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Page 1 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
GradeLevel/Course:Grade7,Grade8,andAlgebra1
Lesson/UnitPlanName:IntroductiontoSolvingLinearInequalitiesinOneVariable
Rationale/LessonAbstract:Thislessonisdesignedtointroducegraphinginequalitiesonanumberlineaswellassolvingone-steplinearinequalitiesinonevariable.
<“lessthan”≤“lessthanorequalto”>“greaterthan”≥“greaterthanorequalto”
Whilethesetranslationsaretechnicallycorrect,itismoreimportanttothinkoftheinequalitysymbolasacomparisonoftwovaluesratherthanasimpletranslationfromlefttoright.Thereforethislessonalsofocusesonhowtoreadinequalitiesfromrighttoleftaswellasfromlefttoright(i.e.thesymbol<canbetranslatedas“greaterthan”iftheinequalityisbeingreadfromrighttoleft).Timeframe:60minutes
CommonCoreStandard(s):
A.REI.3.1Solveone-variableequationsandinequalitiesinvolvingabsolutevalue,graphingthesolutionsandinterpretingthemincontext.
InstructionalResources/Materials:Warmup(pg.10),studentnote-takingguide(pg.11,12).**Pleasenotethatyoutryproblemsonthestudentnote-takingguidearemissingtopreventstudentsfromworkingahead**
AnswerstoWarmUp:Solvetheequationandthengraphyoursolutiononthenumberlinebelow.
2 3 102 3 2 10 2
3 123 123 3
4
xxxx
x
− + =− + + = +
=
=
=
Write< or≥ intheboxtomakethesentencetrue.
) 10 6
) 7 4
) 2 2
a
b
c
− −
Solvetwoways.
245x= −
( )
245
5 5 242 2 510
x
x
x
= −
⎛ ⎞− = − −⎜ ⎟⎝ ⎠− =
( )
( )
245
21 4 15
24525 4 55
20 220 22 210
x
x
x
x
xx
x
= −
⎛ ⎞− = − −⎜ ⎟⎝ ⎠
− =
⎛ ⎞− = ⎜ ⎟⎝ ⎠− =− =
− =
List5possiblesolutionstotheinequalitybelow.Thengraphallpossiblesolutions.
4x ≠ ( x doesn’tequal4)
3, 5,6,10,0,x = − L
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
Page 2 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
PriorKnowledgeStudentsvarywithhowtheywereintroducedtoinequalitiesinpreviousgrades.Herearesomeexamplestohelpyourawarenessasyouworkwithyourstudents:Activity/Lesson:GraphingInequalitiesExample1:Grapha)3 x≥ i.e.
5− 4− 3− 2− 1− 0 1 2 3 4 5
Whenreadingthisfromlefttorightwecansay“3isgreaterthanorequaltox”.Whenreadingthisfromrighttoleftwecansay“xislessthanorequalto3”.Ifxislessthanorequalto3,whataresomepossiblevaluesforxthatmakethisinequalitytrue?(callonstudentsandgraphcorrectvaluesaspointsonthenumberline)
8 6
Pacman,Alligator,oranythingelseeatsthebiggernumber
Thebiggeropeningisonthesideofthebiggernumberandthesmalleropeningisonthesideofthesmallernumber
7 1> Afterdrawinginendpoints,thesidewithtwopointsisonthesideofthebiggernumberandthesidewithonepointisonthesideofthesmallernumber
9 10< Thearrowpointstothesmallernumber
Page 3 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Drawthegraphoverthegraphyouwerebuildingwiththestudentsabove.b) 2n > − Callonarandomstudenttoreadtheinequalityfromlefttoright.“nisgreaterthan–2”Thenanothertoreadfromrighttoleft.“–2islessthann”Checkasolution:
2
4 2
n > −
> −
c) 4w ≤ − Anotherwaytofigureoutwhichwaytoshadeisbytestingapoint.Forinstance,testtoseeif 2x = isasolutionbypluggingitintotheinequality:
Whatisthelargestvaluethatxcanbe?[3]Sincexcanequal3oranythingless,wecanrepresentthesolutionsetbymarkingacloseddotonthethreeandthenshadingeveryvaluetotheleftwherethevaluesareless.
Cannequal–2?[no]
Sincencan’tequal–2,weputanopendotonthe–2toshowthatitisn’tasolution.
Whichvalueswouldbegreaterthan–2:thevaluesontheleftorrightof–2?[right]
x
5− 4− 3− 2− 1− 0 1 2 3 4 5
4− 3− 2− 1− 0 1 2 3 4 5
n
5−
ChoralResponse“Canwequal–4?”[yes]“Shouldweputanopendotorcloseddotonthe–4?[closeddot]
Ourgraphshowsthat 4n = isasolution.Let’schecktoseeifitworksinourinequality.
