ALGEBRA I Interactive Workbook

KELLENBERG MEMORIAL HIGH SCHOOLMATHEMATICS DEPARTMENT

PRE-ALGEBRA REVIEW

Chapter 1

1.1 Operations with Whole Numbers, Decimals, and Fractions1.2 Integers and Signed Numbers1.3 Order of Operations1.4 Basic Definitions and Evaluating Expressions1.5 Sets1.6 Closure1.7 Writing Verbal Statements1.8 Properties of Equality1.9 Like Terms

AddingandSubtractionDecimals

Whenaddingandsubtractingnumberswithdecimals,thenumbersshouldbestackedvertically.Alldecimalpointsshouldbelinedupandanyemptyplacevaluesshouldbe;illedwithzeros.Addorsubtracteachcolumnrememberingtocarryorborrowwhereneeded.Thedecimalpointdropsdownandstaysinthesameplacewhereitstarted.

MultiplyingDecimals

Whenmultiplyingnumberswithdecimals,multiplyasusual.Thedecimalpointsdonotneedtobelinedupformultiplication.Attheendmovethedecimalpointontheanswertotheleft,byaddingupthetotalnumberofplaceseachnumberhasafterthedecimalpoint.

DividingDecimals

Whendividingwithdecimals,ifthedivisor(thenumberyouaredividingby)isnotawholenumber,movethedecimalpointtomakeitawholenumber.Movethedecimalpointonthedividend(thenumberyouaredividing)thesamenumberofplacesasthedivisor.Divideasusual,placingthedecimalpointdirectlyabovewhereitisinthedividend.

ImproperFractionsandMixedNumbers

Tochangeanumberfromamixednumbertoanimproperfraction,multiplythedenominatorbythewholenumberthenaddthenumeratortoit.Placethatnumberovertheoriginaldenominator.

Tochangeanumberfromanimproperfractiontoamixednumber,dividethenumeratorbythedenominator.Thequotientbecomesthewholenumber,theremainderbecomesthenumerator,andtheoriginaldenominatorstaysthesame.

Section 1Operations with Whole Numbers, Decimals and Fractions

2

!Example:!!!4!+!4.64!+!12.892!!!! 4.000!!!!!!!!!!!!04.640!!!!+!!!!!12.892!!!!!!!!!!!17.532!!

!Example:!!!! 12.89!–!11.643!!!!!!!12.890!3!!!11.643!!!!!!!!1.247!!

!Example:!4.812!!x!!3.62!!!!!!!!!!!!!!!!!4.812!!!(3!places!after!the!decimal!point)!!!!!!x!!!!!!!3.62!!!(2!places!after!the!decimal!point)!!!!!!!!!!!!!!9624!!!!!!!!!288720!+!!!1443600! !!!!!!17.41944!!(move!decimal!to!the!left!a!total!of!5!places)!!!

Example: 24.396 ÷ 3.8 24.396 is the dividend, 3.8 is the divisor 006.42 38 243.96 -228 159 - 152 76 - 76 0

!Example:!!!4!+!4.64!+!12.892!!!! 4.000!!!!!!!!!!!!04.640!!!!+!!!!!12.892!!!!!!!!!!!17.532!!

Adding Decimals

!Example:!!4 !! !!"#$!!"!!"#$%#&$!!"#$%&'(!!3!!!4!=!12!!12!+!2!=!14!!=!!"! !!!!

Example: into a mixed number 8 R1 2 17 -16 1

!Example:!!4 !! !!"#$!!"!!"#$%#&$!!"#$%&'(!!3!!!4!=!12!!12!+!2!=!14!!=!!"! !!!!Mixed Number to Improper Fraction

AddingandSubtractingwithFractions(CommonDenominator)

Whenaddingandsubtractingwithfractions,youmusthaveacommondenominator.Firstchangeanymixednumbersintoimproperfractions.Ifyouhaveacommondenominator,addthenumeratorstogetherandkeepthedenominatorthesame.Simplifybyremovinganycommonfactorsthenumeratoranddenominatorshare.

AddingandSubtractingwithFractions(UncommonDenominator)

Whenaddingandsubtractingwithfractions,youmusthaveacommondenominator.Firstchangeanymixednumbersintoimproperfractions.Ifyoudonothaveacommondenominator,;indtheleastcommondenominator(LCD-lowestmultipleofthedenominators).MultiplythenumeratoranddenominatorofeachfractionbyanumberthatwouldmakethedenominatorsequaltotheLCD.Onceyouhaveacommondenominator,addthenumeratorstogetherandkeepthedenominatorthesame.Simplifybyremovinganycommonfactorsthenumeratoranddenominatorshare.

MultiplyingFractions

Allmixednumbersshouldbeturnedintoimproperfractions.Cross-simplifythefractionswherepossible.Multiplytheremainingnumerators.Multiplytheremainingdenominators.Simplifyattheendifneeded.

DividingFractions

Allmixednumbersshouldbeturnedintoimproperfractions.Multiplybythereciprocalofthesecondfraction(KEEPthe;irstfraction,CHANGEthesigntomultiplication,FLIPthesecondfraction).Cross-simplifythefractionswherepossible.Multiplytheremainingnumerators.Multiplytheremainingdenominators.Simplifyattheendifneeded.

PracticeProblems1

PracticeProblems2 3

Example:

!Example:!!!! − !

!!!

!!!!!!!!!!!!!!! − !

!! = !

!!!

!!!

