UNIT TWO: Prealgebra in a Technical World
2.2 Adding Integers
SWBAT 1. Add integers.
2. Solve applied problems with integers.
Add Integers
Kyle gets a new credit card. He does not want to spend more than he can pay off in a
month, so he starts a ledger to keep track of his
expenditures. Kyle uses the card to buy about $130
worth of groceries. Then Kyle pays for a little less than
$45 worth of gas using the card.
Transaction
Amount
Balance
Groceries
−130
−$130
Gas
−45
Kyle has made purchases, so his balance must be negative. He is “putting together” his
two debits (purchases), and this tells us that he will use
addition: −130 + (−45). Since Kyle is putting two
debts together, he is more in debt than when he started.
Transaction Amount
Balance
Groceries
−130
−$130
Gas
−45
−$𝟏𝟕𝟓
Kyle adds: −130 + (−45) = −175.
Since we can use plus signs (+) and minus signs (−) to denote positive and negative
integers, we can talk about these numbers as signed numbers.
In this section we study the addition of signed numbers
by studying the addition of the integers.
When we put together two positive integers, the sum is always positive. This can be
shown on a number line. For instance the sum 3 + 8 is 3 and 8 put together in order, just like
this:
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SECTION 2.2 Adding Integers
When we add two negative numbers, we put them together as well. To add −5 + −4 ,
put together −5 and −4. The result is −9.
Using a number line model can help your brain make sense of why integer operations
produce certain results. Models, and knowing why a process works, help you learn the rules for
the long term, not just until the next test! Identify a model that works for you and use that
model until you do not need it anymore. To get started, use the number line to add.
Example 1: Show the addition −6 + −2.
 Check Point 1
a. Show the addition −7 + −3.
ANSWER: ____________________
b. Show the addition −10 + −1.
ANSWER: _____________________
UNIT TWO: Prealgebra in a Technical World
At this point you may have figured out a rule of your own for adding two negative
integers. To describe the sum of two negative numbers clearly, we use absolute values:
RULE: To add two negative integers, add the absolute values of each number.
The sum is the opposite of this result. The sum will always be negative.
,
Rules written like this leave no room for misunderstanding, but the vocabulary and
symbols needed to be this precise can be hard to understand! As you study, make sense of
why operations and numbers work the way they do, and then study the rule.
Adding negatives gives a negative sum. Adding positives gives a positive sum. But what
is the result when we add a positive to a negative number? We investigate on a number line.
Example 2a: To add −3 + 5 , put together −3 and 5. Start at zero and move left 3 for
−3. From −3, move right 5. The result is 2; −3 + 5 = 2.
Example 2b: Add 8 + (− 11) on the number line and write the sum.
ANSWER: 8 + −11 = − 3
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SECTION 2.2 Adding Integers
 Check Point 2
a. Add 7 + (−11 ) on the number line and write the sum.
ANSWER: _________________________
b. Add (−3 ) + 12 on the number line and write the sum.
ANSWER: _________________________
c. Show the integers being added, write the addition expression and the sum.
ANSWER: ____________________________
We can also use a vertical number line to investigate
these additions. A vertical number line, like the one on the
right, can be constructed quickly on the margin of a sheet of
notebook paper.
Many students have used this as a tool to think
through an operation. To use this number line when adding
integers, go up for positive integers and go down for
negative integers.
UNIT TWO: Prealgebra in a Technical World
Example 3: Use the number line on the previous page to think through these sums:
a. To add −18 + 15, go down 18, from −18 go up 15, end at −3; −18 + 15 = −3
b. To add 8 + (−16), go up 8, from 8 go down 16, end at −8; 8 + (−16) = −8
c. To add −17 + 21, go down 17, from −17 go up 21, end at 4; −17 + 21 = 4
d. To add 5 + (−9), go up 5, from 5 go down 9, end at −4; 5 + (−9) = −4
 Check Point 3
a. −7 + 15 = __________________
b. 19 + (−17 ) = ________________
c. −8 + 4 = ____________________
d. 6 + (−13 ) = _________________
Here is an organized table that lists sums
that are positive on the left and sums that are
negative on the right. Have you seen a pattern?
Write it in the margin so you are prepared to
discuss just how you see it in class.
When adding integers, we always
