2.6: Subtract Integers (Part 2)

Example \(\PageIndex{9}\): simplify

Simplify: \(−74 − (−58)\).

Solution

We are taking 58 negatives away from 74 negatives.	−74 − (−58)
Subtract.	−16

Exercise \(\PageIndex{17}\)

Simplify the expression: \(−67 − (−38)\)

Answer

\(-29\)

Exercise \(\PageIndex{18}\)

Simplify the expression: \(−83 − (−57)\)

Answer

\(-26\)

Example \(\PageIndex{10}\)

Simplify: \(7 − (−4 − 3) − 9\).

Solution

We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right.

Simplify inside the parentheses first.	7 − (−7) − 9
Subtract from left to right.	14 − 9
Subtract.	5

Exercise \(\PageIndex{19}\)

Simplify the expression: \(8 − (−3 − 1) − 9\)

Answer

\(3\)

Exercise \(\PageIndex{20}\)

Simplify the expression: \(12 − (−9 − 6) − 14\)

Answer

\(13\)

Example \(\PageIndex{11}\): simplify

Simplify: \(3 • 7 − 4 • 7 − 5 • 8\).

Solution

We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right.

Multiply first.	21 - 28 - 40
Subtract from left to right.	-7 - 40
Subtract.	-47

Exercise \(\PageIndex{21}\)

Simplify the expression: \(6 • 2 − 9 • 1 − 8 • 9\)

Answer

\(-69\)

Exercise \(\PageIndex{22}\)

Simplify the expression: \(2 • 5 − 3 • 7 − 4 • 9\)

Answer

\(-47\)

Evaluate Variable Expressions with Integers

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

Example \(\PageIndex{12}\): evaluate

Evaluate \(x − 4\) when

1. \(x = 3\)
2. \(x = −6\)

Solution

1. To evaluate \(x − 4\) when \(x = 3\), substitute \(3\) for \(x\) in the expression.

Substitute \(\textcolor{red}{3}\) for x.	\(\textcolor{red}{3} - 4\)
Subtract.	\(-1\)

2. To evaluate \(x − 4\) when \(x = −6\), substitute \(−6\) for \(x\) in the expression.

Substitute \(\textcolor{red}{-6}\) for x.	\(\textcolor{red}{-6} - 4\)
Subtract.	\(-10\)

Exercise \(\PageIndex{23}\)

Evaluate each expression: \(y − 7\) when

1. \(y = 5\)
2. \(y = −8\)

Answer a

\(-2\)

Answer b

\(-15\)

Exercise \(\PageIndex{24}\)

Evaluate each expression: \(m − 3\) when

1. \(m = 1\)
2. \(m = −4\)

Answer a

\(-2\)

Answer b

\(-7\)

Example \(\PageIndex{13}\): evaluate

Evaluate \(20 − z\) when

1. \(z = 12\)
2. \(z = − 12\)

Solution

1. To evaluate \(20 − z\) when \(z = 12\), substitute \(12\) for \(z\) in the expression.

Substitute \(\textcolor{red}{12}\) for z.	\(20 - \textcolor{red}{12}\)
Subtract.	\(8\)

2. To evaluate \(20 − z\) when \(z = −12\), substitute \(−12\) for \(z\) in the expression.

Substitute \(\textcolor{red}{-12}\) for z.	\(20 - (\textcolor{red}{-12})\)
Subtract.	\(32\)

Exercise \(\PageIndex{25}\)

Evaluate each expression: \(17 − k\) when

1. \(k = 19\)
2. \(k = −19\)

Answer a

\(-2\)

Answer b

\(36\)

Exercise \(\PageIndex{26}\)

Evaluate each expression: \(−5 − b\) when

1. \(b = 14\)
2. \(b = −14\)

Answer a

\(-19\)

Answer b

\(9\)

Translate Word Phrases to Algebraic Expressions

When we first introduced the operation symbols, we saw that the expression \(a − b\) may be read in several ways as shown below.