Example:

Adding with a Common Denominator

Adding and Subtracting

Fractions with Uncommon

Denominators

Multiplying Fractions

Dividing Fractions

AddingwithPositiveandNegativeNumbers

Whenaddingwithpositivesandnegatives,ifthesignsarethesame,addthenumberstogetherandkeepthesign.Ifthesignsaredifferent(onepositiveandonenegative)taketheabsolutevalueofeachnumberandsubtractthelargerabsolutevalueminusthesmaller.Keepthesignofwhichevernumberhasthebiggerabsolutevalue.

SubtractingwithPositiveandNegativeNumbers

Whensubtractingwithpositivesandnegatives,turnthesubtractionexpressionintoanadditionexpressionbyusingthemethodkeep,change,change.Oncetheexpressionissetupasaddition,followthesamestepsyouwouldforaddingwithpositivesandnegatives.

MultiplyingandDividingwithPositiveandNegativeNumbers

Whenmultiplyingordividingwithpositivesandnegatives,ifthesignsarethesametheanswerwillbepositive,ifthesignsaredifferenttheanswerwillbenegative.

•Apositivenumbermultipliedordividedbyanotherpositivenumberresultsina positiveanswer.

•Apositivenumbermultipliedordividedbyanegativenumberresultsina negativeanswer.

•Anegativenumbermultipliedordividedbyanothernegativenumberresultsina positiveanswer.

• Anegativenumbermultipliedordividedbyapositivenumberresultsinanegativeanswer.

!Example:!!17!+!-!12!!! !17!+!-!12!=!5!!!!!!!!!!!!!!

!Example:!!19!+!-!27!!! 19!+!-!27!=!-!8!!!!!!!!!!!!!!!

!Example:!!*!4!+!*!16!!! !*!4!+!*!16!=!*!20!!!!!!!!!!!!!!

!Example:!!17!+!-!12!!! !17!+!-!12!=!5!!!!!!!!!!!!!!

Adding Integers

Section 2Integers and Signed Numbers

4

!Example:!!−!47! − !−21!!

−!47!− !−21!!(keep,!change,!change)!

!!!!!!!!!!!! − !47!+ !+!21 = −26!!!!!!!!!!!!!!

!Example:!!37!,!46!!

37!,!46!!(keep,!change,!change)!37!+!,!46!=!,!9!!

!!!!!!!!!!!!!

!Example:!!*!49!*!23!!

*!49!*!23!!(keep,!change,!change)!*!49!+!*!23!=!*!72!

!!!!!!!!!!!!!

!Example:!!−!47! − !−21!!

−!47!− !−21!!(keep,!change,!change)!

!!!!!!!!!!!! − !47!+ !+!21 = −26!!!!!!!!!!!!!!

Subtracting Integers

!Example:!!−16! ∙ !−4!!!!!!!! !!!!!−16! ∙ !−4 = +!64!

!Example:!!32 ∙ !−2!!!!!!!! !!!!!32 ∙ !−2 = −64!

!Example:!!−20!÷ !−4!!!!!!!! !!!!!−20!÷ !−4 = +!5!

!Example:!!−45!÷ !+5!!!!!!!! !!!!!−45!÷ !+5 = −9!

!Example:!!−16! ∙ !−4!!!!!!!! !!!!!−16! ∙ !−4 = +!64!

Multiplying and Dividing Integers

PracticeProblems1

PracticeProblems2

5

Parentheses,Exponents,Multiplication,Division,Addition,andSubtraction

Whencompletingallmathematicalexpressionsthereisanorderofoperationsthatisfollowed.Thisorderismostcommonlyreferredtoas“PEMDAS,”whichstandsforparentheses,exponents,multiplication,division,addition,andsubtraction.

• Parentheses:Anyoperationsinsideofparenthesesshouldbeperformed;irst.Iftherearetwosetsofparentheses,alwaysstartwiththeinnermostparenthesesandworkyourwayout.Next,moveontoexponents.

• Exponents:Performalloperationswithexponentsnext.Rememberthatexponentsstatehowmanytimesanumbershouldbemultipliedbyitself.Next,moveontomultiplicationanddivision.

• MultiplicationandDivision:Multiplicationanddivisionshouldbeperformedfromlefttoright.Ifdivisioncomesbeforemultiplicationinanexpression,itshouldbeperformed;irst,andviceversa.Next,moveontoadditionandsubtraction.

• AdditionandSubtraction:Additionandsubtractionshouldbeperformedfromlefttoright.Ifsubtractioncomesbeforemultiplicationinanexpression,itshouldbeperformed;irst,andviceversa.Attheendofadditionandsubtraction,yourexpressionshouldbecompleted.

PracticeProblems1

PracticeProblems2

!Example:!!16!÷ 4! ∙ !2! − 10− 7 + 5!!16!÷ 4! ∙ !2! − 10− 7 + 5!16!÷ 4! ∙ !2! − 3+ 5!16!÷ 4! ∙ !4− 3+ 5!4! ∙ !4− 3+ 5!16− 3+ 5!13+ 5!=!18!!!!!!!!

Section 3Order of Operations

6

Order of Operations

BasicTermsandDeGinitions

Inthestudyofmathematicstherearetermsthatareseenfrequently.Itisimportanttoknowthede;initionandhaveanunderstandingoftheseterms.

• Sum-theanswerthatisreachedwhenaddingtwoterms.

• Difference-theanswerthatisreachedwhensubtractingtwoterms.

• Product-theanswerthatisreachedwhenmultiplyingtwoterms.

• Quotient-theanswerthatisreachedwhendividingtwoterms.

• Variable-aletterthatisusedtorepresentanumber

• Term-referstoanumber,variable,orbothheldtogetherbymultiplicationor division.