True!!
Page 4 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
42 4x ≤ −≤ −
Since 2x = isnotasolutionanditistotherightofourboundarypoint ( )4x = − ,thenthesolutionstotheinequalitymustbetotheleft:Writethetwoproblemsforstudentstotry(missingfromnotetakingguide):YouTry:Graph.YouTry:Writeaninequalityrepresentedbythegraph.d)1 x< e)Answers:d)1 x< e)“xisgreaterthan1.”“allthevaluesarelessthanorequalto0.” 0z ≤ SolvingInequalitiesWritethefollowingonthewhiteboard:
4 6
4 4 __ 6 4
0 __10
− <
− + +
Since0 10< ,thesignremainsthesameafteraddingthesamevaluetobothsides.
4− 3− 2− 1− 0 1 2 3 4 5 5− X
False!!
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6 6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
Ifweaddorsubtractthesamenumberfrombothsidesofaninequality,willtheinequalitysignremainthesameorhavetobereversed?
Let’sinvestigate.
4− 6 0
4− 6 0 10
10
0
Page 5 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Gobackandwritetheinequalitysigns:
4 6
4 4 __ 6 4
0 __10
− <
− + +
Showstudentsthatsubtractingdoesn’thaveaneffecteitherasitwouldbesimilartoaddinganegativenumber:
4 6
4 4 __ 6 4
8 __ 2
− <
− − −
−
Showthisexampletothestudentswithanopennumberlinetotheright:Example2:Solveandgraph.a)
2
2
2
5
2
7
7
x
x
x
+ +
− > −
− > −
> −
Whenweadded4tobothsides,thetwovaluesmovedtotherightonthenumberline.Thesmallervalue(thepointontheleft)wouldstillbethesmallervaluewhencomparedtotheother,sinceitisstillfurthertotheleft.
Wouldaddinganegativenumberreversethesign?[No,thevalueswouldmovetotheleft,butthebiggervalue(thepointontheright)wouldstillbethebiggervaluewhencomparedtotheother(furthertotheright)]
7−
2x −
7−
2x −
5−
x
If 2x − isgreaterthan 7− ,thenanyofthesepointsaretwolessthan x .
Therefore x wouldhavetobegreaterthan 5− .
Page 6 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Solvethisinequalityfirstandthengraphthesolution.b)
10 10
6 10
6 10
4
g
g
g
− ≥ − +
− ≥ +− +
≥
+
Writetheproblemforstudentstotry(missingfromnotetakingguide):YouTry:Solveandgraph.c) 5 7 n− ≤ − + Walkaroundtheroomandcheckforunderstanding.Ifthemajorityofstudentsseemtobehavingnoproblemshavethemsharetheiranswerswithaneighbor.Ifnot,usethe“youtry”toreteachandgiveasimilarexample(i.e. 10 8x− > − ,buttrytotailorittothemistakesyousee).Answer:
7
5 7
5
2
77
n
n
n
− ≤ − +
− ≤ +− +
≤
+
Callonstudentsrandomly:“WhatdoIneedtodotosolveforg?”[add10tobothsides]“Whatsideisthevariableon?”[right]“Sincethevariableisontheright,howdoIreadtheinequalityfromrighttoleft?”[gislessthanorequalto4]“Doesthegraphhaveanopenorclosedcircleat4?Why?”[closed,gcanequal4]“Isthegraphshadedtotheleftortheright?Why?”[left,becausegislessthanorequalto4]
4 g
2
n
Page 7 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Writebothinequalitiesontheboardside-by-sideandinvestigatewithyourclassmultiplyingbyapositiveandnegative2:Multiplyingbothsidesby2Multiplyingbothsidesby–2
( ) ( )
2
4 6
4 __ 2 6
8 __12
− <
−
−
( ) ( )
2
4 6
4 __ 2 6
8 __ 12
−
− <
− −
−
Gobackandwritetheinequalitysigns:
Multiplyingbothsidesby2Multiplyingbothsidesby–2
( ) ( )
2
4 6
4 __ 2 6
8 __12
− <
−
−
( ) ( )
2
4 6
4 __ 2 6
8 __ 12
−
− <
− −
−
Showstudentsthatdividinghasthesameeffect:
Dividingbothsidesby2Dividingbothsidesby–2
4 6
4 6 __ 2 2
2 __ 3
− <
−
−
4 6
4 6 __2 2
2 __ 3
− <
−− −
−
Ifwemultiplyordividethesamenumberfrombothsidesofaninequality,willtheinequalitysignremainthesameorhavetobereversed?
Let’sinvestigate.
