WRITING

97.Is the sum of a positive and a negative number always positive? Explain why or why not.

98.How do you explain the fact that when asked to add4 and 8, we must actually subtract to obtain the result?

99.Explain why the sum of two negative numbers is a negative number.

100.Write an application problem that will require adding 50 and 60.

101.If the sum of two integers is 0, what can be said about the integers? Give an example.

102.Explain why the expression 6 5 is not written correctly. How should it be written?

REVIEW

103.a. Find the perimeter of the rectangle shown below.

b.Find the area of the rectangle shown below.

5 ft

3 ft

104.What property is illustrated by the statement 5 15 15 5?

105.Prime factor 250. Use exponents to express the result.

106.Divide: 14412

1Use the subtraction rule.

2Evaluate expressions involving subtraction and addition.

3Solve application problems by subtracting integers.

Subtracting Integers

In this section, we will discuss a rule that is helpful when subtracting signed numbers.

1 Use the subtraction rule.

The subtraction problem 6 4 can be thought of as taking away 4 from 6. We can use a number line to illustrate this. Beginning at 0, we draw an arrow of length 6 units long that points to the right. It represents positive 6. From the tip of that arrow, we draw a second arrow, 4 units long, that points to the left. It represents taking away 4. Since we end up at 2, it follows that 6 4 2.

EXAMPLE 1 Subtract and check the result: a. 6 4 b. 3 ( 5) c. 7 23

Strategy To find each difference, we will apply the rule for subtraction: Add the first integer to the opposite of the integer to be subtracted.

WHY It is easy to make an error when subtracting signed numbers. We will probably be more accurate if we write each subtraction as addition of the opposite.

Solution

a.We read 6 4 as “negative six minus four.” Thus, the number to be subtracted is 4. Subtracting 4 is the same as adding its opposite, 4.

Change the subtraction to addition.

6 4 6 ( 4) 10 Use the rule for adding two

integers with the same sign.

Change the number being subtracted to its opposite.

To check, we add the difference, 10, and the subtrahend, 4. We should get the minuend, 6.

Check: 10 4 6 The result checks.

Caution! Don’t forget to write the opposite of the number to be subtracted within parentheses if it is negative.

6 4 6 ( 4)

b.We read 3 ( 5) as “three minus negative five.” Thus, the number to be subtracted is 5. Subtracting 5 is the same as adding its opposite, 5.

Add . . .

3 ( 5) 3 5 8

. . . the opposite

Check: 8 ( 5) 3 The result checks.

Self Check 1

Subtract and check the result:

a.2 3

b.4 ( 8)

c.6 85

Now Try Problems 21, 25, and 29

Self Check 2

a.Subtract 10 from 7.

b.Subtract 7 from 10.

Now Try Problem 33

c.We read 7 23 as “seven minus twenty-three.” Thus, the number to be subtracted is 23. Subtracting 23 is the same as adding its opposite, 23.

Add . . .

Caution! When applying the subtraction rule, do not change the first number.

Strategy We will translate each phrase to mathematical symbols and then perform the subtraction. We must be careful when translating the instruction to subtract one number from another number.

WHY The order of the numbers in each word phrase must be reversed when we translate it to mathematical symbols.

Solution

a.Since 12 is the number to be subtracted, we reverse the order in which 12 and 8 appear in the sentence when translating to symbols.

Subtract 12 from 8

8 ( 12) Write 12 within parentheses.

To find this difference, we write the subtraction as addition of the opposite:

Add . . .

8 ( 12) 8 12 4 Use the rule for adding two

integers with different signs.

. . . the opposite

b.Since 8 is the number to be subtracted, we reverse the order in which 8 and12 appear in the sentence when translating to symbols.

Subtract 8 from 12

12 ( 8) Write 8 within parentheses.

To find this difference, we write the subtraction as addition of the opposite:

Add . . .

12 ( 8) 12 8 4 Use the rule for adding two

integers with different signs.