• Factors-thenumbersmultipliedtogethertoreachaspeci;icproduct.

• Example:thefactorsof8are:1and8,2and4

• Multiples-theproductwhenaspeci;icnumberismultipliedbyaninteger.

• Example:themultipliesof8are:0,8,16,24,32,40

• Coef;icients-thenumberinfrontofavariable.Ifavariabledoesnothavea coef;icient,thecoef;icientisone.

• Example:thecoef;icientof6xis6

EvaluatingExpressions(Substitution)

Whenevaluatinganexpression,substitutethevariableintheexpressionwiththenumberassignedtoit.Ifthevariableappearsmorethanonceintheexpression,replaceiteachtime.Oncethevariablesarereplacedintheexpression,simplifybyfollowingtheorderofoperations.

PracticeProblems

Substitution

Section 4Basic Definitions and Evaluating Expressions

7

Sets

• Sets-agroupofelementsornumbers

• Elements-themembersofaset

• Thesymbol

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∈meansanelementofaset.Thesymbol

€

∉meansnotanelementofaset.

• Example:A={1,2,3,4},4

€

∈A,5

€

∉A:WhengivensetA,4isanelementofthesetand5isnotanelementoftheset.

• EqualSets-havethesameelementsintheirsetsbutmaybewritteninadifferentorder.

• Example:{1,2,3}and{2,1,3}areequalsetsbecausetheyhavethesameelementsintheirsets.

• EquivalentSets-havethesamenumberofelementsintheirsetsbutnotalwaysthesameelements.

• Example:{1,2,3,4}and{M,A,T,H}areequivalentsetsbecausetheybothhavefourelements.

• ContinuousSet-thepatterninasetcontinueswhenasetendswiththreedots.

• Example:{2,4,6…}:Meansallevennumbersfrom2continuingtoin;inity.

• FiniteSet-asetwhoseelementscanbecounted.

• Example:{studentsattendingKellenberg}

• In;initeSet-asetwhoseelementscannotbecounted.

• Example:{allevennumbers}

• Empty/NullSet-therearenoelementsinaset.Canbewrittenas{}or

€

∅ .

• Example:{;ifthgradestudentsattendingKelleneberg}

• Predecessor-thenumberthatprecedes(comesbefore)anothernumber.

• Example:3isthepredecessorof4.

• Successor-thenumberthatcomesafteranothernumber.

• Example:4isthesuccessorof3.RealNumbers

Realnumbersincludeallrationalanirrationalnumbers.

RationalNumbers

Rationalnumbersconsistof:

• NaturalNumbers-Startwithoneandcontinuetopositivein;inity{1,2,3…}.

• WholeNumbers-Startwithzeroandcontinuetopositivein;inity{0,1,2…}.

• Integers-wholenumbersandtheiropposites,continuingtobothnegativeandpositivein;inity{…-2,-1,0,1,2…}.

• TerminatingDecimals-decimalsthatend.

• Example:½=0.5

• RepeatingDecimals-decimalsthatdonotendbutcontinuouslyrepeat.

• Example:

Section 5Sets

8

€

13 = 0.3

IrrationalNumbers

Irrationalnumbersarenumbersthatdonot;itthecriteriaofrationalnumbers.Piisanirrationalnumber.Squarerootsthatdonothaveaperfectsquarearealsoirrational.

• Example:

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π = 3.14159265359...

• Example:

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7 = 2.64575131106...

PracticeProblems9

Closure

Asetisclosed(notopen)underaspeci;icoperationifallanswerstotheoperationareinthesetofgivennumbers.Setscanbeclosedundercertainoperationsandnotclosedunderotheroperations.

**Numbersinasetcanbeadded,subtracted,multiplied,ordividedbythemselves.**

PracticeProblems

Closure

Section 6Closure

10

AdditionTermsinVerbalSentences

Termsusedtoexpressadditioninaverbalsentenceare:sum,plus,morethan,inadditionto,andexceeds.

• Example:Anumber,n,exceededby12iswrittenas:n+12

SubtractionTermsinVerbalSentences

Termsusedtoexpresssubtractioninverbalsentencesare:difference,minus,subtractedfrom,less,andlessthan.Whenwritingexpressionsusinglessthanandsubtractedfromthe;irsttermcomessecondandthesecondtermcomes;irst.

• Example:12minusanumber,n,iswrittenas:12–n

• Example:12lessthananumber,n,iswrittenas:n–12

MultiplicationTermsinVerbalSentences

Termsusedtoexpressmultiplicationinverbalsentencesare:product,multipliedby,andtimes.Twiceanumbermeanstwotimesanumber.

• Example:Theproductof12andanumber,n,iswrittenas:12n

• Example:Twiceanumber,n,iswrittenas:2n

DivisionTermsinVerbalSentences

Termsusedtoexpressdivisioninverbalsentencesare:quotient,anddividedby.

• Example:12dividedbyanumber,n,iswrittenas:

€

12n or12÷ n

OtherCommonTermsinVerbalSentences

• Ismeansequals

• Ofmeansmultiplication

• Halfmeanstomultiplyby½ordivideby2

PracticeProblems

Section 7Writing Verbal Statements

11

AdditionProperties

• CommutativePropertyofAddition-youcanaddnumbersinanyorder.

• Example:4+5=5+4

• AssociativePropertyofAddition-youcangroupnumbersinanyorderwhenadding.

• Example:(4+5)+6=4+(5+6)

• AdditionPropertyofZero(AdditiveIdentity)-whenanynumberisaddedtozero,thenumberstaysthesame.

