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74  Lesson 8

Plug InTheabsolute valueofanumberisitsdistancefrom0onanumberline .

|25|means“theabsolutevalueof25 .”

Youcanuseanumber linetohelpyoufindtheabsolutevalueofanumber .

�5 �4 �3 �2 �1 0 1

Thepointison25onthenumberline .

Thedistancefrom25to0is5units .Thatmeanstheabsolutevalueof25is5 .

�5 �4 �3 �2 �1 0 1

|25|55

absolute valueanumber’sdistancefrom0onanumberline

|23|means“theabsolutevalueof23 .”

number linealinethatshowsnumbersplacedinorder

�4 �3 �2 �1 0 1 2 3 4

Wordsto

Know

Understanding Absolute Value

Absolute ValueLE

SSON 8

DIS

CUSSCananabsolutevaluebe0?Explain .

A Youcanfindtheabsolutevalueofapositivenumber .

Find|7| .

1 Plotthenumber7onanumberline .

2 Countthenumberofunitsto0 .

3 Thisdistanceistheabsolutevalue .

Think:Ineedtofindtheabsolutevalueof 7 .

�1 0 1 2 3 4 6 8 105 7 9

Thereare unitsfrom0to7 .

|7|5

DO

I can think of absolute value as a

distance.Count the number of units from the

point to 0.

I get it! Since a distance is always positive, the

absolute value of a number is always positive.

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B

C

Youcanfindtheabsolutevalueofanegativenumber .

Find|210| .

1 Drawapointonanumberlinefor210 .

2 Countthenumberofunitsto0 .

3 Thisdistanceistheabsolutevalue .

Think:Ineedtofindtheabsolutevalueof .

�10 �9 �8 �7 �6 �5 �3 �1 1�4 �2 0

Thereare unitsfrom210to0 .

|210|5

DO

Youcanfindtheabsolutevalueofanumberbykeepingorchangingitssign .

Find|12|and|250| .

1 Thinkaboutthemeaningofabsolutevalue .

2 Writetheabsolutevalues .

Theabsolutevalueofanumberisalways .

Since12isalreadypositive,tofind|12|removetheabsolutevalue

symbolsand itssign .

Since250isnegative,tofind|250|removetheabsolutevalue

symbolsandchangeitssignfrom to .

|12|5 and|250|5

DO

PRACTICEDraw a point on a number line to show the integer and find its absolute value.

1 |4|5

�5 �4 �3 �2 �1 0 1 2 3 4 5

2 |22|5

�5 �4 �3 �2 �1 0 1 2 3 4 5

Find the absolute value of the integer.

3 |28|5 4 |15|5 5 |2126|5

I get it! Taking the absolute value of a positive number

doesn’t change the number. Taking the absolute value

of a negative number makes it positive.

210

positive

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76  Lesson 8

POWER uP Finding Opposites on a Number LineNegative numbersaretotheleftof0andpositive numbersaretotherightof0 .

|25|55

|5|55

�5 �4 �3 �2 �1 0 1 2 3 4 5

Numbersareoppositeswhentheyhavethesameabsolutevaluebutareondifferentsidesof0 .

Thenumbers5and25bothhaveanabsolutevalueof5 .Theyareboth5unitsawayfrom0 .25and5areopposites .

�5 �4 �3 �2 �1 0 1 2 3 4 5

positive numberanumbergreaterthanzero

15

negative numberanumberlessthanzero

215

readas“negativefifteen”

oppositesnumbersthathavethesameabsolutevaluebutdifferentsigns

15and215areopposites .

Wordsto

Know

Doeseverywholenumber,fraction,integer,anddecimalexcept0haveanopposite?Explain .

DIS

CUSS

A Youcanuseanumberlinetofindtheoppositeofapositivenumber .

Findtheoppositeof7 .

1 Determinewhichsideof0thegivennumberison .Drawapoint .

2 Countthesamenumberofunitsintheoppositedirectionfrom0anddrawapoint .

3 Writetheanswer .

Thenumber7is unitstothe of0 .

�8 �7 �6 �5 �4 �3 �1 1 8�2 0 2 3 4 5 6 7

Tofinditsopposite,count unitstothe of0 .

Theoppositeof7is .

DO

Positive and negative numbers are on opposite

sides of zero on the number line.

I get it! The numbers 5 and 25 are both 5 units

from 0 on the number line, but have opposite signs.

7

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B Youcanuseanumberlinetofindtheoppositeofanegativenumber .

Findtheoppositeof24 .

1 Determinewhichsideof0thegivennumberison .Drawapoint .

2 Countthesamenumberofunitsintheoppositedirectionfrom0anddrawapoint .

3 Writetheanswer .

Thenumber24is units

tothe of0 .

�6 �5 �4 �3 �1 1 6�2 0 2 3 4 5

Tofinditsopposite,count unitstothe of0 .

