3.2 Add Integers

Learning objectives

Learning objectives

• Model addition of integers

• Simplify expressions with integers

• Evaluate variable expressions with integers

• Translate word phrases to algebraic expressions

• Add integers in applications

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 $2-1.$ 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.

We will model four addition facts using the numbers $5,-5\,\mathrm{and}\,3\,,-3.$

Example 3.14

Solution

1. Model the first addend. Draw five blue counters to represent $5$, matching the first figure.
2. Model the second addend. Draw a separate group of three blue counters to represent $3$, matching the second figure.
3. Combine the groups. Place all five counters and all three counters together in one row.
4. Count the total. The combined row contains $5+3=8$ blue counters, as shown in the final figure.
5. State the result. The model shows that $5+3=8$.

Takeaway: When positive counters represent both addends, combine the groups and count the total number of counters.

Example 3.15

Solution

1. Model the first addend. Draw five red counters to represent $-5$, matching the first figure.
2. Model the second addend. Draw a separate group of three red counters to represent $-3$, as in the second figure.
3. Combine the groups. Arrange all five red counters and all three red counters in one row.
4. Count the total. The row contains eight red counters, representing eight negatives as shown in the final figure.
5. State the result. The model shows that $-5+(-3)=-8$.

Takeaway: When negative counters represent both addends, the combined group represents the negative of the total count.

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 $\mathrm{8---either}\,8$ 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 3.16

Solution

1. Model the negative addend. Draw five red counters to represent $-5$, as shown in the first figure.
2. Model the positive addend. Draw three blue counters to represent $3$ above the red row, matching the second figure.
3. Form neutral pairs. Match each of the three blue counters with one red counter, creating three zero pairs.
4. Count what remains. Two unmatched red counters remain, representing $-2$ in the final figure.
5. State the result. There were more negative counters than positive counters, so $-5+3=-2$.

Takeaway: Pair opposite-sign counters first, then use the sign of the unmatched counters to determine the sum.

Example 3.17

Solution

1. Model the positive addend. Draw five blue counters to represent $5$, as shown in the first figure.
2. Model the negative addend. Draw three red counters below them to represent $-3$, matching the second figure.
3. Form neutral pairs. Match each red counter with one blue counter, producing three zero pairs.
4. Count what remains. Two blue counters remain unmatched, as shown in the final figure.
5. State the result. The unmatched counters are positive, so $5+(-3)=2$.

Takeaway: After pairing opposite-sign counters, the unmatched counters determine the magnitude and sign of the sum.

Example 3.18

Solution

1. Model $4+2$. Draw four blue counters and two more blue counters; combining them gives six blue counters, so $4+2=6$.
2. Model $-3+6$. Draw three red counters and six blue counters. Pair three red and blue counters as neutral pairs, leaving three blue counters, so $-3+6=3$.
3. Model $4+(-5)$. Draw four blue counters and five red counters. Pair four of each, leaving one red counter, so $4+(-5)=-1$.
4. Model $-2+(-3)$. Draw two red counters and three more red counters. Since all five are negative, their count represents $-5$, so $-2+(-3)=-5$.
5. State the results. The four models give $6$, $3$, $-1$, and $-5$, respectively. The figures show both the original groups and the unmatched counters after neutral pairs are removed.

Takeaway: Add same-sign counters by counting the whole group, and add opposite-sign counters by canceling neutral pairs.

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.

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 $\mathrm{-101.}$

Look again at the results of Example 3.14 - Example 3.17.

Table

3.1

Addition of Positive and Negative Integers

Example 3.19

Solution

1. Identify the signs. In $19+(-47)$ and $-32+40$, the addends have unlike signs, so subtract the smaller magnitude from the larger.
2. Simplify $19+(-47)$. Since $47>19$, compute $47-19=28$ and use the negative sign of the greater magnitude.
3. State the first sum. The sign rule gives $19+(-47)=-28$, matching the negative result.
4. Simplify $-32+40$. Since $40>32$, compute $40-32=8$ and use the positive sign of the greater magnitude.
5. State the second sum. The same rule gives $-32+40=8$, so both results follow from comparing magnitudes.

