UNIT TWO: Prealgebra in a Technical World
2.3 Subtracting Integers
SWBAT 1. Subtract integers.
2. Simplify integer addition and subtraction expressions.
3. Solve applied problems with integers.
Subtract Integers
The thermometer at the right has reached its limit, but
that does not mean that the temperature cannot drop below
−42℉! When the temperature drops below what a
thermometer can record, people in the North say that the
weather has “buried the needle.”
If the temperature earlier on this day had been −28℉
and the best guess at the current temperature is −42℉, how
much has the temperature changed?
To find the change from −28℉ to −42℉, we need to “find the difference.” To “find the
difference,” we use subtraction.
In this chapter we study the subtraction of integers.
We start with a look at differences in temperatures. If the temperature changes from
50℉ to 75℉, we subtract the original temperature from the final temperature, 75 – 50. This
difference, 25oF, is positive because the temperature increased.
Similarly if the temperature had changed from 75℉ to 50℉, we subtract 50 – 75, (again
subtracting the original temperature from the final temperature), and find that the difference
in temperature is −25℉. This difference, −25℉ , is negative because the temperature
decreased 25 degrees.
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SECTION 2.3 Subtracting Integers
If the temperature starts at −28°F and changes to −42°F, we write the subtraction
−42 – (−28). To see this difference, we can sketch an arrow on a number line. The distance
of the arrow gives the magnitude of the difference. The direction of the arrow gives the sign of
the difference. In this case, the temperature dropped, so the arrow shows a negative
difference. The arrow points to the left:
We write the equation: −42 – (−28) = −14° F. The negative result, −14° F, shows
that the difference in temperature is a decrease of 14 degrees.
Perhaps the very next day, the temperature changes from −42°F to −32°F. We write
this subtraction −32 – (−42), and the resulting change is +10°F. We have the equation:
−32 – (−42) = 10° F. On the sketch below we see that in this case the arrow points toward
the right on the number line and the change is positive:
To subtract, pay attention to the direction of change.
When the change is from a smaller number to a larger
number, the difference is always positive. When the change
is from a larger number to a smaller number, the difference
is always negative.
The operations of subtraction and addition have a
special relationship. Now that we have negative numbers,
we can see the relationship. Study the table on the right.
Notice the pattern?
Subtraction
Addition
5−𝟑=2
5 + (−𝟑) = 2
5−𝟐=3
5 + (−𝟐) = 3
5−𝟏=4
5 + (−𝟏) = 4
5−𝟎=5
5+𝟎=5
5 − (−𝟏) = 6
5+𝟏=6
5 − (−𝟐) = 7
5+𝟐=7
5 − (−𝟑) = 8
5+𝟑=8
UNIT TWO: Prealgebra in a Technical World
Complete this table and use the
Subtraction
pattern you found. Notice that each of the
Addition
4 – (1) =
4 + (–1) = 6
addition problem if we simply add the
4 – (0) =
4 + (0) = 6
opposite.
4 – (–1) =
4 + (1) = 4
4 – (–2) =
4 + (2) = 6
4 – (–3) =
4 + (3) =5
subtraction problems can be changed to an
Mathematicians even say,
“subtraction is adding the opposite.”
RULE: To subtract an integer, add its opposite. Using variables, we record that
this is always true: 𝑎 – 𝑏 = 𝑎 + (−𝑏) and 𝑎 − (−𝑏) = 𝑎 + 𝑏.
We do not need this rule to subtract 5 – 2! We have always done this without changing
signs. We change subtraction to adding the opposite when it makes the calculation easier to
think through.
Before subtracting, find the sign of the difference first. If you are going to change
subtraction to addition, make this change first, and then determine whether your answer will
be positive or negative:
EXAMPLE 1: Change each of these to an addition problem, and then state whether the
difference is positive or negative. Do not complete the subtraction (or addition!)
Subtraction:
Change to addition:
Sign of answer:
(a) −3 − 8
−3 + (−8)
Negative (−)
(b) 7 − 13
7 + (−13)
Negative (−)
(c) 2 – (−7 )
2 + 7
Positive (+)
(d) −4 – (−1)
−4 + 1
Negative (−)
(e) −5 – (−6)
−5 + 6
Positive (+)
Remember to use
a number line to
help you think
problems through
until you know the
rules for signs!
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SECTION 2.3 Subtracting Integers
 Check Point 1
Rewrite these subtractions as addition. Then determine the sign of the answer.
5 − 9
ADDITION: _____________________Sign of answer: ____________________
b. −7 − 8
ADDITION: _____________________Sign of answer: ____________________
a.
c.
