Add Integers

Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of $2 - 1 .$ Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

This figure has a blue circle labeled positive and a red circle labeled negative.

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in Figure 1.

This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0. — A blue counter represents $+ 1 .$ A red counter represents $−1 .$ Together they add to zero.

We will model four addition facts using the numbers $5, −5 and 3, −3 .$

5 + 3 −5 + (−3) −5 + 3 5 + (−3)

Model: $5 + 3 .$

Solution

Interpret the expression.	$5 + 3$ means the sum of $5$ and $3$ .
Model the first number. Start with 5 positives.
Model the second number. Add 3 positives.
Count the total number of counters.
The sum of 5 and 3 is 8.	$5 + 3 = 8$

Model: $−5 + (−3) .$

Solution

Interpret the expression.	$−5 + (−3)$ means the sum of $−5$ and $−3$ .
Model the first number. Start with 5 negatives.
Model the second number. Add 3 negatives.
Count the total number of counters.
The sum of −5 and −3 is −8.	$−5 + −3 = −8$

Example 1 and Example 2 are very similar. The first example adds $5$ positives and $3$ positives—both positives. The second example adds $5$ negatives and $3$ negatives—both negatives. In each case, we got a result of $8—either 8$ positives or $8$ negatives. When the signs are the same, the counters are all the same color.

Now let’s see what happens when the signs are different.

Model: $−5 + 3 .$

Solution

Interpret the expression.	$−5 + 3$ means the sum of $−5$ and $3$ .
Model the first number. Start with 5 negatives.
Model the second number. Add 3 positives.
Remove any neutral pairs.
Count the result.
The sum of −5 and 3 is −2.	$−5 + 3 = −2$

Notice that there were more negatives than positives, so the result is negative.

Model: $5 + (−3) .$

Solution

Interpret the expression.	$5 + (−3)$ means the sum of $5$ and $−3$ .
Model the first number. Start with 5 positives.
Model the second number. Add 3 negatives.
Remove any neutral pairs.
Count the result.
The sum of 5 and −3 is 2.	$5 + (−3) = 2$

This table illustrates the addition problem 4 + 2, detailing each step with a description and its corresponding mathematical or visual representation.
ⓐ
	$4 + 2$
Start with 4 positives.
Add two positives.
How many do you have?	$4 + 2 = 6$

Visual step-by-step demonstration of integer addition for -3 + 6, showing the process and result.
ⓑ
	$- 3 + 6$
Start with 3 negatives.
Add 6 positives.
Remove neutral pairs.
How many are left?
	$3$ . $−3 + 6 = 3$

Visual demonstration of adding 4 and -5, resulting in -1, using a step-by-step approach with illustrations.
ⓒ
	$4 + (−5)$
Start with 4 positives.
Add 5 negatives.
Remove neutral pairs.
How many are left?
	$−1$ . $4 + (−5) = −1$

Step-by-step demonstration of adding negative integers (-2 + -3 = -5) using descriptions, math, and visuals.
ⓓ
	$−2 + (−3)$
Start with 2 negatives.
Add 3 negatives.
How many do you have?	$−5$ . $−2 + (−3) = −5$

Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.

For example, if you want to add $37 + (−53),$ you don’t have to count out $37$ blue counters and $53$ red counters.

Picture $37$ blue counters with $53$ red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because $53 −37 = 16,$ there are $16$ more negative counters.

37 + (−53) = −16

Let’s try another one. We’ll add $−74 + (−27) .$ Imagine $74$ red counters and $27$ more red counters, so we have $101$ red counters all together. This means the sum is $−101.$

−74 + (−27) = −101

Look again at the results of Example 1 - Example 4.

Addition of Positive and Negative Integers
$5 + 3$	$−5 + (−3)$
both positive, sum positive	both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
$−5 + 3$	$5 + (−3)$
different signs, more negatives	different signs, more positives
Sum negative	sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

Simplify:

ⓐ $19 + (−47)$
ⓑ $−32 + 40$

Solution

ⓐ Since the signs are different, we subtract $19$ from $47 .$ The answer will be negative because there are more negatives than positives.

\begin{matrix} 19 + (−47) \\ −28 \end{matrix}

ⓑ The signs are different so we subtract $32$ from $40 .$ The answer will be positive because there are more positives than negatives

\begin{matrix} −32 + 40 \\ 8 \end{matrix}

Simplify: $−14 + (−36) .$

Solution

Since the signs are the same, we add. The answer will be negative because there are only negatives.

\begin{matrix} −14 + (−36) \\ −50 \end{matrix}

The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

Simplify: $−5 + 3 (−2 + 7) .$

Solution

This table illustrates the step-by-step simplification of the mathematical expression -5 + 3(-2 + 7) following the order of operations.
	$−5 + 3 (−2 + 7)$
Simplify inside the parentheses.	$−5 + 3 (5)$
Multiply.	$−5 + 15$
Add left to right.	$10$

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.

