# 3.3: Subtract Integers (Part 2)

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Example 3.38:

Simplify: −74 − (−58).

##### Solution

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

Subtract. | −16 |

Exercise 3.75:

Simplify the expression: −67 − (−38)

Exercise 3.76:

Simplify the expression: −83 − (−57)

Example 3.39:

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 3.77:

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

Exercise 3.78:

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

Example 3.40:

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 3.79:

Simplify the expression: 6 • 2 − 9 • 1 − 8 • 9

Exercise 3.80:

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

### Evaluate Variable Expressions with Integers

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

Evaluate x − 4 when (a) x = 3 (b) x = −6.

##### Solution

(a) 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. | $$-1$$ |

(b) 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 3.81:

Evaluate each expression: y − 7 when (a) y = 5 (b) y = −8

Exercise 3.82:

Evaluate each expression: m − 3 when (a) m = 1 (b) m = −4

Example 3.42:

Evaluate 20 − z when (a) z = 12 (b) z = − 12

##### Solution

(a) 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. | $$8$$ |

(b) 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 3.83:

Evaluate each expression: 17 − k when (a) k = 19 (b) k = −19

Exercise 3.84:

Evaluate each expression: −5 − b when (a) b = 14 (b) b = −14

### 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.

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

Example 3.43:

Translate and then simplify: (a) the difference of 13 and −21 (b) subtract 24 from −19

##### Solution

(a) A *difference *means subtraction. Subtract the numbers in the order they are given.

Translate. | 13 - (-21) |

Simplify. | 34 |

(b) *Subtract *means to take 24 away from −19.

Translate. | -19 - 24 |

Simplify. | -43 |

Exercise 3.85:

Translate and simplify: (a) the difference of 14 and −23 (b) subtract 21 from −17

Exercise 3.86:

Translate and simplify: (a) the difference of 11 and −19 (b) subtract 18 from −11

### 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 3.44:

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 3.87:

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?

Exercise 3.88:

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?

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

Example 3.45:

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 3.89:

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?

Exercise 3.90:

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?

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 3.46:

Leslie has $25 in her checking account and she writes a check for $8. (a) What is the balance after she writes the check? (b) She writes a second check for $20. What is the new balance after this check? (c) 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

(a)

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. |

(b)

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. |

(c)

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 3.91:

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

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

Exercise 3.92:

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

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

ACCESS ADDITIONAL ONLINE RESOURCES

Adding and Subtracting Integers

### Practice Makes Perfect

Model Subtraction of Integers

In the following exercises, model each expression and simplify.

- 8 − 2
- 9 − 3
- −5 − (−1)
- −6 − (−4)
- −5 − 4
- −7 − 2
- 8 − (−4)
- 7 − (−3)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

- (a) 15 − 6 (b) 15 + (−6)
- (a) 12 − 9 (b) 12 + (−9)
- (a) 44 − 28 (b) 44 + (−28)
- (a) 35 − 16 (b) 35 + (−16)
- (a) 8 − (−9) (b) 8 + 9
- (a) 4 − (−4) (b) 4 + 4
- (a) 27 − (−18) (b) 27 + 18
- (a) 46 − (−37) (b) 46 + 37

In the following exercises, simplify each expression.

- 15 − (−12)
- 14 − (−11)
- 10 − (−19)
- 11 − (−18)
- 48 − 87
- 45 − 69
- 31 − 79
- 39 − 81
- −31 − 11
- −32 − 18
- −17 − 42
- −19 − 46
- −103 − (−52)
- −105 − (−68)
- −45 − (−54)
- −58 − (−67)
- 8 − 3 − 7
- 9 − 6 − 5
- −5 − 4 + 7
- −3 − 8 + 4
- −14 − (−27) + 9
- −15 − (−28) + 5
- 71 + (−10) − 8
- 64 + (−17) − 9
- −16 − (−4 + 1) − 7
- −15 − (−6 + 4) − 3
- (2 − 7) − (3 − 8)
- (1 − 8) − (2 − 9)
- −(6 − 8) − (2 − 4)
- −(4 − 5) − (7 − 8)
- 25 − [10 − (3 − 12)]
- 32 − [5 − (15 − 20)]
- 6 • 3 − 4 • 3 − 7 • 2
- 5 • 7 − 8 • 2 − 4 • 9
- 5
^{2}− 6^{2} - 6
^{2}− 7^{2}

#### Evaluate Variable Expressions with Integers

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

- x − 6 when (a) x = 3 (b) x = −3
- x − 4 when (a) x = 5 (b) x = −5
- 5 − y when (a) y = 2 (b) y = −2
- 8 − y when (a) y = 3 (b) y = −3
- 4x
^{2}− 15x + 1 when x = 3 - 5x
^{2}− 14x + 7 when x = 2 - −12 − 5x
^{2}when x = 6 - −19 − 4x
^{2}when x = 5

#### Translate Word Phrases to Algebraic Expressions

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

- (a) The difference of 3 and −10 (b) Subtract −20 from 45
- (a) The difference of 8 and −12 (b) Subtract −13 from 50
- (a) The difference of −6 and 9 (b) Subtract −12 from −16
- (a) The difference of −8 and 9 (b) Subtract −15 from −19
- (a) 8 less than −17 (b) −24 minus 37
- (a) 5 less than −14 (b) −13 minus 42
- (a) 21 less than 6 (b) 31 subtracted from −19
- (a) 34 less than 7 (b) 29 subtracted from −50

#### Subtract Integers in Applications

In the following exercises, solve the following applications.

**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?**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?**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?**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?**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?**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?**Checking Account**John has $148 in his checking account. He writes a check for $83. What is the new balance in his checking account?**Checking Account**Ellie has $426 in her checking account. She writes a check for $152. What is the new balance in her checking account?**Checking Account**Gina has $210 in her checking account. She writes a check for $250. What is the new balance in her checking account?**Checking Account**Frank has $94 in his checking account. He writes a check for $110. What is the new balance in his checking account?**Checking Account**Bill has a balance of −$14 in his checking account. He deposits $40 to the account. What is the new balance?**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

**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?**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

- Explain why the difference of 9 and −6 is 15.
- 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?