• Example:4+0=4

• AdditionPropertyofOpposites(AdditiveInverse)-whenoppositesareaddedtogether,theyarealwayszero.

• Example:-4+4=0

MultiplicationProperties

• CommutativePropertyofMultiplication-youcanmultiplynumbersinanyorder.

• Example:4x5=5x4

• AssociativePropertyofMultiplication-youcangroupnumbersinanyorderwhenmultiplying.

• Example:(4x5)x6=4x(5x6)

• MultiplicationPropertyofZero-whenzeroismultipliedbyanynumber,theanswerisalwayszero.

• Example:4x0=0

• MultiplicationPropertyofOne(MultiplicativeIdentity)-whenanynumberismultipliedbyone,thenumberstaysthesame.

• Example:4x1=4

• MultiplicationPropertyofReciprocals(MultiplicativeInverse)-whenanynumberismultipliedbyitsreciprocal,theanswerisalwaysone.

• Example:2x½=1

• DistributiveProperty-whenanumberisoutsideapairofparenthesesitisdistributedtoalltermsinsidetheparenthesesthroughmultiplication.

• Example:4(x+4)=4(x)+4(4)=4x+16

PropertiesofEqualities

• AdditionPropertyofEquality-thesamenumbercanbeaddedtobothsidesofanequation.

• Example:Ifa=b,thena+c=b+c

• SubtractionPropertyofEquality-thesamenumbercanbesubtractedfrombothsideofanequation.

• Example:Ifa=b,thena-c=b-c

Section 8Properties of Equality

12

• MultiplicationPropertyofEquality-thesamenumbercanbemultipliedonbothsideofanequation.

• Example:Ifa=b,thenaxc=bxc

• DivisionPropertyofEquality-thesamenumbercanbedividedonbothsidesofaequation.

• Example:Ifa=b,thena/c=b/c

• Re;lexiveProperty-anynumberisequaltoitself.

• Example:4=4

• SymmetricProperty-anequationcanbewritteninanyorder.

• Example:x=4and4=xarethesame

• TransitiveProperty-ifa=bandb=c,thena=c

• Example:If

€

12

=24 and

€

24

=36 then

€

12

=36

• SubstitutionProperty-youcansubstitutequantitiesforoneanotherinanexpression.

• Example:ifa=6,then4+a=4+6

PracticeProblems

13

CombiningLikeTerms

Termsarelikeoneanotheriftheyhavethesamevariableandthesameexponent.Whenaddingorsubtractingliketerms,addorsubtractthecoef;icientsandkeepthevariableandexponentthesame.

PracticeProblems1

PracticeProblems2

Combining Like Terms

Section 9Like Terms

14

SOLVING EQUATIONS

Chapter 2

2.1 One-Step Equations2.2 Two-Step Equations2.3 Parentheses and Variables on Both Sides2.4 Word Problems2.5 Working with Formulas2.6 Solving and Graphing Inequalities2.7 Absolute Value Equations2.8 Absolute Value Inequalities

InverseOperations

• AdditionandSubtractionareinverseoperations

• MultiplicationandDivisionareinverseoperations

**FractionalCoefGicients**

-Usethereciprocalastheinverse.

PracticeProblems

!Example:!!!Solve:!!x!+!6!=!14!!! x!+!6!=!14!! !!!2!!6!!!!!2!6!! !!!!!!x!!=!!8!!Check:!! x!+!6!=!14!! 8!+!6!=!14!! !14!=!14!✓!!!

!Example:!!!Solve:!!x!–!13!=!27!!! x!–!13!!!=!27!! !!!!+13!!!+!13!! !!!!!!!!!!x!!=!40!!!Check:!

x!–!13!!!=!27!!!!!!!!!!!!!40!–!13!=!27!!!!!!!! !!!27!=!27!✓!! !!

!Example:!!!Solve:!!4x!=!24!!! 4x!=!24!! !4!!!!!!!4!!! !!x!=!6!!Check:!! 4x!=!24!!!!!!!!!!!!4(6)!=!24!!!!!!!!!!!!24!=!24!✓!!

!Example:!!!Solve:!!x!=!24!! !!!!!!!!!!!!!!!!!!!!6!!!!!!!!!

6!∙!!x!!=!24!! !!!!!!6!!!!!!!!!! !!x!=!144!!Check:!!!!!!!!!!!!!!x!!=!24!!!!!!!!!!!!!!6!!!!!!!!!!!!!!144!=!24!!!!!!!!!!!!!6!!!!!!!!!!!!!!!!!!24!=!24!✓!!!!!!!!!!

!Example:!!!Solve:!!x!+!6!=!14!!! x!+!6!=!14!! !!!2!!6!!!!!2!6!! !!!!!!x!!=!!8!!Check:!! x!+!6!=!14!! 8!+!6!=!14!! !14!=!14!✓!!!

Addition Equation

Section 1One-Step Equations

16

!Example:!!!Solve:!!!! ! = 4!!!! !

! ∙!! ! = 4 ∙ !!!

! !! x!=!!"! !=!10!!Check:!!!!!!!!!!!!!!!!!!! ! = 4!!

!!!! 10 = 4!!!!!"! = 4!!

!!!!!!!!!!!!!!!!4!=!4!✓!!!!!!!!!!

ToSolve:

-Dotheinverseoftheoperationintheequationonbothsidesoftheequal sign.

-Checkyouranswerbackintotheoriginalequation.

-Solveeachsideindependently.

SolvingTwo-StepEquations

PracticeProblems

!Example:!!!Solve:!!!! + 9 = 12!!! !