Figure \(\PageIndex{4}\)

Be careful to get \(a\) and \(b\) in the right order!

Example \(\PageIndex{14}\): translate and simplify

Translate and then simplify:

1. the difference of \(13\) and \(−21\)
2. subtract \(24\) from \(−19\)

Solution

1. A difference means subtraction. Subtract the numbers in the order they are given.

Translate.	13 - (-21)
Simplify	34

2. Subtract means to take \(24\) away from \(−19\).

Translate.	-19 - 24
Simplify	-43

Exercise \(\PageIndex{27}\)

Translate and simplify:

1. the difference of \(14\) and \(−23\)
2. subtract \(21\) from \(−17\)

Answer a

\(-14-(-23)=37\)

Answer b

\(-17-21=-38\)

Exercise \(\PageIndex{28}\)

Translate and simplify:

1. the difference of \(11\) and \(−19\)
2. subtract \(18\) from \(−11\)

Answer a

\(11-(-19)=30\)

Answer b

\(-11-18=-29\)

Subtract Integers in Applications

It’s hard to find something if we don’t know what we’re looking for or what to call it. So when we solve an application problem, we first need to determine what we are asked to find. Then we can write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

HOW TO: SOLVE APPLICATION PROBLEMS.

Step 1. Identify what you are asked to find.

Step 2. Write a phrase that gives the information to find it.

Step 3. Translate the phrase to an expression.

Step 4. Simplify the expression.

Step 5. Answer the question with a complete sentence.

Example \(\PageIndex{15}\): simplify

The temperature in Urbana, Illinois one morning was \(11\) degrees Fahrenheit. By mid-afternoon, the temperature had dropped to \(−9\) degrees Fahrenheit. What was the difference between the morning and afternoon temperatures?

Solution

Step 1. Identify what we are asked to find.	the difference between the morning and afternoon temperatures
Step 2. Write a phrase that gives the information to find it.	the difference of 11 and −9
Step 3. Translate the phrase to an expression. The word difference indicates subtraction.	11 − (−9)
Step 4. Simplify the expression.	20
Step 5. Write a complete sentence that answers the question.	The difference in temperature was 20 degrees Fahrenheit.

Exercise \(\PageIndex{29}\)

The temperature in Anchorage, Alaska one morning was \(15\) degrees Fahrenheit. By mid-afternoon the temperature had dropped to \(30\) degrees below zero. What was the difference between the morning and afternoon temperatures?

Answer

\(45\) degrees Fahrenheit

Exercise \(\PageIndex{30}\)

The temperature in Denver was \(−6\) degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to \(−15\) degree Fahrenheit. What was the difference between the lunchtime and sunset temperatures?

Answer

\(9\) degrees Fahrenheit

Geography provides another application of negative numbers with the elevations of places below sea level.

Example \(\PageIndex{16}\): simplify

Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is \(14,497\) feet above sea level and the elevation of Death Valley is \(282\) feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley?

Solution

Step 1. What are we asked to find?	The difference in elevation between Mt. Whitney and Death Valley
Step 2. Write a phrase.	elevation of Mt. Whitney−elevation of Death Valley
Step 3. Translate.	14,497 − (−282)
Step 4. Simplify.	14,779
Step 5. Write a complete sentence that answers the question.	The difference in elevation is 14,779 feet.

Exercise \(\PageIndex{31}\)

One day, John hiked to the \(10,023\) foot summit of Haleakala volcano in Hawaii. The next day, while scuba diving, he dove to a cave \(80\) feet below sea level. What is the difference between the elevation of the summit of Haleakala and the depth of the cave?

Answer

\(10,103\) feet

Exercise \(\PageIndex{32}\)

The submarine Nautilus is at \(340\) feet below the surface of the water and the submarine Explorer is \(573\) feet below the surface of the water. What is the difference in the position of the Nautilus and the Explorer?

Answer

\(233\) feet

Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)

Example \(\PageIndex{17}\): simplify

Leslie has \(\$25\) in her checking account and she writes a check for \($8\).

