Subtracting Integers

Chapter 1: Variables, Expressions and Integers

Lesson 1.6: Subtracting Integers

Using additive inverses to subtract positive and negative integers

Learning Objectives

By the end of this lesson, you should be able to:

Subtract integers by adding the opposite of the number being subtracted.
Find and interpret changes in temperature, elevation, money, and other quantities.

Asking God for Wisdom

Subtracting integers can seem confusing when negative numbers are involved. We learn to slow down, rewrite the problem carefully, and follow a reliable rule. In the same way, God invites us to ask Him for wisdom when we do not understand something. Careful thinking and faithful effort help us grow.

Memory verse: “If any of you lacks wisdom, you should ask God, who gives generously to all without finding fault.” — James 1:5

1. Subtracting an Integer Means Adding Its Opposite

To subtract an integer, add its opposite.

$a-b=a+(-b)$

The first number stays the same. Change the subtraction sign to addition, and change the sign of the second number.

Example 1: Find $3-7$ .

The opposite of 7 is -7.

$3-7=3+(-7)=-4$

Example 2: Find $5-(-2)$ .

The opposite of -2 is +2.

$5-(-2)=5+2=7$

Picture space for subtracting integers on a number line — Rewrite subtraction as addition of the opposite before moving on a number line.

2. Examples with Positive and Negative Integers

Example 3: Find $-4-6$ .

The opposite of 6 is -6.

$-4-6=-4+(-6)=-10$

Example 4: Find $-11-(-8)$ .

The opposite of -8 is +8.

$-11-(-8)=-11+8=-3$

Remember: Subtracting a positive number moves you left on a number line. Subtracting a negative number is the same as adding a positive number, so it moves you right.

3. Finding the Change in a Quantity

To find how much a quantity has changed, subtract the original value from the final value.

$\text{Change}=\text{final value}-\text{original value}$

A positive change means the quantity increased. A negative change means the quantity decreased.

Worked Example: Kick-’em-Jenny Volcano

The summit elevation of Kick-’em-Jenny was -235 metres in 1962 and -182 metres in 2002. Find the change in elevation.

$\text{Change}=-182-(-235)$

$=-182+235=53$

The elevation increased by 53 metres.

Picture space for changes in temperature and elevation — Learning for a happy Life

Common Mistake to Avoid

Do not change the sign of the first number. In $-12-(-5)$ , only the second number changes when subtraction is rewritten:

$-12-(-5)=-12+5=-7$

Practice Questions

Rewrite subtraction as addition of the opposite when helpful. Show important steps in the application problems.

1. Skill Practice

A. Rewrite each subtraction problem as addition. Then find the difference.

$8-12$
$-5-7$
$9-(-4)$
$-6-(-10)$
$0-15$
$-20-4$

B. Find each difference.

$-41+30$
$8-9$
$1-(-8)$
$-10-6$
$-5-(-17)$
$24-41$
$-39-32$
$79-(-98)$
$-86-(-34)$
$-20-4$

C. Evaluate each expression when $m=-6$ .

$17-m$
$4-m$
$m-7$
$-16-m$
$m-19$
$m-3-10$
$20-m-5$
$14-30-m$

D. Find the change.

From $-15^\circ\text{C}$ to $10^\circ\text{C}$
From $-5^\circ\text{F}$ to $-13^\circ\text{F}$
From -120 feet to -90 feet
From 30 metres to -70 metres

2. Applications

Volcano elevation: The summit elevation of an underwater volcano was -235 metres in 1962 and -182 metres in 2002. Find the change in elevation. Did the elevation increase or decrease?
Temperature: The temperature of a laboratory sample changed from $-18^\circ\text{C}$ to $90^\circ\text{C}$ . Find the change in temperature.
Daily temperatures: The temperature at sunrise was . By noon it was , and by midnight it was .
1. What was the change from sunrise to noon?
2. What was the change from noon to midnight?
Ice cream production: The temperatures at four stages of production are shown below.

Stage Temperature (°C)

Pasteurization 80

Aging -5

Hardening -40

Storage -15
1. Find the change between each pair of consecutive stages.
2. Find the absolute value of each change.
3. Between which consecutive stages was the absolute temperature change greatest?

Stage	Temperature (°C)
Pasteurization	80
Aging	-5
Hardening	-40
Storage	-15

Bank account: A bank account has a balance of -€28. Later, the balance is €17. What was the change in the account balance?
Elevation: A hiker moves from an elevation of -45 metres to an elevation of 125 metres. What is the change in elevation?
Challenge: Find a number $n$ such that $n-(-9)=15$ . Explain how you found $n$ .

Answers

Show answers: Skill Practice

$8+(-12)=-4$
$-5+(-7)=-12$
$9+4=13$
$-6+10=4$
$0+(-15)=-15$
$-20+(-4)=-24$
-11
-1
9
-16
12
-17
-71
177
-52
-24
23
10
-13
-10
-25
-19
21
-10
25°C
-8°F
30 feet
-100 metres

Show answers: Applications

$-182-(-235)=-182+235=53$ ; the elevation increased by 53 metres.
$90-(-18)=90+18=108^\circ\text{C}$ ; the temperature increased by 108°C.
a. $4-(-12)=16^\circ\text{C}$ ; b. $-3-4=-7^\circ\text{C}$ .
a. Pasteurization to aging: $-5-80=-85^\circ\text{C}$ ; aging to hardening: $-40-(-5)=-35^\circ\text{C}$ ; hardening to storage: $-15-(-40)=25^\circ\text{C}$ . b. The absolute changes are 85°C, 35°C, and 25°C. c. The greatest change was from pasteurization to aging.
$17-(-28)=45$ ; the balance increased by €45.
$125-(-45)=170$ metres.
$n+9=15$ , so $n=6$ .