! + 9 = 12!! !!!!-!9!!!!!!!-!9!!!!!!!!!!!!!!!2 ∙ !!! = 3 ∙ 2!! !!!!!!!!x!=!6!!!!!!!!!!!!!!!!Check:!!!!!!!!!!!!!!!! + 9 = 12!!!!!!!!!!!!!!!!! + 9 = 12!!! 3!+!9!=!12!!

!!!!12!=!12!✓!!!!!!!!!!

!Example:!!!Solve:!!4! − 7 = 9!!! 4! − 7 = 9!! !!!!!!+!7! !!+!7!! !!4x!=!16!! !!!4!!!!!!!4!! !!!!!x!=!4!Check:!!!!!!!!!!!!!!4! − 7 = 9!!!!!!!!!!!!!4(4)!–!7!=!9!! 16!–!7!=!9!

!!!!9!=!9!✓!!!!!!!!!!

!Example:!!!Solve:!!!! + 9 = 12!!! !

! + 9 = 12!! !!!!-!9!!!!!!!-!9!!!!!!!!!!!!!!!2 ∙ !!! = 3 ∙ 2!! !!!!!!!!x!=!6!!!!!!!!!!!!!!!!Check:!!!!!!!!!!!!!!!! + 9 = 12!!!!!!!!!!!!!!!!! + 9 = 12!!! 3!+!9!=!12!!

!!!!12!=!12!✓!!!!!!!!!!

Solving 2-Step Equations

Section 2Two-Step Equations

17

ToSolve:

-Dotheinverseofadditionandsubtraction;irst.

-Thentheinverseofmultiplicationanddivision.

-Check.

EquationswithParenthesesandVariablesonBothSides

FractionBarsinEquations

PracticeProblems

!Example:!!!Solve:!!– !4(!!– 2) = 16!!! – !4 !!– 2 = 16!! !!!–!4x!+!8!=!16!! !!!!!!!!!!!!!4!8!!!!4!8!! !!!–!4x!=!!8!!!!!!!! !!!!–4!!!!!–4!! !!!!!!!x!=!–!2!Check:!!!!!!!!!!!!– 4(! − 2) = 16!!!!!!!!!!!!–!4(–2!–!2)!=!16!!!!!!!!!!!!!!!–!4(–!4)!=!16!! !!!!16!=!16!✓!!!!!!!!!!

!Example:!!!Solve:!!2x− 7 = 4x+ 13!!! 2x− 7 = 4x+ 13!!!!!!!!!!!!-!2x! !!!-!2x!!!!!!!! !!!!!!!-!7!=!2x!+!13!! !!!!-!13!!!!!!!!!!!-!!13!!!!!!!!!!!!!!!!!!!-!20!=!!2x!!!!!!!!! !!!!!!!2!!!!!!!!!2!! !!!!!!-10!=!!x!Check:!!!!!!!!!!!!2x− 7 = 4x+ 13!!!!!!2(-10)!-!7!=!4!(-10)!+!13!!!!!!!!!!!!!!!-!20!-!7=!-!40+13!! !!!!-!27!=!-!27!✓!!!!!!!!!!

!Example:!!!Solve:!!– !4(!!– 2) = 16!!! – !4 !!– 2 = 16!! !!!–!4x!+!8!=!16!! !!!!!!!!!!!!!4!8!!!!4!8!! !!!–!4x!=!!8!!!!!!!! !!!!–4!!!!!–4!! !!!!!!!x!=!–!2!Check:!!!!!!!!!!!!– 4(! − 2) = 16!!!!!!!!!!!!–!4(–2!–!2)!=!16!!!!!!!!!!!!!!!–!4(–!4)!=!16!! !!!!16!=!16!✓!!!!!!!!!!

Parentheses Example

Section 3Parentheses and Variables on Both Sides

18

ToSolve:

-Distributeanynumbersoutsideparenthesestoalltermsinside.

-Combineanyliketerms.

-Isolatevariablesononesideoftheequationbyusinginverseoperationsof additionandsubtractiontobringthemovertotheotherside.

-Isolateallothernumbersontheothersidebyusinginverseoperationsof additionandsubtractiontobringthemovertotheotherside.

-Useinverseoperationstosolveforthevariable.

-Check.

Parentheses and Variables on Both Sides

ToSolve:

-UndoanyadditionorsubtractionNOTincludedinthefraction.

-Undothedivisionofthefraction.

-Finishlikearegularequation.

-Check.

Fraction Bar in Equation

WordProblems

PracticeProblems

ToSolve:

-SetupaLetStatementtorepresentthebasicunknown.

-Everythingelseisrepresentedintermsofthatvariable.

-Writeanequation.

-Solve.

Section 4Word Problems

19

Word Problems

ReplacingVariablesinaFormula:

-Pluginthegivenvaluesintothecorrectformula.

-Solveforthemissingvariable.

WritingaFormulawithGivenVariables:

-Usethevariablesinthewordproblemtosetupaformula.

SolvinginTermsofaVariable:

-Isolatethevariableyouare solvingintermsofbyusing inverseoperationstogetthe desiredvariablealone.

PracticeProblems

!Example:!!!A!=!LW,!Find!W!when!A=!36cm2!and!L=!9cm!! !! ! A!=!LW!! !!!!!!!!!!!!36!=!9L!! ! 9!!!!!!!9!! ! 4!cm!=!L!!!!!!!!!!

Section 5Working with Formulas

20

!Example:!!!The!distance,!d,!is!equal!to!the!rate,!r,!times!the!time,!t.!!! ! ! d!=!rt!!!!!!!!!! !

