$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 3.4: Add Integers (Part 2)

• • Contributed by OpenStax
• Mathematics at OpenStax CNX
$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## 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 $$\PageIndex{9}$$: evaluate

Evaluate $$x + 7$$ when

1. $$x = −2$$
2. $$x = −11$$

Solution

1. Evaluate $$x + 7$$ when $$x = −2$$
 Substitute $$\textcolor{red}{-2}$$ for x. $$\textcolor{red}{-2} + 7$$ Simplify. $$5$$
1. Evaluate $$x + 7$$ when $$x = −11$$
 Substitute $$\textcolor{red}{-11}$$ for x. $$\textcolor{red}{-11} + 7$$ Simplify. $$-4$$

Exercise $$\PageIndex{17}$$

Evaluate each expression for the given values: $$x + 5$$ when

1. $$x = −3$$
2. $$x = −17$$

$$2$$

$$-12$$

Exercise $$\PageIndex{18}$$

Evaluate each expression for the given values: $$y + 7$$ when

1. $$y = −5$$
2. $$y = −8$$

$$2$$

$$-1$$

Example $$\PageIndex{10}$$: evaluate

When $$n = −5$$, evaluate

1. $$n + 1$$
2. $$−n + 1$$

Solution

 Evaluate n + 1 when n = −5 $$n + 1$$ Substitute $$\textcolor{red}{-5}$$ for n. $$\textcolor{red}{-5} + 1$$ Simplify. $$-4$$
 Evaluate −n + 1 when n = −5 $$-n + 1$$ Substitute $$\textcolor{red}{-5}$$ for n. $$-(\textcolor{red}{-5}) + 1$$ Simplify. $$5 + 1$$ Add. $$6$$

Exercise $$\PageIndex{19}$$

When $$n = −8$$, evaluate

1. $$n + 2$$
2. $$−n + 2$$

$$-6$$

$$10$$

Exercise $$\PageIndex{20}$$

When $$y = −9$$, evaluate

1. $$y + 8$$
2. $$−y + 8$$

$$-1$$

$$17$$

Next we'll evaluate an expression with two variables.

Example $$\PageIndex{11}$$: evaluate

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

Solution

 Substitute $$\textcolor{red}{12}$$ for a and $$\textcolor{blue}{-30}$$ for b. $$3(\textcolor{red}{12}) + (\textcolor{blue}{-30})$$ Multiply. $$36 + (-30)$$ Add. $$6$$

Exercise $$\PageIndex{21}$$

Evaluate the expression: $$a + 2b$$ when $$a = −19$$ and $$b = 14$$.

$$9$$

Exercise $$\PageIndex{22}$$

Evaluate the expression: $$5p + q$$ when $$p = 4$$ and $$q = −7$$.

$$13$$

Example $$\PageIndex{12}$$: evaluate

Evaluate $$(x + y)^2$$ when $$x = −18$$ and $$y = 24$$.

Solution

This expression has two variables.

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

Exercise $$\PageIndex{23}$$

Evaluate: $$(x + y)^2$$ when $$x = −15$$ and $$y = 29$$.

$$196$$

Exercise $$\PageIndex{24}$$

Evaluate: $$(x + y)^3$$ when $$x = −8$$ and $$y = 10$$.

$$8$$

## 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 $$\PageIndex{13}$$: translate

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

Solution

 The sum of −9 and 5 indicates addition. the sum of −9 and 5 Translate. −9 + 5 Simplify. −4

Exercise $$\PageIndex{25}$$

Translate and simplify the expression: the sum of $$−7$$ and $$4$$

$$-7+4=-3$$

Exercise $$\PageIndex{26}$$

Translate and simplify the expression: the sum of $$−8$$ and $$−6$$

$$-8+(-6)=-14$$

Example $$\PageIndex{14}$$: translate

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

Solution

The phrase increased by indicates addition.

 Translate. [8 + (−12)] + 3 Simplify. −4 + 3 Add. −1

Exercise $$\PageIndex{27}$$

Translate and simplify: the sum of $$9$$ and $$−16$$, increased by $$4$$.

$$[9+(-16)]+4=-3$$

Exercise $$\PageIndex{28}$$

Translate and simplify: the sum of $$−8$$ and $$−12$$, increased by $$7$$.

$$[-8+(-12)]+7=-13$$

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 $$\PageIndex{15}$$: evaluate

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 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.