3 – (−5 )
ADDITION: _____________________Sign of answer: ____________________
d. −10 – (−12) ADDITION: _____________________Sign of answer: ____________________
e. −16 – (−7)
ADDITION: _____________________Sign of answer: ____________________
Now we are ready to subtract integers!
EXAMPLE 2: Subtract the following by adding.
a. −3 − 8 = ___________
THINK: −𝟑 − 𝟖 = −𝟑 + (−𝟖) and the answer is negative.
ANSWER: −𝟏𝟏
b. 7 − 13 = ___________
THINK: 𝟕 − 𝟏𝟑 = 𝟕 + (−𝟏𝟑) and the answer is negative.
ANSWER: − 𝟔
c. 2 – (−7 ) = ___________
THINK: 𝟐 – (−𝟕) = 𝟐 + (𝟕) and the answer is positive.
ANSWER: 𝟗
d. −4 – (−1) = _________
THINK: −𝟒 – (−𝟏) = −𝟒 + (𝟏) and the answer is negative.
ANSWER: −𝟑
e. −5 – (−6) = _________
THINK: −𝟓 – (−𝟔) = −𝟓 + (𝟔) and the answer is positive.
ANSWER: 𝟏
 Check Point 2
Subtract the following by adding.
a.
5 − 9 = ______________________________ ANSWER: ________________________
b. −7 − 8 = ______________________________ ANSWER: ________________________
c.
3 – (−5 ) =_____________________________ ANSWER: ________________________
d. −10 – (−12) =____________________________ ANSWER: ________________________
e. −16 – (−7) =_____________________________ ANSWER: _______________________
S
UNIT TWO: Prealgebra in a Technical World
s
Simplify Integer Addition and Subtraction Expressions
When we balance our checkbook, read our bank statement, or add up the transactions
on our debit card, we add positive and negative numbers.
Example 3: Marjorie had $300 in her debit card account when she left to go shopping and run
errands. She spent $83 at a shoe sale, $147 for groceries, and $23 for gas. Then she picked up
her babysitting check and ran by the bank where she deposited another $88 into her debit card
account. On the way home she realized that she did not have any cash, so she stopped at an
ATM and withdrew $20 for the week.
When Marjorie returned home, one pair of shoes did not fit her, so she returned the
pair and received a reimbursement to her debit card for $40.
She jotted down these transactions on a piece of paper:
300 – 83 − 147 − 23 + 88 − 20 + 40
Marjorie computed the approximate balance in her account by adding two numbers at a
time in the order that made her work the easiest. Here is one way she might do the math:
Start with the numbers on paper: 300 – 83 − 147 − 23 + 88 − 20 + 40
1. Change all subtraction to addition
= 300 + (−83) + (−147) + (−23) + 88 + (−20) + 40
2. Regroup to make the addition easier by using the commutative and associative properties:
= [300 + 40] + [88 + (−83)] + [(−147) + (−23)] + (−20)
3. Simplify:
= [340] + [5] + [−170] + (−20)
4. Continue to regroup to make the addition easier until the balance is reached:
= [340 + (−170)] + [5 + (−20)]
= 170 + (−15)
= 155
Marjorie’s method saved paper, and she finds that her balance is $155.
Perhaps you have already regrouped to simplify long lists of additions without thinking
about negative numbers. Perhaps Marjorie’s choice of just how to sum up her numbers would
not be your choice!
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SECTION 2.3 Subtracting Integers
No matter which way we group additions, we are guaranteed that we can make these
rearrangements by the commutative property. Once we change subtractions to adding the
opposite, we can simplify calculations.
RULE: To simplify expressions involving subtraction, change subtractions to
addition and use the commutative and associative properties to simplify
calculations.
EXAMPLE 4: Simplify by changing to addition and regrouping: 7 − 21 − 9 + 11 − 12 + 8.
One Possible Answer:
7 − 21 − 9 + 11 − 12 + 8
= 7 + (−21) + (−9) + 11 + (−12) + 8
Think step: Add the opposite.
= 7 + [(−21) + (−9)] + [11 + (−12)] + 8
Regroup to simplify the calculation.
[−30]
=7+
=
+
[−1]
−𝟏𝟔
+ 8
Add.
Continue adding.
 Check Point 3
Simplify by changing to addition and regrouping. SHOW YOUR STEPS.
a. 5 − 13 + 18 − 15 − 7 − 8
ANSWER: _____________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
b. 85 – (−8) − 37 + (−15) + 12 + (−93)
ANSWER: _____________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
UNIT TWO: Prealgebra in a Technical World
Applied Problems
Businesses working with money and inventory often find occasion to subtract signed
numbers.