Example:!!!Solve!in!terms!of!y.!!! ! 6y!+!x!=!!z!! ! !!!!!!9!x!!!!9!x!! ! !!6y!=!x!–!z!! ! !!!6!!!!!!!!!6!! ! !!!y!=!x!–!z!! ! !!!!!!!!!!!!6!!!!!!!!!!

GraphingonaNumberLine:

• SeparatePoints:Individualpointsonanumberline.

• Ex:Wholenumbersbetween1and5.

• ContinuousPoints:Notjustindividualpoints,butallrealnumbersinbetween.

• <and>haveanopenpointonthegraph.

• ≤and≥haveaclosedpointonthegraph.

• Ex:Realnumbersgreaterthan7.

• Ex:Realnumbersbetween2and5.

• CompoundInequalities:twoinequalitiesthatcometogethertoformoneinequality.

• Ex:Realnumbersgreaterthan2butlessthanorequalto5.

SetNotation:

• Asymbolicwayofwritingananswer.Thinkofitasamathsentence.

• Ex:“thesetofallnumbersgreaterthan2”becomes{x:x>2}

• Ex:“thesetofrealnumbersbetween1and5”becomes{x:1<x<5}

• Ex:“thesetofrealnumbersgreaterthanorequalto-3andlessthan4”becomes{x:-3≤x<4}

SolvingInequalities:

-Useinverseoperationstosolveforthevariable. **WhenmultiplyingordividingbyaNEGATIVENUMBER,youmustchange theinequalitysign.** So<to>or≤to≥andviceversa. -Ifyourinequalityendsupwiththeconstantcomingbeforetheinequalitysign, reversetheinequalitysothatitstartswiththevariable. Ex.Change7>xtox<7 -Graphyourinequality. -Checkyouranswer. -Writeyour;inalanswerinsetnotation.

Section 6Solving and Graphing Inequalities

21

https://sites.google.com/a/kellenberg.org/mrs-farrell/home

Solving Inequalities 1

Solving Inequalities 2

PracticeProblems1

PracticeProblems2

22

AbsoluteValueEquations

AbsoluteValue:anumber’sdistancefromzero.

Ex.:|-8|=8and|+8|=8

PracticeProblems

ToSolve:

-Isolatetheabsolutevalueexpressionononesideoftheequation. **YoumustgetridofEVERYTHINGELSE** -Splittheequationintotwoseparateequations,droppingtheabsolutevalue bars. -FirstEquation:Droptheabsolutevaluebars.Leavetherestofthe equationalone. -SecondEquation:Droptheabsolutevaluebars.NEGATEthe OTHERside(changethesigns). -Solveeachequationindependently. -TwoChecks -CheckeachoftheanswersintotheORIGINALabsolutevalue equation. -FinalanswerinSetNotation. -SmallerNumbergoes;irst.

Section 7Absolute Value Equations

23

Absolute Value Equation 1

Absolute Value Equation 2

AbsoluteValueInequalities

PracticeProblems

ToSolve:

-Isolatetheabsolutevalueexpressionononesideoftheinequality. **YoumustgetridofEVERYTHINGELSE** **Remembertherulesformultiplyingordividingbyanegative number** -Splittheequationintotwoseparateinequalities,droppingtheabsolute valuebars. -FirstInequality:Droptheabsolutevaluebars.Leavetherestof theinequalityalone. -SecondInequality:Droptheabsolutevaluebars.FLIPthe inequalitysign.NEGATEtheentireOTHERside.(changeall ofthesigns) -Solveeachinequalityindependently. -Graphtheinequalities. -Check. **IftheanswersfaceAWAYfromeachother,2checksneeded.This happenswithGREATERTHANinequalities.** **IftheanswersfaceTOWARDSeachother,only1checkneeded. ThishappenswithLESSTHANinequalities.Youneeda compoundinequalityinyoursetnotationhere. -FinalanswerinSetNotation.

Section 8Absolute Value Inequalities

24

Absolute Value Inequality 1

Absolute Value Inequality 2

Whentwoinequalitiesonagraphfacetowardseachother,acompoundinequalitymustbewritten.

Towriteacompoundinequality,putyouranswersinorderfromleasttogreatestwithavariableinbetweenthem.Allsignsshouldbelessthanor

lessthanorequalto.

WORD PROBLEMS

Chapter 3

3.1 Consecutive Integer Problems3.2 Money-Value Problems3.3 Investment Problems3.4 Motion Problems3.5 Perimeter Word Problems3.6 Percent Mixture Problems

ConsecutiveIntegers:

Therearethreetypesofconsecutiveintegers:

Everyproblemwillstatehowmanyandwhattypeofconsecutiveintegerstouse.

TherearetwotypesofLetstatements:

PracticeProblems1

PracticeProblems2

1)ConsecutiveIntegers(i.e.3,4,5,6)

2)ConsecutiveEvenIntegers(i.e.4,6,8,10)

3)ConsecutiveOddIntegers(i.e.3,5,7,9)

Section 1Consecutive Integer Problems

26

Forconsecutiveintegers:

Letx=1stconsecutiveinteger

Letx+1=2ndconsecutiveinteger

Andsoon,adding1toeach.

Forconsecutiveevenoroddintegers:

Letx=1stconsecutiveeven(orodd)integer

Letx+2=2ndconsecutiveeven(orodd)integer

Andsoon,adding2toeach.

Note:Evenandoddnumbersareboth2apart.

Tosolve:

-Notehowmanyandwhattypeofconsecutiveintegersinordertowrite theletstatements.

