# 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? #### Questions & Answers what is the stm Brian Reply is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.? Rafiq industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong Damian How we are making nano material? LITNING Reply what is a peer LITNING Reply What is meant by 'nano scale'? LITNING Reply What is STMs full form? LITNING scanning tunneling microscope Sahil how nano science is used for hydrophobicity Santosh Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq Rafiq what is differents between GO and RGO? Mahi what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq Rafiq what is Nano technology ? Bob Reply write examples of Nano molecule? Bob The nanotechnology is as new science, to scale nanometric brayan nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale Damian Is there any normative that regulates the use of silver nanoparticles? Damian Reply what king of growth are you checking .? Renato What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ? Stoney Reply why we need to study biomolecules, molecular biology in nanotechnology? Adin Reply ? Kyle yes I'm doing my masters in nanotechnology, we are being studying all these domains as well.. Adin why? Adin what school? Kyle biomolecules are e building blocks of every organics and inorganic materials. Joe anyone know any internet site where one can find nanotechnology papers? Damian Reply research.net kanaga sciencedirect big data base Ernesto Introduction about quantum dots in nanotechnology Praveena Reply what does nano mean? Anassong Reply nano basically means 10^(-9). nanometer is a unit to measure length. Bharti do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment? Damian Reply absolutely yes Daniel how to know photocatalytic properties of tio2 nanoparticles...what to do now Akash Reply it is a goid question and i want to know the answer as well Maciej characteristics of micro business Abigail for teaching engĺish at school how nano technology help us Anassong How can I make nanorobot? Lily Do somebody tell me a best nano engineering book for beginners? s. Reply there is no specific books for beginners but there is book called principle of nanotechnology NANO how can I make nanorobot? Lily what is fullerene does it is used to make bukky balls Devang Reply are you nano engineer ? s. fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball. Tarell what is the actual application of fullerenes nowadays? Damian That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes. Tarell A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
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