Since 8 12− < ,thesignremainsthesametomakea
truestatement.
Since8 12> − ,thesignneedstobereversedtomakeatruestatement.
Page 8 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Example3:Solve.a) 2 10x− > b) 2 10x > − Beforeworkingitout,doaThink-Pair-Share:Whenwesolvetheseinequalities,arewegoingtoreversethesignforoneinequalityorboth?Why?Afterlettingafewpartnersshare,worktheproblemsoutside-by-sidea) 2 10x− > b)2 10x > −
2 102 2
5
x
x
− <− −
< −
2 102 2
5
x
x
−>
> −
Example4:Solveandgraph.Letstudentsguideyouinsolvingtheinequality.Thenshowstudentsthesecondwaywhereyoudon’thavetoreversetheinequality.
( )
( )
123
1 12 13
123
3 12 33
36
x
x
x
x
x
− ≥ −
⎛ ⎞− − ≤ − −⎜ ⎟⎝ ⎠
≤
⎛ ⎞≤ ⎜ ⎟⎝ ⎠≤
Whatdoweneedtodotosolveparta)?[Dividebothsidesby–2]
Howaboutpartb)?[Dividebothsidesby2]
Wedividedbya–2sowereverse
thesign
Wedividedbya+2sowedonotreversethesign
36
x ( )
123
123 3 3
12 03
12 12 0 123
123
3 3 123
36
x
x x x
x
x
x
x
x
− ≥ −
− + ≥ − +
− + ≥
− + + ≥ +
≥
⎛ ⎞ ≥⎜ ⎟⎝ ⎠≥
Page 9 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Writetheproblemforstudentstotry(missingfromnotetakingguide):YouTry:SolveandGraph.
( )
5 102
52 2 102
5 20
5 205 5
4
y
y
y
y
y
− <
⎛ ⎞− <⎜ ⎟⎝ ⎠− <
− >− −
> −
Havestudentschecktheiranswerwithapartner.
Example5:Useaninequalitytorepresentthesituationandsolve.Youarecheckingabagattheairport.Bagscanweighnomorethan50lbs.Ifyourbagalreadyweighs20lbs,findallpossibleweights,w,thatyoucanaddtothebag?
w =weightyoucanaddtothebag
20 50
20 20 50 2030
w
ww
+ ≤
+ − ≤ −≤
4−
y
**Studentswillsaytheansweris30,soremindthemthatwearenotlookingforthemostyoucanadd.
Setuptheinequalitywithyourstudentsandsolve.Whenyougraphthesolution,talkaboutwhatvaluesofw arepossible(eventhoughitappears–40isasolution,thereareonly20lbs.inthesuitcasesoitwouldn’tmakesenseevenifyouthoughtofnegativesasremovingweight).Whateveryoudecideispossibleornot,adjustyourminimumvalueonthegraph.
30
w
Page 10 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
Warm-Up Algebra:A.REI.3.1 Review
Solvetheequationandthengraphyoursolutiononthenumberlinebelow.
2 3 10x− + =
Write< or≥ intheboxtomakethesentencetrue.
) 10 6
) 7 4
) 2 2
a
b
c
− −
x
y
Algebra:A.REI.3.1
List5possiblesolutionstotheinequalitybelow.Thengraphallpossiblesolutions.
4x ≠ ( x doesn’tequal4)
___, ___, ___, ___, ___,x = L
Algebra:A.REI.3.1
Solvetwoways.
245x= −
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
Page 11 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
IntroductiontoSolvingInequalities:Note-TakingGuide
GraphingInequalitiesExample1:Graph.a)3 x≥ b) 2n > − c) 4w ≤ − YouTry:Graph.YouTry:Writetheinequalityrepresentedbythegraph.d)e)SolvingInequalitiesExample2:Solveandgraph.a) 2 7x − > − b) 6 10 g− ≥ − + YouTry:Solveandgraph.c)
5− 4− 3− 2− 1− 0 1 2 3 4 5
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
5− 4− 3− 2− 1− 0 1 2 3 4 5 5− 4− 3− 2− 1− 0 1 2 3 4 5
6− 5− 4− 3− 2− 1− 0 1 2 3 4 5 6
Page 12 of 12 MCC@WCCUSD(WCCUSD) 03/09/16
SolvingInequalities(cont.)Example3:Solve.a) 2 10x− > b) 2 10x > −
Example4:Solveandgraph.
123x− ≥ −
YouTry:Solveandgraph.
Example5:Useaninequalitytorepresentthesituationandsolve.Youarecheckingabagattheairport.Bagscanweighnomorethan50lbs.Ifyourbagalreadyweighs20lbs,findallpossibleweights,w,thatyoucanaddtothebag?