Takeaway: When adding integers with unlike signs, subtract the smaller absolute value from the larger and keep the sign of the larger magnitude, giving $-28$ and $8$.

Example 3.20

Solution

1. Recognize the common sign. Both $-14$ and $-36$ are negative, so add their magnitudes and retain a negative sign.
2. Add the magnitudes. Compute $14+36=50$ to find the total distance on the negative side of the number line.
3. Apply the negative sign. The sum is therefore $-(14+36)=-50$.
4. Check the direction. Moving $14$ units left and then $36$ more units left from zero ends at $-50$.

Takeaway: Two negative integers add by adding their absolute values and keeping the negative sign, so $-14+(-36)=-50$.

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

Example 3.21

Solution

1. Simplify the parentheses first. In $-5+3(-2+7)$, compute $-2+7=5$ before multiplying.
2. Rewrite the product. The expression becomes $-5+3(5)$, with the parenthesized value now a single number.
3. Multiply. Compute $3(5)=15$, giving $-5+15$.
4. Add the remaining terms. Compute $-5+15=10$, which is the simplified expression.
5. Check the sign. The positive product $15$ is greater in magnitude than $5$, so the final sum is positive.

Takeaway: Simplify the parenthetical expression before multiplying, so $-5+3(-2+7)=-5+3(5)=-5+15=10$.

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 3.22

Solution

1. Substitute the first value. Replace $x$ with $-2$ in $x+7$ to get $-2+7$, matching the first substitution figure.
2. Add for the first case. Since $-2+7=5$, the first value is $5$, as shown in the result figure.
3. Substitute the second value. Replace $x$ with $-11$ to get $-11+7$, as shown by the later substitution figures.
4. Add for the second case. Since $-11+7=-4$, the second value is $-4$, matching the final result figure.
5. State the results. The expression $x+7$ equals $5$ when $x=-2$ and equals $-4$ when $x=-11$.

Takeaway: Substitution must occur before addition so that the sign of the given variable value is used correctly.

Example 3.23

Solution

1. Use the given value. For $n+1$, substitute $n=-5$ to obtain $-5+1$, matching the first substitution figures.
2. Evaluate the first expression. Adding $1$ gives $-5+1=-4$, as shown in the first result figure.
3. Substitute for the second expression. For $-n+1$, replace $n$ with $-5$ to obtain $-(-5)+1$, as shown by the later figures.
4. Simplify carefully. The opposite of $-5$ is $5$, so $5+1=6$.
5. State the results. The expressions evaluate to $-4$ and $6$, respectively.

Takeaway: A negative sign before a substituted variable negates that value before the remaining operations are performed.

Next we'll evaluate an expression with two variables.

Example 3.24

Solution

1. Write the expression. The expression to evaluate is $3a+b$, as shown in the first figure.
2. Substitute both values. Replace $a$ with $12$ and $b$ with $-30$ to obtain $3(12)+(-30)$, matching the next two figures.
3. Multiply. Compute $3(12)=36$, so the expression becomes $36+(-30)$.
4. Add the integers. The positive magnitude $30$ is smaller than $36$, so $36+(-30)=6$.
5. State the result. The expression $3a+b$ equals $6$ when $a=12$ and $b=-30$, as shown in the final figure.

Takeaway: When evaluating two-variable expressions, substitute each value for its correct variable before multiplying or adding.

Example 3.25

Solution

1. Substitute both values. Replace $x$ with $-18$ and $y$ with $24$ in $(x+y)^2$, as shown in the accompanying figure.
2. Simplify the base. Compute the sum inside the parentheses: $-18+24=6$.
3. Apply the exponent. Square the result to obtain $6^2$.
4. Multiply. Compute $6^2=6\cdot6=36$.
5. State the result. The expression $(x+y)^2$ equals $36$ when $x=-18$ and $y=24$.

Takeaway: In a powered grouped expression, simplify the grouped base completely before applying the exponent.