. . . the opposite

The Language of Mathematics When we change a number to its opposite, we say we have changed (or reversed) its sign.

2 Evaluate expressions involving subtraction and addition.

Expressions can involve repeated subtraction or combinations of subtraction and addition.To evaluate them, we use the order of operations rule discussed in Section 1.9.

EXAMPLE 3 Evaluate: 1 ( 2) 10

Strategy This expression involves two subtractions. We will write each subtraction as addition of the opposite and then evaluate the expression using the order of operations rule.

WHY It is easy to make an error when subtracting signed numbers. We will probably be more accurate if we write each subtraction as addition of the opposite.

Solution We apply the rule for subtraction twice and then perform the additions, working from left to right. (We could also add the positives and the negatives separately, and then add those results.)

1 ( 2) 10 1 2 ( 10) Add the opposite of 2, which is 2. Add the opposite of 10, which is 10.

1 ( 10) Work from left to right. Add 1 2 using the rule for adding integers that have different signs.

9 Use the rule for adding integers that have different signs.

EXAMPLE 4 Evaluate: 80 ( 2 24)

Strategy We will consider the subtraction within the parentheses first and rewrite it as addition of the opposite.

WHY By the order of operations rule, we must perform all calculations within parentheses first.

Solution

Strategy This expression involves one addition and two subtractions. We will write each subtraction as addition of the opposite and then evaluate the expression.

Self Check 3

Evaluate: 3 5 ( 1)

Now Try Problem 37

Self Check 4

Evaluate: 72 ( 6 51)

Now Try Problem 49

Self Check 5

Evaluate:

( 3) ( 16) 9 ( 28)

Now Try Problem 55

Self Check 6

THE GATEWAY CITY The record high temperature for St. Louis, Missouri, is 107ºF. The record low temperature is 18°F. Find the temperature range for these extremes. (Source: The World Almanac and Book of Facts,

2009)

Now Try Problem 101

Self Check 7

CRUDE OIL On Wednesday, the level of crude oil in a storage tank was 5 feet above standard capacity. Thursday, after a large refining session, the level fell to a mark 76 feet below standard capacity. Find the change in the crude oil level from Wednesday to Thursday.

Now Try Problem 103

162 Chapter 2 The Integers

4.The difference between the maximum and the minimum value of a collection of measurements is

CONCEPTS

Fill in the blanks.

5. To subtract two integers, add the first integer to the (additive inverse) of the integer to be

subtracted.

7.Subtracting 3 is the same as adding .

8.Subtracting 6 is the same as adding .

9. We can find the in a quantity by subtracting

the earlier value from the later value.

10.After rewriting a subtraction as addition of the opposite, we then use one of the rules for the

of signed numbers discussed in the previous section to find the result.

11.In each case, determine what number is being subtracted.

12.Fill in the blanks to rewrite each subtraction as addition of the opposite of the number being subtracted.

a.2 7 2

b.2 ( 7) 2

c.2 7 2

d.2 ( 7) 2

13.Apply the rule for subtraction and fill in the three blanks.

3 ( 6) 3

14.Use addition to check this subtraction: 14 ( 2) 12. Is the result correct?

NOTATION

15.Write each phrase using symbols.

a.negative eight minus negative four

b.negative eight subtracted from negative four

16.Write each phrase in words.

a.7 ( 2)

b.2 ( 7)

Complete each solution to evaluate each expression.

17.1 3 ( 2) 1 ( ) 2

2

18.6 5 ( 5) 6 5

5

19.( 8 2) ( 6) [ 8 ( )] ( 6)

( 6)

10

20.( 5) ( 1 4) [ 1 ( )]

5 ( )

5

GUIDED PRACTICE

Subtract. See Example 1.

Perform the indicated operation. See Example 2.

33.a. Subtract 1 from 11.

b.Subtract 11 from 1.

34.a. Subtract 2 from 19.

b.Subtract 19 from 2.

35.a. Subtract 41 from 16.

b.Subtract 16 from 41.