Exercise $$\PageIndex{29}$$

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

$$4$$ degrees Celsius

Exercise $$\PageIndex{30}$$

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

$$-33$$ feet

Example $$\PageIndex{16}$$: evaluate

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.

Exercise $$\PageIndex{31}$$

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

$$14$$ yard line

Exercise $$\PageIndex{32}$$

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

$$37$$ yard line

## Key Concepts

Addition of Positive and Negative Integers

 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.

## Practice Makes Perfect

In the following exercises, model the expression to simplify.

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

### Simplify Expressions with Integers

In the following exercises, simplify each expression.

1. −21 + (−59)
2. −35 + (−47)
3. 48 + (−16)
4. 34 + (−19)
5. −200 + 65
6. −150 + 45
7. 2 + (−8) + 6
8. 4 + (−9) + 7
9. −14 + (−12) + 4
10. −17 + (−18) + 6
11. 135 + (−110) + 83
12. 140 + (−75) + 67
13. −32 + 24 + (−6) + 10
14. −38 + 27 + (−8) + 12
15. 19 + 2(−3 + 8)
16. 24 + 3(−5 + 9)

### Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

1. 87. x + 8 when
1. x = −26
2. x = −95
2. y + 9 when
1. y = −29
2. y = −84
3. y + (−14) when
1. y = −33
2. y = 30
4. x + (−21) when
1. x = −27
2. x = 44
5. When a = −7, evaluate:
1. a + 3
2. −a + 3
6. When b = −11, evaluate:
1. b + 6
2. −b + 6
7. When c = −9, evaluate:
1. c + (−4)
2. −c + (−4)
8. When d = −8, evaluate:
1. d + (−9)
2. −d + (−9)
9. m + n when, m = −15 , n = 7
10. p + q when, p = −9 , q = 17
11. r−3s when, r = 16 , s = 2
12. 2t + u when, t = −6 , u = −5
13. (a + b)2 when, a = −7 , b = 15
14. (c + d)2 when, c = −5 , d = 14
15. (x + y)2 when, x = −3 , y = 14
16. (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.

1. The sum of −14 and 5
2. The sum of −22 and 9
3. 8 more than −2
4. 5 more than −1
7. 6 more than the sum of −1 and −12
8. 3 more than the sum of −2 and −8
9. the sum of 10 and −19, increased by 4
10. the sum of 12 and −15, increased by 1

In the following exercises, solve.

1. 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?
2. Temperature The temperature in Chicago was −15 °F at 6 am. By afternoon the temperature had risen 28 °F. What was the afternoon temperature?
3. Credit Cards Lupe owes $73 on her credit card. Then she charges$45 more. What is the new balance?
4. Credit Cards Frank owes $212 on his credit card. Then he charges$105 more. What is the new balance?
5. Weight Loss Angie lost 3 pounds the first week of her diet. Over the next three weeks, she lost 2 pounds, gained 1 pound, and then lost 4 pounds. What was the change in her weight over the four weeks?
6. Weight Loss April lost 5 pounds the first week of her diet. Over the next three weeks, she lost 3 pounds, gained 2 pounds, and then lost 1 pound. What was the change in her weight over the four weeks?
7. 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?
8. 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?
9. Calories Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of 20 minutes, she burned 90 calories. The frozen yogurt she ate was 110 calories. What was her total calorie gain or loss?
10. Calories Ozzie rode his bike for 30 minutes, burning 168 calories. Then he had a 140-calorie iced blended mocha. Represent the change in calories as an integer.

## Everyday Math

1. 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: Monday −504, Tuesday +142, Wednesday −449, Thursday +410, Friday +369. What was the overall change for the week?
2. Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was: Monday −201, Tuesday −16, Wednesday −23, Thursday +172, Friday −34. What was the overall change for the week?

## Writing Exercises

1. Explain why the sum of −8 and 2 is negative, but the sum of 8 and −2 and is positive.
2. Give an example from your life experience of adding two negative numbers.

## Self Check

(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. (b) After reviewing this checklist, what will you do to become confident or all objectives?