-Writeanequationwhichfollowsthewords.

-Solveandanswerthequestionintheproblem.Makesureyouranswer makessense.

Consecutive Integer Problem

1

Consecutive Integer Problem

2 (Even/Odd)

MONEY-VALUECHART:

To;indthetotalvalueofanumberofacertainitem,thefollowingformulaisused:

PracticeProblems

Kind or Type Value of One Item

Number of Items

Total Value of Item

Section 2Money-Value Problems

27

Tosolve:

-WriteLetstatementsforeachitem(thesewilldescribehowmanyofeach item).

-Puttheinformationinthechart(theletstatementswillbeusedforthe numbercolumn).Thelastcolumnwillbetheproductofthevalueand numbercolumns(usetheformula).

-Writetheequationbyaddingthelastcolumnforeachitemandsetit equaltothetotalmoneygivenintheproblem.

-Solvetheequationandchecktoseethatthetotalvaluesadduptothe totalgiven.

(valueofoneitem)x(numberofthatitem)=(totalvalueoftheitem)

Money-Value Problems 1

Money-Value Problem 2

INVESTMENTPROBLEMCHART:

• Principal–theamountbeinginvestedatacertainrate

• Rate–thepercentusedfortheinvestment(principal)

• Income–themoneyearnedontheinvestment

Tosolveinvestmentproblemsusethefollowingformula:

PracticeProblems1

PracticeProblems2

Type Principal Rate Income

Section 3Investment Problems

28

PrincipalxRate=Income

Tosolve:

-Writeletstatementsforeachprincipalbeinginvested.

-Puttheinformationinthechart.Thelastcolumnwillbetheproductof theprincipalandtherate.

-Writetheequationbyaddingtheincomesorfollowingthewordsofthe problem.

Note:totalannualincomeisthesumforalltheincomesinthe chart.

-Solvetheequationandchecktoseeiftheincomesmakesense.

-Answerthequestionintheproblem.

Investment Problem 1

Investment Problem 2

MOTIONPROBLEMCHART:

Tosolvemotionproblemsusethefollowingformulatogetlastcolumn:

PracticeProblems

Kind or Type Rate or Speed of One Time Traveled Distance of

One

Section 4Motion Problems

29

(rateofone)x(timetraveled)=(distancetraveled)

Therearethreedifferentsetupsforthe;ivemotionscenarios:

ToSolve:

-Writealetstatement(s)tode;inetheunknown(thisiseithertherateor thetime).

-Puttheinformationinthechart.

-Determinewhichtypeofequationtouse.Usethedatafromthelast columnintheequation.

-Solvetheequationandchecktoseethatthedistancesmakesenseinthe problem.

-Answerthequestionintheproblem.

Motion Problem 1

Motion Problem 2

Motion Problem 3

AdditionSetup:

Usedwhenthesumofthetwodistances

isgiven.

SubtractionSetup:

Usedwhenthedifferencebetweenthetwodistancesis

given.

EqualSetup:

Usedinroundtripandcatch-upproblems.

PerimeterWordProblems:

Forperimeterproblemsusetheformulas:

.

PracticeProblems

2L+2W=perimeterofrectangle

4s=perimeterofasquare

a+b+c=perimeteroftriangle

Section 5Perimeter Word Problems

30

Tosolve:

-Readtheproblemtodeterminehowmany;iguresareintheproblemand ifmorethanone,howtheycompare.

-Writeletstatementsforlengthandwidthoforiginalandanyother;igure.

-Ifa;igurechangesdimensions,setupanoriginalandanew

-Theequationwillfollowthewordsoftheproblemusingtheformulas.

-Solveandanswerthequestionintheproblem.

Perimeter-Triangle

Perimeter-Rectangle

MixtureProblemsChart:

To;indthetotalofeachtype,usethefollowingformula:

Section 6Percent Mixture Problems

31

Type Number of Units Price per Unit Total

Type 1

Type 2

Mixture

(#ofunits)x($perunit)=(Total)

The#ofunitscolumnshouldadddowntototalthe#ofunitsinthemixture.

The$perunitofthemixtureshouldbebetweenthe$perunitofthetwotypes.

ToSolve:

-WriteLetStatementforeachitem.

-FillintheChart.

-MultiplyAcrosstogettheTotalColumn.

-WriteanequationusingtheTotalColumn.

Type1Total+Type2Total=MixtureTotal

-Solvetheequation.

-Check.

-Answerthequestionintheproblem

Mixture Problem

PercentMixtureProblems:

To;indthetotalofeachtype,usethefollowingformula:

PracticeProblems32

Type Number of Units Solution

Percent Pure Solution Total

Type 1

Type 2

Mixture

**Remember:PureSolutionis100%whilePureWateris0%**

(#ofunits)x(%perunit)=(Total)

The#ofunitscolumnshouldadddowntototalthe#ofunitsinthemixture.

The%perunitofthemixtureshouldbebetweenthe%perunitofthetwotypes.

ToSolve:

-WriteLetStatementforeachitem.

-FillintheChart.

-MultiplyAcrosstogettheTotalColumn.

-WriteanequationusingtheTotalColumn.

Type1Total+Type2Total=MixtureTotal

-Solvetheequation.

-Check.

-Answerthequestionintheproblem.