EXAMPLE 5: Dusty and Theresa own a construction business. Each month they take inventory
of what they have in stock. If they have the item on hand they record the amount they have as
a positive number. Whenever they need something that they do not have, they use a negative
number. When material has been backordered, the “Have” column may also show a negative
amount. To find the balance, they add. For example:
Item
Have
Need
Crown molding
3600 feet
-780 feet
Picture molding
-150 feet
-600 feet
¾ interior plywood
48 sheets
-120 sheets
Balance
ANSWER:
Item
Have
Need
Balance
Crown molding
3600 feet
-780 feet
+2,820 feet
Picture molding
-150 feet
-600 feet
-750 feet
¾ interior plywood
48 sheets
-120 sheets
-72 sheets
 Check Point 4
Complete the balance sheet by adding the “Have” and “Need” columns.
Item
Deck screws
Have
Need
40 lbs.
-100 lbs.
6” x 1” RW decking
4,200 lin. ft
-2400 lin. ft
8 ft, 2” by 6” Cedar
126 boards -274 boards
Railing
-160 ft
-580 ft
Deck wash
80 gal.
-150 gal.
7 gal.
-18 gal.
Deck stain (m1)
Balance
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SECTION 2.3 Subtracting Integers
EXAMPLE 6: Aaron overpaid his phone bill last month. At the beginning of the month, his cell
phone account had a balance (credit) of $18. During the month he accumulated $23 in extra
charges for texting in addition to his monthly bill of $40. What was his cell phone bill at the end
of the month?
Think it through: Start with an $18 positive balance, then subtract the texting charges and
monthly bill: 𝟏𝟖 − 𝟐𝟑 – 𝟒𝟎.
Change these subtractions to adding the opposite: 𝟏𝟖 + (−𝟐𝟑) + (−𝟒𝟎).
Use the commutative property to add the negative numbers first (if you
wish): 𝟏𝟖 + (−𝟔𝟑)
Then add the positive and negative numbers: 𝟏𝟖 + (−𝟔𝟑) = −𝟒𝟓.
ANSWER: Aaron’s cell phone bill at the end of the month is $45.
Notice in Example 6 that using the commutative property is always an option. Two
students may add in different orders, but in the end they will have the same sum.
Differences in elevations and temperature are found by subtracting integers.
EXAMPLE 7: Marie has been to the shores of the Dead Sea (between Israel and Jordan). The
elevation of the Dead Sea is −1,377 feet. Marie has also climbed the 11,429 foot Mt. Hood, in
Oregon, USA. What is the difference in elevation between the highest point and the lowest
point that Marie visited?
Think it through: Draw a diagram to “see” the problem.
𝟏𝟏, 𝟒𝟐𝟗 – (−𝟏, 𝟑𝟕𝟕) = 𝟏𝟏, 𝟒𝟐𝟗 + 𝟏, 𝟑𝟕𝟕
𝟏𝟐, 𝟖𝟎𝟔 feet
ANSWER: The difference between the highest point and
lowest point Marie has visited is 12,806 feet.
UNIT TWO: Prealgebra in a Technical World
 Check Point 5
Bonnie camps at the top of Stevens Pass, WA, elevation 4,061 feet. On Saturday she rides her
bike to Leavenworth, WA, elevation 1,166 feet. What is her change in elevation?
ANSWER: _____________________________________________________________________
______________________________________________________________________________
 Check Point 6
The temperature at which helium boils is −452°F. Water boils at 212°F. What is the difference
between the boiling point of helium and the boiling point of water?
ANSWER: _____________________________________________________________________
______________________________________________________________________________
STUDY SKILLS: Subtraction is the most difficult of the four operations to think about.
The most common error occurs when problems like −3 − 8 and −10 − (−11) aren't
rewritten as adding the opposite. For example when we change −3 − 8 to −3 +
(−8) then it is easier to “see” that the sum is -11. Change −10 − (−11) to −10 +
(11) and it is easier to “see” that this sum is 1. Unless the difference is easy to see,
change all subtraction to adding the opposite and use the number line to help you
visualize the difference between the numbers. Do many, many subtraction problems
until you can automatically subtract integers. You have extra problems in this book
and on your class Web site, or feel free to write and solve your own subtraction
problems.
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SECTION 2.3 Subtracting Integers
UNIT TWO: Prealgebra in a Technical World
2.3 Exercise Set
Name _______________________________
Skills
Simplify. Change subtraction to addition of the opposite when it makes calculating easier.
1. −9 − 2
2. 5 − (−3)
3. −3 − 3
4. 3 − (−2)
5. 6 − (−6)
6. −4 − 4
7. −10 − (−3)
8. −2 − (−9)
9.