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 3.26

Solution

1. Interpret the key phrase. The phrase the sum of identifies addition as the operation connecting the two quantities.
2. Write the expression. Place the numbers in the stated order to obtain $-9+5$.
3. Compare the magnitudes. The magnitudes are $9$ and $5$, so subtract the smaller from the larger.
4. Apply the sign rule. Since $9>5$, the sum takes the negative sign and equals $-(9-5)=-4$.

Takeaway: The sum of $-9$ and $5$ translates to $-9+5$, which simplifies to $-4$.

Example 3.27

Solution

1. Identify the inner operation. The sum of $8$ and $-12$ is the inner quantity, so write it as $8+(-12)$.
2. Translate increased by. The phrase increased by $3$ means that $3$ is added after the inner sum.
3. Combine the translations. Use parentheses to preserve the original grouping: $[8+(-12)]+3$.
4. Simplify the inner sum. The magnitudes $12$ and $8$ differ by $4$, so $8+(-12)=-4$.
5. Add the final amount. Compute $-4+3=-1$, which is the simplified value of the original phrase.

Takeaway: A quantity increased by $3$ requires adding $3$ after the original sum, so $[8+(-12)]+3=-1$.

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 $\text{\textdollar 5}$ could be represented as $\mathrm{-\textdollar 5.}$ 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 3.28

Solution

1. Represent the starting temperature. A temperature $7$ degrees below zero is the initial value $-7$ degrees Fahrenheit.
2. Represent the temperature change. Warming up by $12$ degrees is a positive change, so add $12$.
3. Write one expression. Combine the initial value and the change as $-7+12$.
4. Simplify the sum. Because the signs differ and $12>7$, the result is $12-7=5$ and it is positive.
5. Answer in context. The temperature at noon was $5$ degrees Fahrenheit.

Takeaway: A temperature $7$ degrees below zero begins at $-7$, so warming by $12$ degrees gives $-7+12=5$ degrees.

Example 3.29

Solution

1. Record the initial position. The team begins on the $42$-yard line, so the running total starts at $42$.
2. Assign signs to the plays. Losing $6$ yards contributes $-6$, gaining $4$ yards contributes $4$, and losing $8$ yards contributes $-8$.
3. Write the combined expression. The three changes give $42-6+4-8$ on the yard line.
4. Simplify from left to right. Compute $42-6=36$, then $36+4=40$, and finally $40-8=32$.
5. Answer with the unit. At the end of the three plays, the ball is on the $32$-yard line.

Takeaway: Track each change of possession with its signed yardage, giving $42-6+4-8=32$ on the final yard line.

Section 3.2 Exercises

Section 3.2 Exercises

Model Addition of Integers

In the following exercises, model the expression to simplify.

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Translate Word Phrases to Algebraic Expressions

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

Add Integers in Applications

In the following exercises, solve.

Everyday Math

Writing Exercises

Self Check

ⓐ 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?