36.a. Subtract 57 from 15.

b.Subtract 15 from 57.

Evaluate each expression. See Example 3.

Evaluate each expression. See Example 5.

53.( 5) ( 15) 6 ( 48)

54.( 2) ( 30) 3 ( 66)

55.( 3) ( 41) 7 ( 19)

56.( 1) ( 52) 4 ( 21)

Use a calculator to perform each subtraction. See Using Your Calculator.

TRY IT YOURSELF

75.( 9) ( 20) 14 ( 3)

76.( 8) ( 33) 7 ( 21)

77.[ 4 ( 8)] ( 6) 15

78.[ 5 ( 4)] ( 2) 22

79.Subtract 6 from 10.

80.Subtract 4 from 9.

APPLICATIONS

Use signed numbers to solve each problem.

89.SUBMARINES A submarine was traveling

2,000 feet below the ocean’s surface when the radar system warned of a possible collision with another sub. The captain ordered the navigator to dive an additional 200 feet and then level off. Find the depth of the submarine after the dive.

90.SCUBA DIVING A diver jumps from his boat into the water and descends to a depth of 50 feet. He pauses to check his equipment and then descends an additional 70 feet. Use a signed number to represent the diver’s final depth.

91.GEOGRAPHY Death Valley, California, is the lowest land point in the United States, at 282 feet below sea level. The lowest land point on the Earth is the Dead Sea, which is 1,348 feet below sea level. How much lower is the Dead Sea than Death Valley?

92.HISTORY Two of the greatest Greek mathematicians were Archimedes (287–212 B.C.) and Pythagoras (569–500 B.C.).

a.Express the year of Archimedes’ birth as a negative number.

b.Express the year of Pythagoras’ birth as a negative number.

c.How many years apart were they born?

93.AMPERAGE During normal operation, the ammeter on a car reads 5. If the headlights are turned on, they lower the ammeter reading 7 amps. If the radio is turned on, it lowers the reading

6 amps. What number will the ammeter register if they are both turned on?

what is his new score?

95.FOOTBALL A college football team records the outcome of each of its plays during a game on a stat sheet. Find the net gain (or loss) after the third play.

96.ACCOUNTING Complete the balance sheet below. Then determine the overall financial condition of the company by subtracting the total debts from the total assets.

W a l k e r C o r p o r a t i o n

Balance Sheet 2010

97.OVERDRAFT PROTECTION A student forgot that she had only $15 in her bank account and wrote a check for $25, used an ATM to get $40 cash, and used her debit card to buy $30 worth of groceries. On each of the three transactions, the bank charged her a $20 overdraft protection fee. Find the new account balance.

98.CHECKING ACCOUNTS Michael has $1,303 in his checking account. Can he pay his car insurance premium of $676, his utility bills of $121, and his rent of $750 without having to make another deposit? Explain.

99.TEMPERATURE EXTREMES The highest and lowest temperatures ever recorded in several cities are shown below. List the cities in order, from the largest to smallest range in temperature extremes.

Extreme Temperatures

100.EYESIGHT Nearsightedness, the condition where near objects are clear and far objects are blurry, is measured using negative numbers. Farsightedness, the condition where far objects are clear and near objects are blurry, is measured using positive numbers. Find the range in the measurements shown in the next column.

WRITING

105.Explain what is meant when we say that subtraction is the same as addition of the opposite.

106.Give an example showing that it is possible to subtract something from nothing.

107.Explain how to check the result: 7 4 11

108.Explain why students don’t need to change every subtraction they encounter to an addition of the opposite. Give some examples.

REVIEW

109.a. Round 24,085 to the nearest ten.

b.Round 5,999 to the nearest hundred.

110.List the factors of 20 from least to greatest.

111.It takes 13 oranges to make one can of orange juice. Find the number of oranges used to make 12 cans.

112.a. Find the LCM of 15 and 18.

b.Find the GCF of 15 and 18.