Download - SSON 5 Adding and Subtracting Rational Numbers · 2019. 12. 22. · The additive inverse of 26 is 6 . So, 210 2 ( 6) 5 210 1 6 Since 210 and 6 have opposite signs, subtract the lesser

LESSON

30 Chapter 2: The Number System

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UNDERSTAND Everyrational numberhasalocationonthenumberline .Numberstotheleftof0onthenumberlinearenegative .Numberstotherightof0arepositive .Anynumbersnand2nareopposites,oradditive inverses .Thesumofnand2nis0 .

Youcanuseanumberlinetoaddorsubtractrationalnumbers .Itmayhelpyoutothinkofthesumofp1q asthenumberlocatedadistance|q|unitsfromp onthenumberline .

Findthesum:2313

Useanumberline .23isnegative,somove3unitstotheleftof0,asshownbythebluearrow .Toaddpositive3tothatnumber,move3unitstotherightasshownbythegreenarrow .Thegreenarrowpointsto0,whichisthesum .

▸ 231350 .Thismakessensebecause23and3areadditiveinverses .

–5 –4 –3 –2 –1 0 1 2 3 4 5

� 3

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5 Adding and Subtracting Rational Numbers

Matthasabalanceof2$2 .50inhisbankaccount .Headds$2 .50 .Findhisnewbalance .

Thebluearrowshowsmovingfrom0toapointhalfwaybetween23and22 .Thatpointshows22 .50 .Thegreenarrowmodelsadding2 .50tothatamount .Thesumis0 .

–3 –2 –1 0 1 2 3

� 2.5

� 2.5

▸Mattnowhasa$0balanceinhisbankaccount .22 .50and2 .50areadditiveinverses .

Insteadofusinganumberlinetoaddintegers,youcanlookatthesignsoftheaddends:

• Iftheyhavethesamesign,addtheirabsolutevaluesandgivethesumthatsign .

• Iftheyhaveoppositesigns,subtractthelesserabsolutevaluefromthegreaterabsolutevalue .Givethesumthesignoftheaddendwiththegreaterabsolutevalue .

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Connect

Lesson 5: Adding and Subtracting Rational Numbers 31

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Usetwodifferentmethodstofindthesum:261(24)

Useanumberlineandtheruleforaddingintegerswiththesamesign .

Youcoulduseanumberlinetodeterminethat(26)1(24)5(210) .

Since26and24havethesamesign,youcouldalsoaddthemlikethis:

|26|1|24|5614510

Since26and24arenegative,givethesumanegativesign .

▸Thesumis210 .Thisshowsthat(2)1(2)5(2) .

� 6

� (�4)

–10 –8 –6 –4 –2 0 2 4 6 8 10

Useanumberlineandtheruleforaddingintegerswithoppositesigns .

Youcoulduseanumberlinetodeterminethat(9)1(23)56 .

Since9and23haveoppositesigns,findtheirdifference .

|9|59 |23|53

92356

Theaddend9hasthegreaterabsolutevalue,sogivethesumapositivesign .

▸Thesumis6 .Thisshowsthatsometimes(1)1(2)5(1) .

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� (�3)

–10 –8 –6 –4 –2 0 2 4 6 8 10

Usetwodifferentmethodstofindthesum:91(23)

Chooseonemethod(numberlineorrulesforaddingintegers) .Useittofindthesum:41(29) .Usethisexampletoshowthatsometimes(1)1(2)5(2) .

TRY

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EXAMPLE A Findthedifference:42(26)

EXAMPLE B Findthedifference:2102(26)

Rewritetheproblemasanadditionproblem .

Theadditiveinverseof26is6 .

So,42(26)5416 .

Since4and6havethesamesign,addtheirabsolutevalues .

|4|1|6|5416510

Bothaddendsarepositive,sotheanswerispositive,10 .

▸42(26)510

Rewritetheproblemasanadditionproblem .

Theadditiveinverseof26is6 .

So,2102(26)521016

Since210and6haveoppositesigns,subtractthelesserabsolutevaluefromthegreaterabsolutevalue .

|210|.|6|because10.6,so:

|210|2|6|5102654

Givetheanswerthesignofthenumberwiththegreaterabsolutevalue .

Since210hasthegreaterabsolutevalue,theansweris24 .

▸ 2102(26)524

Drawamodeltofindthedifference:2325 .

MODEL

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Problem Solving


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read

Thehightemperatureinacityonedaywas2 .6°C .Thelowtemperaturewas22 .8°C .Whatwasthetemperaturerangeinthecitythatday?

plan

Thetemperaturerangeforthedayisequaltothedistancebetweenthehigh

temperatureandthelowtemperature .

Thedistancebetweentworationalnumbersonthenumberlineisequaltotheabsolutevalueoftheirdifference .Useabsolutevaluetofindtheanswer .

solve

Findtheabsolutevalueofthedifference:2 .62(22 .8) .

|2 .62(22 .8)|5|2 .61 |5| |5

So,thetemperaturerangeinthecitythatdaywas degreesCelsius .

check

Onewaytocheckthatthedistancebetween2 .6and22 .8isactuallytheansweryoufoundaboveistofindthedistancebetweenthosetwonumbersonthenumberline .

Plotpointsfor2 .6and22 .8onthenumberlinebelow .Thencounttheunitsbetweenthemtocheckthatyouransweriscorrect .

–3 –2 –1 0 1 2 3

Thedistancebetween2 .6and22 .8onthenumberlineis units .

Isthisthesameansweryoufoundabove?

▸Therangeofthetemperaturesonthatdaywas °C .

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Practice


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Identify the additive inverse, or opposite, of each number.

1. 7 2. 26

Change the sign of each number to find its opposite.

HINT

3. 9 .53

Rewrite each subtraction expression as an addition expression.

4. 1025

1

5. 211 .1212 .6

1

REMEMBER The expression p 2 q is the same as p 1 (2q ).

6. 1__82 211__5 1

Use the number line to find each sum or difference. Show your work on the number line.

7. 241(25)

–10 –8 –6 –4 –2 0 2 4 6 8 10

9. 62(24)

–10 –8 –6 –4 –2 0 2 4 6 8 10

8. 22110

–10 –8 –6 –4 –2 0 2 4 6 8 10

10. 5211

–10 –8 –6 –4 –2 0 2 4 6 8 10

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The following thermometers show the range in temperatures in a given day. Find the temperature ranges. Show or explain your work.

11.

–10

0

10

20

30

Temperature (in °F)

hightemperature

lowtemperature

12. 543210

–1–2–3–4–5

Temperature (in °C)

hightemperature

lowtemperature

13.

–1

–2

0

1

2

3

Temperature (in °C)

hightemperature

lowtemperature

Choose the best answer.

14. Aheliumatomhas2electrons,eachofwhichhasa21charge .Eachprotonhasa11charge .Overall,theheliumatomhasnocharge .Howmanyprotonsmusttheatomhave?

A. 22 B. 0

C. 2 D. 3

15. OnJanuary1,Rose’sbankbalancewas$200 .Duringthemonth,shewrotechecksfor$115 .25and$350 .00andmadeonedepositof$150 .50 .Whichbestrepresentshercheckingaccountbalanceattheendofthemonth?

A. 2$114 .75 B. 2$114 .25

C. $114 .25 D. $115 .50

Solve.

16. Show Irawritesthatthesumofa1bisanumberthatislocatedadistanceof|b|unitsfromaonthenumberline .Choosefourvaluesforaandbandshowthatthisistrue .Usethefournumberlinesontherighttoshowyourworkandsupportyouranswer .

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

–5 –4 –3 –2 –1 0 1 2 3 4 5

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