1. What is the balance after she writes the check?
2. She writes a second check for \(\$20\). What is the new balance after this check?
3. Leslie’s friend told her that she had lost a check for \(\$10\) that Leslie had given her with her birthday card. What is the balance in Leslie’s checking account now?

Solution

What are we asked to find?	The balance of the account
Write a phrase.	$25 minus $8
Translate.	$25 - $8
Simplify.	$17
Write a sentence answer.	The balance is $17.

What are we asked to find?	The new balance
Write a phrase.	$17 minus $20
Translate.	$17 - $20
Simplify.	-$3
Write a sentence answer.	She is overdrawn by $3.

What are we asked to find?	The new balance
Write a phrase.	$10 more than −$3
Translate.	-$3 + $10
Simplify.	$7
Write a sentence answer.	The balance is now $7.

Exercise \(\PageIndex{33}\)

Araceli has \(\$75\) in her checking account and writes a check for \(\$27\).

1. What is the balance after she writes the check?
2. She writes a second check for \(\$50\). What is the new balance?
3. The check for \(\$20\) that she sent a charity was never cashed. What is the balance in Araceli’s checking account now?

Answer a

\(\$48\)

Answer b

\(-\$2\)

Answer c

\(\$18\)

Exercise \(\PageIndex{34}\)

Genevieve’s bank account was overdrawn and the balance is \(−\$78\).

1. She deposits a check for \(\$24\) that she earned babysitting. What is the new balance?
2. She deposits another check for \(\$49\). Is she out of debt yet? What is her new balance?

Answer a

\(-\$54\)

Answer b

No, \(-\$5\)

Access Additional Online Resources

• Adding and Subtracting Integers
• Subtracting Integers with Color Counters
• Subtracting Integers Basics
• Subtracting Integers
• Integer Application

Key Concepts

• Subtraction of Integers

5–3	–5–(–3)
2	–2
2 positives	2 negatives
When there would be enough counters of the color to take away, subtract.
–5–3	5–(–3)
–8	8
5 negatives, want to subtract 3 positives	5 positives, want to subtract 3 negatives
need neutral pairs	need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.

• Subtraction Property
  • \(a-b = a+(-b)\)
  • \(a-(-b) = a+b\)
• Solve Application Problems
  • Step 1. Identify what you are asked to find.
  • Step 2. Write a phrase that gives the information to find it.
  • Step 3. Translate the phrase to an expression.
  • Step 4. Simplify the expression.
  • Step 5. Answer the question with a complete sentence.

Practice Makes Perfect

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

127. 8 − 2
128. 9 − 3
129. −5 − (−1)
130. −6 − (−4)
131. −5 − 4
132. −7 − 2
133. 8 − (−4)
134. 7 − (−3)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

135. (a) 15 − 6 (b) 15 + (−6)
136. (a) 12 − 9 (b) 12 + (−9)
137. (a) 44 − 28 (b) 44 + (−28)
138. (a) 35 − 16 (b) 35 + (−16)
139. (a) 8 − (−9) (b) 8 + 9
140. (a) 4 − (−4) (b) 4 + 4
141. (a) 27 − (−18) (b) 27 + 18
142. (a) 46 − (−37) (b) 46 + 37

In the following exercises, simplify each expression.

143. 15 − (−12)
144. 14 − (−11)
145. 10 − (−19)
146. 11 − (−18)
147. 48 − 87
148. 45 − 69
149. 31 − 79
150. 39 − 81
151. −31 − 11
152. −32 − 18
153. −17 − 42
154. −19 − 46
155. −103 − (−52)
156. −105 − (−68)
157. −45 − (−54)
158. −58 − (−67)
159. 8 − 3 − 7
160. 9 − 6 − 5
161. −5 − 4 + 7
162. −3 − 8 + 4
163. −14 − (−27) + 9
164. −15 − (−28) + 5
165. 71 + (−10) − 8
166. 64 + (−17) − 9
167. −16 − (−4 + 1) − 7
168. −15 − (−6 + 4) − 3
169. (2 − 7) − (3 − 8)
170. (1 − 8) − (2 − 9)
171. −(6 − 8) − (2 − 4)
172. −(4 − 5) − (7 − 8)
173. 25 − [10 − (3 − 12)]
174. 32 − [5 − (15 − 20)]
175. 6.3 − 4.3 − 7.2
176. 5.7 − 8.2 − 4.9
177. 5² − 6²
178. 6² − 7²

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression for the given values.

