 61. 63. �8.1 65. 16 67. 19

69. 71. 73. 14 75. 20.6 77. �12 79. 8 81. 2

83. �7 85. 3.2 87. �9.491 89. �1.0155 91. 2.352293. 5 95. 7 97. 22 99. 13 101. 10 103. 11 105. $128107. 23 yards 109. 70 111. $95.50 113. 1175 115. 24

117. 119. 14F 121. a. �4 b. 15 c. 27

d. �30 e. �25 f. �32

17

14

9

4

�11

10

�7

20

1

4

�7

8

13

10

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1.3 Adding and Subtracting Signed Numbers · Adding and Subtracting Signed Numbers 1.3 ... mum and maximum. ... The median is one-half the sum of the middle two numbers. 2 3 2 1 2