# 3.2 Add integers  (Page 4/4)

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The temperature in Chicago at 5 A.M. was $10\phantom{\rule{0.2em}{0ex}}\text{degrees}$ below zero Celsius. Six hours later, it had warmed up $\text{14 degrees Celsius.}$ What is the temperature at 11 A.M.?

4 degrees Celsius

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

−33 feet

A football team took possession of the football on their $\text{42-yard line.}$ In the next three plays, they lost $\text{6 yards,}$ gained $\text{4 yards,}$ and then lost $\text{8 yards.}$ 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.

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

14-yard line

The Chargers began with the football on their $\text{25-yard line.}$ They gained $\text{5 yards,}$ lost $\text{8 yards}$ and then gained $\text{15 yards}$ 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  $5+3$ $-5+\left(-3\right)$ 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. $-5+3$ $5+\left(-3\right)$ 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.

$7+4$

11

$8+5$

$-6+\left(-3\right)$

−9

$-5+\left(-5\right)$

$-7+5$

−2

$-9+6$

$8+\left(-7\right)$

1

$9+\left(-4\right)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

$-21+\left(-59\right)$

−80

$-35+\left(-47\right)$

$48+\left(-16\right)$

32

$34+\left(-19\right)$

$-200+65$

−135

$-150+45$

$2+\left(-8\right)+6$

0

$4+\left(-9\right)+7$

$-14+\left(-12\right)+4$

−22

$-17+\left(-18\right)+6$

$135+\left(-110\right)+83$

108

$140+\left(-75\right)+67$

$-32+24+\left(-6\right)+10$

−4

$-38+27+\left(-8\right)+12$

$19+2\left(-3+8\right)$

29

$24+3\left(-5+9\right)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

$x+8$ when

1. $\phantom{\rule{0.2em}{0ex}}x=-26$
2. $\phantom{\rule{0.2em}{0ex}}x=-95$

1. ⓐ −18
2. ⓑ −87

$y+9$ when

1. $\phantom{\rule{0.2em}{0ex}}y=-29$
2. $\phantom{\rule{0.2em}{0ex}}y=-84$

$y+\left(-14\right)$ when

1. $\phantom{\rule{0.2em}{0ex}}y=-33$
2. $\phantom{\rule{0.2em}{0ex}}y=30$

1. ⓐ −47
2. ⓑ 16

$x+\left(-21\right)$ when

1. $\phantom{\rule{0.2em}{0ex}}x=-27$
2. $\phantom{\rule{0.2em}{0ex}}x=44$

When $a=-7,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}a+3$
2. $\phantom{\rule{0.2em}{0ex}}-a+3$

1. ⓐ −4
2. ⓑ 10

When $b=-11,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}b+6$
2. $\phantom{\rule{0.2em}{0ex}}-b+6$

When $c=-9,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}c+\left(-4\right)$
2. $\phantom{\rule{0.2em}{0ex}}-c+\left(-4\right)$

1. ⓐ −13
2. ⓑ 5

When $d=-8,$ evaluate:

1. $\phantom{\rule{0.2em}{0ex}}d+\left(-9\right)$
2. $\phantom{\rule{0.2em}{0ex}}-d+\left(-9\right)$

$m+n$ when, $m=-15,\phantom{\rule{0.2em}{0ex}}\text{}n=7$

−8

$p+q$ when, $p=-9,\phantom{\rule{0.2em}{0ex}}\text{}q=17$

$r-3s$ when, $r=16,$ $s=2$

10

$2t+u$ when, $t=-6,$ $u=-5$

${\left(a+b\right)}^{2}$ when, $a=-7,$ $b=15$

64

${\left(c+d\right)}^{2}$ when, $c=-5,\phantom{\rule{0.2em}{0ex}}\text{}d=14$

${\left(x+y\right)}^{2}$ when, $x=-3,\phantom{\rule{0.2em}{0ex}}\text{}y=14$

121

${\left(y+z\right)}^{2}$ when, $y=-3,\phantom{\rule{0.2em}{0ex}}\text{}z=15$

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 $5$

−14 + 5 = −9

The sum of $-22$ and $9$

$8$ more than $-2$

−2 + 8 = 6

$5$ more than $-1$

$-10$ added to $-15$

−15 + (−10) = −25

$-6$ added to $-20$

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

[−1 + (−12)] + 6 = −7

$3$ more than the sum of $-2$ and $-8$

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

[10 + (−19)] + 4 = −5

the sum of $12$ and $-15,$ increased by $1$

In the following exercises, solve.

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

7°F

Temperature The temperature in Chicago was $-15\text{°F}$ at 6 am. By afternoon the temperature had risen $\text{28°F.}$ What was the afternoon temperature?

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

−\$118

Credit Cards Frank owes $\text{212}$ on his credit card. Then he charges $\text{105}$ more. What is the new balance?

Weight Loss Angie lost $\text{3 pounds}$ the first week of her diet. Over the next three weeks, she lost $\text{2 pounds,}$ gained $\text{1 pound,}$ and then lost $\text{4 pounds.}$ What was the change in her weight over the four weeks?

−8 pounds

Weight Loss April lost $\text{5 pounds}$ the first week of her diet. Over the next three weeks, she lost $\text{3 pounds,}$ gained $\text{2 pounds,}$ and then lost $\text{1 pound.}$ What was the change in her weight over the four weeks?

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

25-yard line

Football The Cowboys began with the ball on their own $\text{20-yard line.}$ They gained $\text{15 yards,}$ lost $\text{3 yards}$ and then gained $\text{6 yards}$ on the next three plays. Where was the ball at the end of these plays?

Calories Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of $\text{20 minutes,}$ she burned $\text{90 calories.}$ The frozen yogurt she ate was $\text{110 calories.}$ What was her total calorie gain or loss?

20 calories

Calories Ozzie rode his bike for $\text{30 minutes,}$ burning $\text{168 calories.}$ Then he had a $\text{140-calorie}$ iced blended mocha. Represent the change in calories 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:

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

What was the overall change for the week?

−32

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

$\begin{array}{cccccc}\text{Monday}\hfill & -201\hfill & \text{Tuesday}\hfill & -16\hfill & \text{Wednesday}\hfill & -23\hfill \\ \text{Thursday}\hfill & +172\hfill & \text{Friday}\hfill & -34\hfill & \end{array}$

What was the overall change for the week?

## Writing exercises

Explain why the sum of $-8$ and $\text{2}$ is negative, but the sum of $\text{8}$ and $-2$ and is positive.

Sample answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive.

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

## 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?