179. x − 6 when (a) x = 3 (b) x = −3
180. x − 4 when (a) x = 5 (b) x = −5
181. 5 − y when (a) y = 2 (b) y = −2
182. 8 − y when (a) y = 3 (b) y = −3
183. 4x² − 15x + 1 when x = 3
184. 5x² − 14x + 7 when x = 2
185. −12 − 5x² when x = 6
186. −19 − 4x² when x = 5

Translate Word Phrases to Algebraic Expressions

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

187. (a) The difference of 3 and −10 (b) Subtract −20 from 45
188. (a) The difference of 8 and −12 (b) Subtract −13 from 50
189. (a) The difference of −6 and 9 (b) Subtract −12 from −16
190. (a) The difference of −8 and 9 (b) Subtract −15 from −19
191. (a) 8 less than −17 (b) −24 minus 37
192. (a) 5 less than −14 (b) −13 minus 42
193. (a) 21 less than 6 (b) 31 subtracted from −19
194. (a) 34 less than 7 (b) 29 subtracted from −50

Subtract Integers in Applications

In the following exercises, solve the following applications.

195. Temperature One morning, the temperature in Urbana, Illinois, was 28° Fahrenheit. By evening, the temperature had dropped 38° Fahrenheit. What was the temperature that evening?
196. Temperature On Thursday, the temperature in Spincich Lake, Michigan, was 22° Fahrenheit. By Friday, the temperature had dropped 35° Fahrenheit. What was the temperature on Friday?
197. Temperature On January 15, the high temperature in Anaheim, California, was 84° Fahrenheit. That same day, the high temperature in Embarrass, Minnesota was −12° Fahrenheit. What was the difference between the temperature in Anaheim and the temperature in Embarrass?
198. Temperature On January 21, the high temperature in Palm Springs, California, was 89°, and the high temperature in Whitefield, New Hampshire was −31°. What was the difference between the temperature in Palm Springs and the temperature in Whitefield?
199. Football At the first down, the Warriors football team had the ball on their 30-yard line. On the next three downs, they gained 2 yards, lost 7 yards, and lost 4 yards. What was the yard line at the end of the third down?
200. Football At the first down, the Barons football team had the ball on their 20-yard line. On the next three downs, they lost 8 yards, gained 5 yards, and lost 6 yards. What was the yard line at the end of the third down?
201. Checking Account John has $148 in his checking account. He writes a check for $83. What is the new balance in his checking account?
202. Checking Account Ellie has $426 in her checking account. She writes a check for $152. What is the new balance in her checking account?
203. Checking Account Gina has $210 in her checking account. She writes a check for $250. What is the new balance in her checking account?
204. Checking Account Frank has $94 in his checking account. He writes a check for $110. What is the new balance in his checking account?
205. Checking Account Bill has a balance of −$14 in his checking account. He deposits $40 to the account. What is the new balance?
206. Checking Account Patty has a balance of −$23 in her checking account. She deposits $80 to the account. What is the new balance?

Everyday Math

207. Camping Rene is on an Alpine hike. The temperature is −7°. Rene’s sleeping bag is rated “comfortable to −20°”. How much can the temperature change before it is too cold for Rene’s sleeping bag?
208. Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to −100 feet. She is diving at −45 feet on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

Writing Exercises

209. Explain why the difference of 9 and −6 is 15.
210. Why is the result of subtracting 3 − (−4) the same as the result of adding 3 + 4?

Self Check

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

(b) What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Contributors and Attributions

• Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.