Evaluate

x + 7 when

ⓐ $x = −2$
ⓑ $x = −11 .$

Solution

ⓐ Evaluate $x + 7$ when $x = −2$


Simplify.

ⓑ Evaluate $x + 7$ when $x = −11$


Simplify.

When

n = −5,

evaluate

ⓐ $n + 1$
ⓑ $- n + 1 .$

Solution

ⓐ Evaluate $n + 1$ when $n = −5$


Simplify.

ⓑ Evaluate $- n + 1$ when $n = −5$


Simplify.
Add.

Next we'll evaluate an expression with two variables.

Evaluate $3 a + b$ when $a = 12$ and $b = −30 .$

Solution



Multiply.
Add.

Evaluate ${(x + y)}^{2}$ when $x = −18$ and $y = 24 .$

Solution

This expression has two variables. Substitute

−18

for

x

and

24

for

y .

	${(x + y)}^{2}$
	${(−18 + 24)}^{2}$
Add inside the parentheses.	${(6)}^{2}$
Simplify	$36$

Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

Translate and simplify: the sum of $−9$ and $5 .$

Solution

Steps to translate a verbal mathematical phrase into an expression and then simplify it.
The sum of −9 and 5 indicates addition.	the sum of $−9$ and $5$
Translate.	$−9 + 5$
Simplify.	$−4$

Translate and simplify: the sum of $8$ and $−12,$ increased by $3 .$

Solution

The phrase increased by indicates addition.

Steps to translate and simplify a mathematical phrase into its numerical result.
	The sum of $8$ and $−12$ , increased by $3$
Translate.	$[8 + (−12)] + 3$
Simplify.	$−4 + 3$
Add.	$−1$

Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of $$5$ could be represented as $−$5.$ Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

The temperature in Buffalo, NY, one morning started at $7 degrees$ below zero Fahrenheit. By noon, it had warmed up $12 degrees .$ What was the temperature at noon?

Solution

We are asked to find the temperature at noon.

This table illustrates the step-by-step process of solving a temperature word problem, including phrasing, translation to math notation, simplification, and final answer sentence.
Write a phrase for the temperature.	The temperature warmed up 12 degrees from 7 degrees below zero.
Translate to math notation.	−7 + 12
Simplify.	5
Write a sentence to answer the question.	The temperature at noon was 5 degrees Fahrenheit.

A football team took possession of the football on their $42-yard line.$ In the next three plays, they lost $6 yards,$ gained $4 yards,$ and then lost $8 yards.$ On what yard line was the ball at the end of those three plays?

Solution

We are asked to find the yard line the ball was on at the end of three plays.

Illustrates problem-solving steps: converting a word phrase to math notation, simplifying, and stating the final answer for a ball's position.
Write a word phrase for the position of the ball.	Start at 42, then lose 6, gain 4, lose 8.
Translate to math notation.	42 − 6 + 4 − 8
Simplify.	32
Write a sentence to answer the question.	At the end of the three plays, the ball is on the 32-yard line.

Key Concepts

Addition of Positive and Negative Integers

This table demonstrates the rules for adding integers, illustrating examples and results for sums involving numbers with both same and different signs.
$5 + 3$	$−5 + (−3)$
both positive, sum positive	both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
$−5 + 3$	$5 + (−3)$
different signs, more negatives	different signs, more positives
Sum negative	sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

Practice Makes Perfect

Model Addition of Integers

In the following exercises, model the expression to simplify.

$7 + 4$

Solution

This figure shows a row of 11 light pink circles, representing positive counters. They are separated into a group of seven and a group of four.
11

$8 + 5$

$−6 + (−3)$

Solution

This figure shows a row of 9 dark pink circles, representing negative counters. They are separated into a group of six and a group of three.
−9

$−5 + (−5)$

$−7 + 5$

Solution

This figure shows two rows of circles. The top row shows 7 dark pink circles, representing negative counters. The bottom row shows 5 light pink circles, representing positive counters.
−2

$−9 + 6$

$8 + (−7)$

Solution

This figure shows two rows of circles. The top row shows 8 light pink circles, representing positive counters. The bottom row shows 7 light pink circles, representing negative counters.
1

$9 + (−4)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$−21 + (−59)$

Solution

−80

$−35 + (−47)$

$48 + (−16)$

Solution

32

$34 + (−19)$

$−200 + 65$

Solution

−135

$−150 + 45$

$2 + (−8) + 6$

Solution

0

$4 + (−9) + 7$

$−14 + (−12) + 4$

Solution

−22

$−17 + (−18) + 6$

$135 + (−110) + 83$

Solution

108

$140 + (−75) + 67$

$−32 + 24 + (−6) + 10$

Solution

−4

$−38 + 27 + (−8) + 12$

$19 + 2 (−3 + 8)$

Solution

29

$24 + 3 (−5 + 9)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

x + 8

when

ⓐ $x = −26$
ⓑ $x = −95$

Solution

ⓐ −18
ⓑ −87

y + 9

when

ⓐ $y = −29$
ⓑ $y = −84$

y + (−14)

when

ⓐ $y = −33$
ⓑ $y = 30$

Solution

ⓐ −47
ⓑ 16

x + (−21)

when

ⓐ $x = −27$
ⓑ $x = 44$

When

a = −7,

evaluate:

