Add Integers

3.2 Add Integers

The topics covered in this section are:

  1. Model addition of integers
  2. Simplify expressions with integers
  3. Evaluate variable expressions with integers
  4. Translate word phrases to algebraic expressions
  5. Add integers in applications

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.

This figure has a blue circle labeled positive and a red circle labeled 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.

This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.
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

Interpret the expression.$5+3$ means the sum of $5$ and $3$.
Model the first number. Start with $5$ positives..
Model the second number. Add $3$ positives..
Count the total number of counters..
The sum of $5$ and $3$ is $8$.$5+3=8$

Example 2

Model: $5+(-3)$.

Solution

Interpret the expression.$-5+(-3)$ means the sum of $-5$ and $-3$.
Model the first number. Start with $5$ negatives..
Model the second number. Add $3$ negatives..
Count the total number of counters..
The sum of $-5$ and $-3$ is $-8$.$-5+-3=-8$

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

Interpret the expression.$-5+3$ means the sum of $-5$ and $3$.
Model the first number. Start with $5$ negatives..
Model the second number. Add $3$ positives..
Remove any neutral pairs..
Count the result..
The sum of $-5$ and $3$ is $-2$.$-5+3=-2$

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

Example 4

Model: $5+(-3)$.

Solution

Interpret the expression.$5+(-3)$ means the sum of $5$ and $-3$.
Model the first number. Start with $5$ positives..
Model the second number. Add $3$ negatives..
Remove any neutral pairs..
Count the result..
The su of $5$ and $-3$ is $2$.$5+(-3)=2$

Example 5

Modeling Addition of Positive and Negative Integers

Model each addition.

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

Part 1.$4+2$
Start with $4$ positives..
Add two positives..
How many do you have?$4+2=6$
Part 2.$-3+6$
Start with $3$ negatives..
Add $6$ positives..
Remove neutral pairs..
How many are left?.
$3$$-3+6=3$
Part 3.$4+(-5)
Start with $4$ positives..
Add $5$ negatives..
Remove neutral pairs..
How many are left?.
$-1$$4+(-5)=-1$
Part 4.$-2+(-3)$
Start with $2$ negatives..
Add $3$ negatives..
How many do you have?$-2+(-3)=-5$

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$$-53(-3)$
both positive, sum positiveboth negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
$-5+3$$5+(-3)$
different signs, more negativesdifferent signs, more positives
Sum negativesum 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

$-5+3(-2+7)$
Simplify inside the parentheses.$-5+3(5)$
Multiply.$-5+15$
Add left to right.$10$

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

Part 1. Evaluate $x+7$ when $x=-2$$-2+7$
Substitute $-2$ for $x$.$-2+7$
Simplify.$5$
Part 2. Evaluate $x+7$ when $x=-11$$x+7$
Substitute $-11$ for $x$.$-11+7$
Simplify.$-4$

Example 10

When $n=-5$, evaluate

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

Part 1. Evaluate $n+1$ when $n=-5$$n+1$
Substitute $-5$ for $n$.$-5+1$
Simplify.$-4$
Part 2. Evaluate $-n+1$ when $n=5$$-n+1$
Substitute $-5$ for $n$.$-(-5)+1$
Simplify.$5+1$
Add.$6$

Next we’ll evaluate an expression with two variables.

Example 11

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

Solution

$3a+b$
Substitute $12$ for $a$ and $-30$ for $b$.$3(12)+(-30)$
Multiply.$36+(-30)$
Add.$6$

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 $y$.

$(x+y)^{2}$
Substitute $-18$ for $x$ and $24$ for $y$.$(-18+24)^{2}$
Add inside the parentheses.$(6)^{2}$
Simplify.$36$

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

The sum of $-9$ and $5$ indiciates addition.the sum of $-9$ and $5$
Translate.$-9+5$
Simplify.$-4$

Example 14

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

Solution

The phrase increased by indicates addition.

The sum of $8$ and $-12$, increased by $3$
Translate.$[8+(-12)]+3$
Simplify.$-4+3$
Add.$-1$

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.

Write a phrase for the temperature.The temperature warmed up to $12$ degrees from $7$ degrees below zero.
Translate to math notation.$-7+12$
Simplify.$5$
Write a sentence to answer the question.The temperature at noon was $5$ degrees Fahrenheit.

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.

Write a word phrase for the position of the ball.Start at $42$, then lose $6$, gain $4$, lose $8$.
Translate to math notation.$42-6+4-8$
Simplify.$32$
Write a sentence to answer the question.At the end of the three plays, the ball is on the $32$-yard line.
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