By the end of this section, you will be able to:

- Model addition of integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate word phrases to algebraic expressions
- Add integers in applications

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 2−1.2−1. 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.

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 3.17.

**Figure 3.17** A blue counter represents +1.+1. A red counter represents −1.−1. Together they add to zero.

Example 3.14 and Example 3.15 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—either8 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.

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 3.14 – Example 3.17.

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. |

**Table3.1** Addition of Positive and Negative Integers

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

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.

Next we’ll evaluate an expression with two variables.

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.

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$5 could be represented as −$5.−$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.

**Model Addition of Integers**

In the following exercises, model the expression to simplify.

- 7+4
- 8+5
- −6+(−3)
- −5+(−5)
- −7+5
- −9+6
- 8+(−7)
- 9+(−4)

Simplify Expressions with Integers

In the following exercises, simplify each expression.

- −21+(−59)
- −35+(−47)
- 48+(−16)
- 34+(−19)
- −200+65
- −150+45
- 2+(−8)+6
- 4+(−9)+7
- −14+(−12)+4
- −17+(−18)+6
- 135+(−110)+83
- 140+(−75)+67
- −32+24+(−6)+10
- −38+27+(−8)+12
- 19+2(−3+8)
- 24+3(−5+9)

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

- x+8 when

ⓐx=−26

ⓑx=−95 - y+9 when

ⓐy=−29

ⓑy=−84 - y+(−14) when

ⓐy=−33

ⓑy=30 - x+(−21) when

ⓐx=−27

ⓑx=44 - When a=−7, evaluate:

ⓐa+3

ⓑ−a+3 - When b=−11, evaluate:

ⓐb+6

ⓑ−b+6 - When c=−9, evaluate:

ⓐc+(−4)

ⓑ−c+(−4) - When d=−8, evaluate:

ⓐd+(−9)

ⓑ−d+(−9) - m+n when, m=−15, n=7
- p+q when, p=−9, q=17
- r−3s when, r=16, s=2
- 2t+u when, t=−6, u=−5
- (a+b)2 when, a=−7, b=15
- (c+d)2 when, c=−5, d=14
- (x+y)2 when, x=−3, y=14
- (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
- The sum of −22 and 9
- 8 more than −2
- 5 more than −1
- −10 added to −15
- −6 added to −20
- 6 more than the sum of −1 and −12
- 3 more than the sum of −2 and −8
- the sum of 10 and −19, increased by 4
- 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?
- 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?
- 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?
- 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?
- 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?
- 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:

MondayThursday−504+410TuesdayFriday+142+369Wednesday−449

What was the overall change for the week?

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

MondayThursday−201+172TuesdayFriday−16−34Wednesday−23

What was the overall change for the week?

**Writing Exercises**

125.

Explain why the sum of −8−8 and 22 is negative, but the sum of 88 and −2−2 and is positive.126.

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

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

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

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