Theoppositeof24is .

DO

Onthenumberlinebelow,locateanumber(otherthan0)anditsopposite .Whatistheoppositeoftheoppositeofyouroriginalnumber?Explain .

�8 �7 �6 �5 �4 �3 �2 �1 80 1 2 3 4 5 6 7

DIS

CUSS

PRACTICEFind the opposite of the number shown on the number line.

1 �5 �4 �3 �2 �1 0 1 2 3 4 5

Theoppositeof23is .

2 �5 �4 �3 �2 �1 0 1 2 3 4 5

Theoppositeof2is .

Plot the number on the number line. Then, find its opposite and plot it on the number line.

3 28 Theoppositeof28is .

�10 �9 �8 �6�7 �5 �4 �3 �2 �1 100 1 2 3 4 5 76 8 9

4 6 Theoppositeof6is .

�10 �9 �8 �6�7 �5 �4 �3 �2 �1 100 1 2 3 4 5 76 8 9

Use the number line to find the opposite of the opposite of the given number.

5 Theoppositeoftheoppositeof12is .

�15 �14 �13 �12 �11 �10 �9 �8 �7 �6 �5 �4 �3 �2 �1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The opposite of a positive number is negative, and

the opposite of a negative number

is positive.4

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78  Lesson 8

READY TO gO Absolute ValueYoucanfindtheabsolutevaluesofallrational numbers,whichincludeintegers .

Thenumbersyouworkwithintherealworldareoftenrationalnumbers .

Youcanuseabsolutevaluetofindandcomparedistancesfromareferencepoint .

Theelevation35 .5feetis35 .5feetabovesealevel .

Theelevation235 .5feetis35 .5feetbelowsealevel .

Thereferencepointis0feet,orsealevel .

Youcanusetheabsolutevalueofanumbertodescribeitssize,oritsdistancefromareferencepoint .

35 .5feetand235 .5feetareboth35 .5feetfromsealevel,butinoppositedirections .

rational numberanumberthatcanbewrittenasaratiooftwointegers

0 .75,215,0,71__2,22 .5,7

__9

Wordsto

Know

Describethetemperature24°F,usingitsabsolutevalue .Whatisthereferencepointinthiscase?

DIS

CUSS

Plug In POWER uP gO!lEssOn lInk

A number line can help you find the absolute value of a number.

�3 �2 �1 0 1 2 3

|22|52

A number line can help you find opposite numbers.

�3 �2 �1 0 1 2 3

22and2areopposites .

The absolute value of a rational

number is its distance from 0!

I get it! I can use the absolute value of a real-world quantity to describe it in real terms.

I see! I can use absolute value to describe positive

and negative rational numbers in real-world

situations!

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WORk TOgEThERYoucandescribeareal-worldsituationusingabsolutevalue .

• Abalanceof2$35 .23meansnegative35dollarsand23cents .

• Theabsolutevalueof235 .23is35 .23 .

• Thereferencepointisabalanceof0,whereJeannewouldhavenomoneyinheraccountandnotowethebankanymoney,either .

• Jeannehasadebtof$35 .23 .

Jeannespentmoremoneythanshehadinherbankaccount,andherbalanceisnow2$35 .23 .HowmuchdoesJeanneowethebank?

|235 .23|535 .23

Anegativebalancemeansmoneyowed .Thebalanceof2$35 .23meansthatJeanneowesthebank$35 .23 .

A Useabsolutevaluetodescribethesituation .

Atreasurechestisburiedat225feet .Howmanyfeetbelowthegroundisthetreasurechest?

1 Determinethereferencepoint .

2 Findthedistancefromthereferencepoint .

3 Describethesituation .

Thereferencepointis .

Thedistancefromthereferencepointistheabsolutevalueof225 .

|225|5

Thetreasurechestis .

DO

B Writeapositiveornegativenumberforthesituationdescribed .

Bjornwroteacheckfor$45 .85totheelectriccompany .WriteanumbertorepresentthechangeinBjorn’scheckingaccount .

1 Determinethereferencepoint .

2 Determinethedistancefromthereferencepoint .

3 Determinewhetherthedistanceisinapositiveornegativedirection .

4 Writeanumber .

Thereferencepointis$ ,whichwouldrepresentnochangetoBjorn’saccount .

Thedistancefromthereferencepointis .

TheamountonthecheckwillbetakenfromBjorn’scheckingaccountandgiventotheelectriccompany .So,Bjorn’saccountislosingthemoney .Thenumbershouldbe .

Thenumbertodescribethesituationis .

DO

Whendealingwithabankaccount,acreditismoneyaddedtotheaccountandadebitismoneytakenout .Forthetermscreditanddebit,whichwouldbedescribedbyapositivenumberandwhichwouldbedescribedbyanegativenumber?