determine the sign of the sum first, and then we
Positive sum
Negative sum
−17 + 21 = (+)
8 + −11 = (−)
−7 + 15 = (+)
6 + (−13 ) = (−)
19 + (−17 ) = (+)
−8 + 4 = (−)
−3 + 12 = (+)
5 + (−19) = (−)
−5 + 9 = (+)
8 + (−16) = (−)
−1 + 6 = (+)
−18 + 15 = (−)
add. With practice and attention to patterns, we
can learn to add integers as quickly as we add whole numbers.
RULE: To add integers with opposite signs, subtract their absolute values. The
sign of the sum is the sign of the integer with the largest absolute value.
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SECTION 2.2 Adding Integers
Example 4: Tell whether the sum is positive or negative, then determine the sum:
a. −17 + 21
Think it through: 21 is farther from zero than -17, so the sum will be positive.
−𝟏𝟕 + 𝟐𝟏 = | 𝟐𝟏| – |𝟏𝟕| = 𝟒
ANSWER: The sum is positive. 𝟒
b. 6 + (−13)
Think it through: −𝟏𝟑 is farther from zero than 6, so the sum will be negative.
−(|−𝟏𝟑| − |𝟔|) = −(𝟏𝟑 − 𝟔) = −𝟕
ANSWER: The sum is negative. −𝟕
We show all the steps our brains think through to add, but you do not have to
write all of these steps to add integers. Single digit integers, like whole
numbers, should soon be mental math calculations!
 Check Point 4
Tell whether the sum is positive or negative, then compute the sum.
a. −8 + 6
____________________________________________________
b. 15 + (−12) ____________________________________________________
c. −9 + 4
____________________________________________________
d. 4 + (−7)
____________________________________________________
e. −20 + 4
____________________________________________________
f. 8 + (−2)
____________________________________________________
Remember that zero is neither positive nor negative. When we add zero to a negative
integer, the sum is still that integer. For instance −9 + 0 = −9.
UNIT TWO: Prealgebra in a Technical World
The number line also makes it clear what happens when we add opposites:
PROPERTIES
Additive Property of Zero: Let 𝑎 be any integer, then 𝑎 + 0 = 0 + 𝑎 = 𝑎.
The Add-Op Property: Let 𝑎 be any integer, then 𝑎 + −𝑎 = 0.
Solving Applications by Adding Integers
Bookkeepers keep track of financial transactions. Since we all receive and spend money,
we can use a personal ledger system to keep track of our own income and expenses. With
ATMs, debit cards, and credit cards, keeping our own books may be a good idea! On a ledger
we write positive numbers for credits, and negative numbers for debits, then use addition to
find each balance.
Example 5: Randy received his debit card statement. He couldn’t believe he had $50 in
overdraft fees! Randy decided
to estimate to check the bank
results. He rounds up to the
dollar for each of his debits.
Randy rounds down to the
dollar for each of his credits.
He records the amount and the
balance in his ledger.
The sad news is that the
bank was correct. Randy had two overdrafts. You can see this in the two negative balances.
Notice that Randy started with a $827 available, and this was shown as a positive.
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SECTION 2.2 Adding Integers
 Check Point 5
Complete Randy’s ledger for February. Overdrafts are a $25 charge each.
Date
Transaction
2/1
Balance
2/5
Gas
-38
2/16
Cash
-500
2/20
Groceries
-459
2/25
Electricity
-396
2/26
Overdrafts
Amount
Balance
1031
(Perhaps Randy should start taking a little less
cash out of his account each month!)
UNIT TWO: Prealgebra in a Technical World
2.2 Exercise Set
Name _______________________________
Skills
Add integers.
1. −9 + 5
2. 2 + (−5)
3.
−9 + 2
4. 2 + (−7)
5. 2 + (−5)
6. −8 + 2
7.
−10 + (−3)
8. −4 + (−8)
9. −12 + (−63)
10. −64 + (−3)
11. 5 + (−45)
12. −64 + 33
13. 36 + (−42)
14. −17 + (−32)
15. 2 + (−3)
16. −16 + 16
17. 102 + (41)
18. 16 + (−16)
19. −17 + (−17)
20. −49 + (−6)
21. 476 + (−29)
22. 3 + (−15)
23. −13 + 43
24. −12 + 19
25. −8 + 5
26. −4 + (−4)
27. 58 + (−32)
28. −4 + 6
29. 13 + (−14)
30. −217 + (−26)
31. −24 + (−70)
32. −11 + (−3)
33. −14 + 5 + (−12)
34.
19 + (−7) + 7
35.
28 + (−32) + 37
36. If 𝑥 is a negative integer, then −𝑥 is
negative /
positive .
(Circle One)
37. If −𝑥 is a negative integer, then 𝑥 is
negative /
positive .
(Circle One)
38. If 𝑥 is a positive integer, then −𝑥 is
negative /
positive .
(Circle One)
39. If −𝑥 is a positive integer, then 𝑥 is
negative /
positive .
(Circle One)
40. If a negative is added to a negative,
the sum will always be
negative /
positive .
(Circle One)
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SECTION 2.2 Adding Integers
Applications
Use UPS for each. If you need more paper, use your own and attach. Write a sentence using
your answer.
41.