ⓐ $a + 3$
ⓑ $- a + 3$

Solution

ⓐ −4
ⓑ 10

When

b = −11,

evaluate:

ⓐ $b + 6$
ⓑ $- b + 6$

When

c = −9,

evaluate:

ⓐ $c + (−4)$
ⓑ $- c + (−4)$

Solution

ⓐ −13
ⓑ 5

When

d = −8,

evaluate:

ⓐ $d + (−9)$
ⓑ $- d + (−9)$

$m + n$ when, $m = −15$ , $n = 7$

Solution

−8

$p + q$ when, $p = −9$ , $q = 17$

$r −3 s$ when, $r = 16$ , $s = 2$

Solution

10

$2 t + u$ when, $t = −6$ , $u = −5$

${(a + b)}^{2}$ when, $a = −7$ , $b = 15$

Solution

64

${(c + d)}^{2}$ when, $c = −5$ , $d = 14$

${(x + y)}^{2}$ when, $x = −3$ , $y = 14$

Solution

121

${(y + z)}^{2}$ when, $y = −3$ , $z = 15$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

The sum of $−14$ and $5$

Solution

−14 + 5 = −9

The sum of $−22$ and $9$

$8$ more than $−2$

Solution

−2 + 8 = 6

$5$ more than $−1$

$−10$ added to $−15$

Solution

−15 + (−10) = −25

$−6$ added to $−20$

$6$ more than the sum of $−1$ and $−12$

Solution

[−1 + (−12)] + 6 = −7

$3$ more than the sum of $−2$ and $−8$

the sum of $10$ and $−19,$ increased by $4$

Solution

[10 + (−19)] + 4 = −5

the sum of $12$ and $−15,$ increased by $1$

Add Integers in Applications

In the following exercises, solve.

Temperature The temperature in St. Paul, Minnesota was $−19 °F$ at sunrise. By noon the temperature had risen $26°F.$ What was the temperature at noon?

Solution

7°F

Temperature The temperature in Chicago was $−15 °F$ at 6 am. By afternoon the temperature had risen $28°F.$ What was the afternoon temperature?

Credit Cards Lupe owes $$73$ on her credit card. Then she charges $$45$ more. What is the new balance?

Solution

−$118

Credit Cards Frank owes $$212$ on his credit card. Then he charges $$105$ more. What is the new balance?

Football A team lost $3 yards$ the first play. Then they lost $2 yards,$ gained $1 yard,$ and then lost $4 yards.$ What was the change in overall yardage over the four plays?

Solution

−8 yards

Card Games April lost $5 cards$ the first turn. Over the next three turns, she lost $3 cards,$ gained $2 cards,$ and then lost $1 card.$ What was the change in cards over the four turns?

Football The Rams took possession of the football on their own $35-yard line.$ In the next three plays, they lost $12 yards,$ gained $8 yards,$ then lost $6 yards.$ On what yard line was the ball at the end of those three plays?

Solution

25-yard line

Football The Cowboys began with the ball on their own $20-yard line.$ They gained $15 yards,$ lost $3 yards$ and then gained $6 yards$ on the next three plays. Where was the ball at the end of these plays?

Scuba Diving A scuba diver swimming $8 feet$ below the surface dove $17 feet$ deeper; the pressure got to them and they rose five feet. What is their new depth?

Solution

20 feet

Gas Consumption: Ozzie rode their motorcycle for $30 minutes,$ using $168 fluid ounces of gas.$ Then they stopped and got $140-fluid ounces of gas.$ Represent the change in gas amount as an integer.

Everyday Math

Stock Market The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:

$\begin{matrix} Monday & −504 & Tuesday & + 142 & Wednesday & −449 \\ Thursday & + 410 & Friday & + 369 \end{matrix}$

What was the overall change for the week?

Solution

−32

Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:

$\begin{matrix} Monday & −201 & Tuesday & −16 & Wednesday & −23 \\ Thursday & + 172 & Friday & −34 \end{matrix}$

What was the overall change for the week?

Writing Exercises

Explain why the sum of $−8$ and $2$ is negative, but the sum of $8$ and $−2$ and is positive.

Solution

Sample answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive.

Give an example from your life experience of adding two negative numbers.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment table for math students, allowing them to rate their confidence in skills such as modeling integer addition, simplifying expressions, evaluating variables, and translating word phrases.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Model Addition of Integers

Solution

Solution

Solution

Solution

Modeling Addition of Positive and Negative Integers

Simplify Expressions with Integers

Solution

Solution

Solution

Evaluate Variable Expressions with Integers

Solution

Solution

Solution

Solution

Translate Word Phrases to Algebraic Expressions

Solution

Solution

Add Integers in Applications

Solution

Solution

Key Concepts

Practice Makes Perfect

Everyday Math

Writing Exercises

Self Check