DIS

CUSS

The absolute value of the negative

balance describes the size of her debt!

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80  Lesson 8

READY TO gO

PRACTICE

REMEMBER The absolute value of a number is always positive.

Find the absolute value of the number represented by the point on the number line.

1 �1 100 1 2 3 4 5 76 8 9

Theabsolutevalueis .

2 �10 1�9 �8 �7 �6 �5 �4 �2�3 �1 0

Theabsolutevalueis .

Find the absolute value of the number.

3 |45 .2|5 4 |21241__2|5

Determine the reference point. Use absolute value to describe the situation.

5 Akitefliesup30feet .Howfaristhekitefromtheground?

Thereferencepointis .

|30|5

Thekiteis feetfromtheground .

6 Agiantsquidswimsat2800feet .Howfaristhesquidfromthesurfaceoftheocean?

Thereferencepointis .

|2800|5

Thesquidis feetfromthesurface .

Determine the reference point. Describe the situation.

7 Hafidagreestopayhisneighbor$25formowinghislawn .

Thereferencepointis .

8 Jannabakesacakeinanoventhatis350°F .

Thereferencepointis .

HINT The distance is the absolute value of the number.

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Putting It Together

Ahikerandascubadiverbothstartatsealevel .Afteranhour,thehikerisatanelevationof67 .5feet,andthescubadiverisatanelevationof288 .1feet .Howcanyouuseabsolutevaluetoexplainwhetherthehikerordiverisfartherfromsealevel?

DIS

CUSS

Write a positive or negative number for the situation described.

9 Thetemperatureis12°Fbelowzero .

Thetemperatureis °F .

10 Ahot-airballoonis150feetintheair .

Thehot-airballoon’selevationis feet .

Write a positive or negative number to describe each person’s account balance.

11 Coleowesthebank$21 .15 .

12 Dominicopenedabankaccountbydepositingacheckfor$250 .

Use absolute value to represent each situation. Then solve.

13 Gail’scheckingaccountshowsabalanceof265 .34 .HowmuchdebtdoesGailshowinhercheckingaccount?

14 Thedepthofalakewaswrittenas2281__2feet .Howmanyfeetbelowthesurfaceisthebottomofthelake?

I know! A deposit means putting

money into the bank.

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READY TO gO

PROblEm sOlvIng

82  Lesson 8

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ChECk

sOlvE

READ

PlAn

Tamilahad$75 .80inherbankaccount .Thebankcreditedheraccount$35 .45andthendebitedtheaccount$40 .Doesheraccounthavelessthanormorethan$75 .80afterthesetransactions?

• Whatistheproblemaskingyoutofind?

IfTamila’saccounthas thanor than$ now

• Whatdoyouneedtoknowtosolvethisproblem?

Thereferencepointis .

Acreditisa numberandadebitisa number .

Ifthe|credit|ismorethanthe|debit|,thentheaccounthasmorethanthereferencepoint .Ifthe|credit|islessthanthe|debit|,thentheaccounthaslessthanthereferencepoint .

• Howcanyousolvethisproblem?

Comparetheabsolutevaluesofthe and .

Thecreditcanbedescribedbythenumber .

Thedebitcanbedescribedbythenumber .

Findtheabsolutevalues .

|35 .45|5 |240|5

Comparetheabsolutevalues .

The|credit|is thanthe|debit| .

Tamila’saccounthas thanthereferencepointnow .

Checktheproblembyusingadditionandsubtraction .

$75 .801$35 .455

2$405 .

DoesTamila’saccounthavemorethanorlessthan$75 .80?

dEBIT oR CREdIT?

If I compare the absolute values

of the credit and debit, I can solve the problem with

mental math!

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PRACTICEUse the problem-solving steps to help you.

1 Thegroundfloorofabuildingisfloor0 .Jeremygotintotheelevatorandpushedthebuttonmarked3 .Alanawentdownoneflightofstairstothebasement .Writeanumbertodescribeeachperson’spositioninthebuilding,relativetothegroundfloor .

Jeremy’spositioncanbedescribedbythenumber .

Alana’spositioncanbedescribedbythenumber .

2 TaliaandRobertopenedaccountsatthesamebankonthesameday .Afterthefirstweek,Talia’sbalancewas$55 .67andRobert’sbalancewas2$2 .17 .Useabsolutevaluetodescribeeachperson’saccountbalance .

3 Afootballcoachusesnegativenumbersforyardsawayfromtheotherteam’sgoalline .Heusespositivenumbersforyardsclosertotheotherteam’sgoalline .Afterthefirstplay,thecoachsaysthattheball’spositionispositive12 .Whatdoesthatmean?

CHECKLIST

READ

PLAN

SOLVE

CHECk

CHECKLIST

READ

PLAN

SOLVE

CHECk

CHECKLIST

READ

PLAN

SOLVE

CHECk

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