43.
45.
Connie charged $45.98 and $62.97 on
her credit card. She also returned two
items: one for $12.49 and the other for
$24.15. How much does she owe on her
credit card after these 4 transactions?
42. A family keeps a weekly budget. During
a 4-week period, they were $16.77 over
budget, $9.84 under budget, $24.31 over
budget, and $6.23 under budget. How
are they doing?
Sentence:_________________________
Sentence:_________________________
_________________________________
A robot is programmed to turn using
angle measure. A full circle is a 360o
angle. An angle is positive if it turns in a
counterclockwise direction (left) and
negative if it turns in a clockwise
direction (right). A certain robot turns
180°, −45°, −225°, and finally 90°.
What is the robot’s total turn?
_________________________________
44. Read problem 43 for an explanation of
robot turns. If a certain robot turns
47°, −88°, 107°, −111°, and finally 11°,
what is the robot’s total turn?
Sentence:_________________________
Sentence:_________________________
______________________________
At 6 PM, the temperature in Fairbanks
Alaska was 10℉ . By midnight, the
temperature had dropped 24°F. What
was the temperature at midnight?
_________________________________
46. At 9 a.m. the temperature in Anchorage,
Alaska, was −11℉. During the next
2 hours the temperature rose 9°. What
was the temperature 2 hours later?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
UNIT TWO: Prealgebra in a Technical World
47.
A football team needed 19 yards to
reach the end zone and score. In
3 plays, they lost 13 yards, gained
12 yards, and then gained 21 yards.
Was that enough to score?
Sentence:_________________________
49.
51.
48. A football team was penalized 5 yards
and then gained 5 yards on the next play.
If the ball was on the 24-yard-line before
the penalty, where was the ball after the
next play?
Sentence:_________________________
_________________________________
_________________________________
The base of Mauna Kea in the Hawaiian
50. The highest point in California, Mt.
Trough, lies 19,680 feet below sea level.
Whitney, is 14,787 feet higher than the
Its summit is 13,796 feet above sea level.
lowest point in California, Death Valley.
Determine the height of this volcanic
Given that the elevation of Death Valley
mountain. (Hint: Draw a picture.)
is 282 feet below sea level (negative!),
find the elevation of Mt. Whitney.
(Hint: Draw a picture.)
Sentence:_________________________
Sentence:_________________________
_________________________________
A hiker climbed 2,250 feet up a
mountain, then realized that he had left
his water bottle at the last stop. He
went back down 1,940 feet and
retrieved it. He then climbed up another
1,500 feet. At what elevation was he
then?
_________________________________
52. A tourist on vacation in Las Vegas won
$304 on the blackjack table. That same
evening, he lost $989 and then won an
additional $131. What were his net
winnings/losses for the evening?
Sentence:_________________________
Sentence:_________________________
_________________________________
_________________________________
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SECTION 2.2 Adding Integers
Review and Extend
Negative integers can also be rounded up, down and to the nearest. The first two integers have
been completed for you in the tables below.
53.
−82
−80
82
Round up to the leading digit 99000
Round down to the leading digit
Round to the nearest ten
Round up to the hundred
54.
80
830
1300
$5.45
Round to the nearest dollar
Round up to the leading digit
Round down to the dime
$53
$5
$5.40
−90
−80
0
908
1,000
900
910
1,000
−908
−900
−1,000
−910
−900
−$5.45 $50.43
$−5
$50
$−5
$50
$−5.50 $50.40
−8,532
−8,000
−9,000
−8,530
−8500
−$50.43
$-50
$-50
$-50.40
8,532
9,000
8,000
8,530
8,600
$97.72 −$97.72
$98
$100
$-98
$-90
$97.70
$−97.80
For problems 55 to 60, round each integer to the leading digit and write these in the blanks provided.
Then calculate your estimate and write it in the final blank. Problem 55 is done for you.
55.
57.
59.
−78 + (−29) 
−51 + 13
−80
+ −30

−110

______ + ______

______
19 + (−87) 

______ + ______
______
56. −95 + (−64) 

______ + ______
______
58. 173 + (−52) 
______ + ______

______
70
______ + ______
60. −25 + 789 

______
Fill in the blanks for the following patterns:
61.
−34, −27, −20, ______, ______, ______
62. −34, −47, −60, ______, ______, ______
63.
102, −3, −108, ______, ______, ______
64. -−34, −227, −420, ______, ______, ______
65.
1, −2, 3, −4, 5, ______, ______, ______
66. −20, 32, −44, 56, ______, ______, ______