−22 − (−49)
13. −8 − 41
10. −34 − (−6)
11. 6 − (−25)
12. −62 − 39
14. −23 − (−34)
15. 6 − (−3)
16. −29 − 29
19. −17 − (−17)
20.
17. 125 − (−25)
18.
14 − (−15)
−22 − (−22)
21. 302 − (−62)
22.
4 − (−16)
23.
−9 − 34
24.
−24 − 26
25. −8 − 2
26.
−3 − (−16)
27.
90 − (−24)
28.
−29 − 67
29. 4 − 6 − (−13)
30. −3 + (−2) + 6
31.
4 + (−2) + 9
32. −8 + (−2) − (−3)
33. 0 − (−17) + 9
34.
−28 + 6 + (−5)
35. 92 − (−64) − 128
36. −22 − (−20) + 36
37.
−481 + 130 − 40
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SECTION 2.3 Subtracting Integers
38.
−2 − 4 − 14 − 5
41.
−66 − 16 − (−2) + 9
39. −19 + (−28) − 16 − 5
40.
27 − (−21) + (−2) − 9
42. 26 − 14 − 8 + (−13)
Applications UPS
43.
45.
The highest point in the Sahara Desert
has an elevation of 11,000 feet. The
lowest point is 440 feet below sea level.
What is the difference between the
highest and the lowest points in the
Sahara Desert? (Draw a picture.)
44. In 2007, a small business reported a loss
of $38,152. In 2008, it posted a profit of
$20,743. By how many dollars did the
earning power of the business increase
from 2007 to 2008?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
On Monday, the low temperature was
−6℉. On Tuesday, the low temperature
was 2℉. How much warmer or cooler
was it on Tuesday than Monday?
46. On Saturday, the low temperature was
−4℉. On Sunday, the low temperature
was −14℉. How much warmer or
cooler was it on Sunday than on
Saturday?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
UNIT TWO: Prealgebra in a Technical World
47.
49.
51.
A motorist sees a milepost marker that
reads 139 miles. A few hours later, she
travels by a marker reading 641 miles.
How far has she traveled?
48. Jay has his credit card paid off, then
returns an item that cost $19. What is
the balance on his credit card after the
return?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
Sam has been exercising to lose weight.
He went from 12 lbs. above the
recommended weight on his health
chart, to 7 lb below it. How many
pounds did Sam lose?
50. Susan has been exercising to lose weight.
She went from 31 lbs. above the
recommended weight on her chart, to
17 lb below it. How many pounds did
Susan lose?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
The highest temperature ever recorded
on Earth was 136℉, and the lowest ever
recorded was −129℉. What is the
difference between the highest and the
lowest temperatures recorded?
52. The temperature one Wednesday at
midnight in Montana was −12℉. On
Thursday at midnight it was −4℉. What
was the difference in temperatures
between Wednesday and Thursday?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
53. With an elevation of 535 feet above sea level, Driskill Mountain in Louisiana is the highest
point in the state. The lowest point in Louisiana is in New Orleans, which has an elevation
of 8 feet below sea level. What is the range (difference) between these two elevations?
Sentence: _________________________________________________________________
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SECTION 2.3 Subtracting Integers
54. The highest point on earth is Mt. Everest at 29,028 feet above sea level. The lowest point
(not covered with water) is the Dead Sea Depression at 1,385 feet below sea level.
a. What is the difference between the highest and lowest points on Earth?
Sentence: ______________________________________________________________
b. If you consider the lowest point in the Earth's crust, it would be the Mariana Trench in
the Pacific Ocean, with a depth of 35,840 feet below the sea. Given this new
information, what is the difference in elevation between Mt. Everest and the Mariana
Trench?
Sentence: ______________________________________________________________
Skills
Add mentally from left to right or by using compensation.
55. 57 + (−29) = ______
56. −57 + 29 = ______
57.
79 + 57 = ______
58. (−79) + (−57) = _____
59. −26 + 77 = ______
60.
73 + (− 29) = _____
61. 193 + (−165) = ________
62.
−132 + 89 = _______
Some elementary students figure out that they do not have to borrow to subtract! Instead they
use integers. Perhaps you are one of these people. We hope your teacher did not tell you that
you were wrong, because this method is just another way to subtract. Here is the method:
−84
−57
−27
=
−80 + −4
−50
−7
−30 + −3 = 27
In fact they subtract mentally:
(80 – 50) + (4 – 7) is 27
“30”
“+−3”
Use this “elementary” method to complete problems 63 to 66 without borrowing.
63.
−51
−17
64.
−76
−48
65. −475
−357
66.
−617
−572