1. Simplify each expression:

1. ⓐ $\,15+(-32)\,$

2. ⓑ $\,-19+76$

2. Simplify each expression:

1. ⓐ $\,-55+9\,$

2. ⓑ $\,43+(-17)$

3. Simplify the expression:

$-31+(-19)$

1. Simplify the expression:

$-42+(-28)$

2. Evaluate each expression for the given values:

$x+5\,\mathrm{when}\,$

1. ⓐ $\,x=-3\,\mathrm{and}\,$

2. ⓑ $\,x=-17\,$

3. Evaluate each expression for the given values: $y+7\,$ when

1. ⓐ $\,\,y=-5\,$

2. ⓑ $\,y=-8\,$

1. When $n=-8,$ evaluate

1. ⓐ $\,n+2\,\,$

2. ⓑ $\,-n+2\,$

2. $\mathrm{When}\,y=-9,\,\mathrm{evaluate}\,$

1. ⓐ $\,y+8\,$

2. ⓑ $\,-y+8.$

3. Translate and simplify the expression:

the sum of $-7$ and $4$

1. Translate and simplify the expression:

the sum of $-8$ and $-6$

2. Translate and simplify:

the sum of $9$ and $-16,$ increased by $4.$

3. Translate and simplify:

the sum of $-8$ and $-12,$ increased by $7.$

1. $-21+(-59)$

2. $-200+65$

3. $-14+(-12)+4$

1. $-17+(-18)+6$

2. $-32+24+(-6)+10$

3. $x+8$ when

1. ⓐ $\,x=-26$

2. ⓑ $\,x=-95$

1. $y+(-14)$ when

1. ⓐ $\,y=-33$

2. ⓑ $\,y=30$

2. When $a=-7,$ evaluate:

1. ⓐ $\,a+3$

2. ⓑ $\,-a+3$

3. When $c=-9,$ evaluate:

1. ⓐ $\,c+(-4)$

2. ⓑ $\,-c+(-4)$

1. $m+n$ when, $m=-15$, $n=7$

2. The sum of $-14$ and $5$

3. $8$ more than $-2$

1. $-10$ added to $-15$

2. $6$ more than the sum of $-1$ and $-12$

3. the sum of $10$ and $-19,$ increased by $4$

1. Model the expression.

$-2+(-4)$

2. Model the expression.

$-2+(-5)$

3. Model the expression, and then simplify:

$2+(-4)$

1. Model the expression, and then simplify:

$2+(-5)$

2. Model each addition.

1. ⓐ 3 + 4

2. ⓑ −1 + 4

3. ⓒ 4 + (−6)

4. ⓓ −2 + (−2)

3. 1. ⓐ 5 + 1

2. ⓑ −3 + 7

3. ⓒ 2 + (−8)

4. ⓓ −3 + (−4)

1. The temperature in Chicago at 5 A.M. was $10\,\mathrm{degrees}$ below zero Celsius. Six hours later, it had warmed up $\mathrm{14 degrees Celsius.}$ What is the temperature at 11 A.M.?

2. A scuba diver was swimming $\mathrm{16 feet}$ below the surface and then dove down another $\mathrm{17 feet.}$ What is her new depth?

3. The Bears took possession of the football on their $\mathrm{20-yard line.}$ In the next three plays, they lost $\mathrm{9 yards,}$ gained $\mathrm{7 yards,}$ then lost $\mathrm{4 yards.}$ On what yard line was the ball at the end of those three plays?

1. The Chargers began with the football on their $\mathrm{25-yard line.}$ They gained $\mathrm{5 yards,}$ lost $\mathrm{8 yards}$ and then gained $\mathrm{15 yards}$ on the next three plays. Where was the ball at the end of these plays?

2. $-6+(-3)$

3. $-7+5$

1. Temperature The temperature in St. Paul, Minnesota was $-19\mathrm{^{\circ}F}$ at sunrise. By noon the temperature had risen $\mathrm{26^{\circ}F.}$ What was the temperature at noon?

2. Credit Cards Lupe owes $\text{\textdollar 73}$ on her credit card. Then she charges $\text{\textdollar 45}$ more. What is the new balance?

3. Football A team lost $\mathrm{3 yards}$ the first play. Then they lost $\mathrm{2 yards,}$ gained $\mathrm{1 yard,}$ and then lost $\mathrm{4 yards.}$ What was the change in overall yardage over the four plays?

1. Football The Rams took possession of the football on their own $\mathrm{35-yard line.}$ In the next three plays, they lost $\mathrm{12 yards,}$ gained $\mathrm{8 yards,}$ then lost $\mathrm{6 yards.}$ On what yard line was the ball at the end of those three plays?

2. Scuba Diving A scuba diver swimming $\mathrm{8 feet}$ below the surface dove $\mathrm{17 feet}$ deeper; the pressure got to them and they rose five feet. What is their new depth?

3. 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:

$\begin{array}{llllll} \mathrm{Monday} & -504 & \mathrm{Tuesday} & +142 & \mathrm{Wednesday} & -449 \\ \mathrm{Thursday} & +410 & \mathrm{Friday} & +369 & \\ \end{array}$

What was the overall change for the week?

1. Explain why the sum of $-8$ and $2$ is negative, but the sum of $8$ and $-2$ and is positive.