The topics covered in this section are:

## 3.2.1 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 2−1.2−1. 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 the figure below. 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 $3,-3$.

• $5+3$
• $-5+(-3)$
• $-5+3$
• $5+(-3)$

#### Example 1

Model: $5+3$

Solution

#### Example 2

Model: $5+(-3)$.

Solution

Examples 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.

#### Example 3

Modle: $-5+3$

Solution

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

#### Example 4

Model: $5+(-3)$.

Solution

#### Example 5

##### Modeling Addition of Positive and Negative Integers

1. $4+2$
2. $-3+6$
3. $4+(-5)$
4. $-2+(-3)$
Solution

## 3.2.2 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 $−101$.

$-74+(-27)=-101$

Loog again at the results of Example 1 through Example 4.

 $5+3$ $-53(-3)$ 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.

#### Example 6

Simplify:

1. $19+(-47)$
2. $-32+40$
Solution

Part 1. Since the signs are different, we subtract $19$ from $47$. The answer will be negative because there are more negatives than positives.

$19+(-47)$

$-28$

Part 2. The signs are different so we subtract $32$ from $40$. The answer will be positive because there are more positives than negatives.

$-32+40$

$8$

#### Example 7

Simplify:

$-14+(-36)$.

Solution

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

$-14+(-36)$

$-50$

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

#### Example 8

Simplify: $-5+3(-2+7)$.

Solution

## 3.2.3 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.

#### Example 9

Evaluate $x+7$ when

1. $x=-2$
2. $x=-11$.
Solution

#### Example 10

When $n=-5$, evaluate

1. $n+1$
2. $-n+1$.
Solution

Next we’ll evaluate an expression with two variables.

#### Example 11

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

Solution

#### Example 12

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

Solution

This expression has two variables. Substitute $-18$ for $x$ and $24$ for $y$.

## 3.2.4 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.

#### Example 13

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

Solution

#### Example 14

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

Solution

The phrase increased by indicates addition.

## 3.2.5 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.

#### Example 15

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.

#### Example 16

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.