Percent-Mixture Problem 1

Percent-Mixture Problem 2

FRACTIONAL AND DECIMAL COEFFICIENTS

Chapter 4

4.1 Ratio4.2 Proportion4.3 Lever and Pulley Problems 4.4 Variation4.5 Solving Equations with Fractional Coefficients4.6 Solving Equations with Decimal Coefficients4.7 Work Word Problems

Ratios

Aratioisarelationshipoftwonumbersbydivision.Therearethreewaysthatratioscanbewritten:withacolon,usingthewordto,orasafraction.Ratiosareexpressedusingthesameunitandalwaysexpressedinsimplestterms.Denominatorsofoneremainintheratio.

RatioWordProblems

Whenexpressingratioinawordproblem,usetheratioandavariabletoexpresstheletstatements.

Rates

Arateisarelationshipoftwonumbersbydivision.Unlikeratios,rateshavetwodifferentunits.Ratesareexpressedinsimplesttermsthatusuallyhaveadenominatorofone.Ifthedenominatorisone,theoneisomittedandtheunitsarewrittenasasinglerate.

Example:45milesinonehouriswrittenas:45milesperhour.

PracticeProblems1

PracticeProblems2

Section 1Ratio

34

Ratio Word Problem

Rate Word Problem

Example:Inaclass,thereare15boysand12girls.Theratioofboystogirlsis15:12,whichsimpli;iesto5:4.

Proportions

Aproportionconsistsoftworatiosthatareequaltooneanother.

SolvingforaMissingVariableinaProportion

Tosolveforamissingvariableinaproportion,usecross-multiplication.Solvetheequationto;indthemissingvariable.

ProportionWordProblems

Whensettingupproportionwordproblemsmakesuretheunitsoneachsideoftheequalsignaresetupthesameway.Cross-multiplyproportionwordproblemstosolve.

FindingtheOriginalFractionWordProblems

Usingonevariablesetupanoriginalfractiontorepresentthenumeratoranddenominatorofthefraction.Ifthenumeratorand/ordenominatorarechangedsetupanewfraction.Cross-multiplytosolve.Forthe;inalanswerplugbackintotheoriginalfractionbutdonotsimplifytheanswer.

PracticeProblems1

PracticeProblems2

PracticeProblems3

Ratiosthatmakeupaproportioncanbeprovedequivalenttoeachotherinoneofthefollowingways:

1)Whencross-multiplyingequivalentfractions,theproductofthemeansandtheproductoftheextremes(thetwodiagonals)willequaleachother.

2)Whenputtingbothfractionsinsimplestterms,theyareequaltoeachother.

Section 2Proportion

35

Proportion 1 Proportion 2

Proportion with a Missing Variable

Proportion Word Problem

Finding Original Fraction

LeverWordProblems

Leversareusedinmanydifferentways.Aleverisusuallyusedtoassistinliftingheavyobjects.Aleverismadeupofabarthatisattachedtoafulcrum.Seesawsareanexampleoflevers.Theweightoneachsideofthelever,andthedistancethatweightisfromthefulcrumeffectshowtheleverworks.Inorderforalevertoworkproperlytheweightanddistanceononesidemustbeequaltotheweightanddistanceontheotherside.

PulleyWordProblems

Apulleyisasystemoftwowheelsthatworkinconjunctionwitheachother.Acordrunsbetweentothetwowheels.Elevatorsareanexampleofsomethingthatrunsonapulleysystem.Thediameterofthewheeloneachsideofthepulley,andthespeedatwhichtheyturneffectshowthepulleyworks.Inorderforapulleytoworkproperlythediameterandspeedononesidemustbeequaltothediameterandspeedontheotherside.

PracticeProblems

LeverFormula:(weight1)x(distance1)=(weight2)x(distance2)

Section 3Lever and Pulley Problems

36

PulleyFormula:(speed1)x(diameter1)=(speed2)x(diameter2)

Lever Pulley

DirectVariation

Directvariationoccurswhentwovariableshaveaconstantratio.Whendirectvariationoccursasonevariableincreasessodoestheother.

InverseVariation

Inversevariationoccursasonevariableincreasestheotherdecreases.

PracticeProblems

Setupasaproportion(x/y=x/y)

Section 4Variation

37

Setupasanequation(xy=xy)

**ThinkLeverandPulleyProblems**

Direct Variation

Inverse Variation

EquationswithFractionalCoefGicients

Ifthereisonefractiononeachsideofanequationusecrossmultiplicationtosolve.Whenthereismorethanonefractiononeachsideoftheequationyoumustuseleastcommonmultiplestoarriveattheanswer.

PracticeProblems1

PracticeProblems2

ToSolve:

-Findtheleastcommonmultipleofallthedenominators.

-Multiplytheentireequationbytheleastcommonmultiple.

-Cross-simplifyeachfractionandleastcommonmultipleinorderto eliminatethedenominators.

-Solvetheequationandcheck.

Section 5Solving Equations with Fractional Coefficients

38

Fractional Coefficient Equation

Fractional Coefficient Word

Problem

EquationswithDecimalCoefGicients

Whensolvingequationswithdecimalsuseleastcommonmultiplestoarriveattheanswer.

WordProblemswithDecimalCoefGicients

Anydecimalwordproblemthatconsistsofapercentrequiresyoutoturnthepercentintoanumber.Sinceapercentisoutof100,divideapercentby100(movethedecimalpointtoplacestotheleft)tomakeitanumber.

PracticeProblems1

PracticeProblems2

ToSolve:

-Findtheleastcommonmultipleofthedecimals.Thisisdeterminedby choosingthenumberwiththemostplacesafterthedecimalpoint.

-Determinehowmanyplacesthedecimalpointneedstobemovedto makeitawholenumber.Eachtimeyoumovethedecimalpointtothe rightyouaremultiplyingbyten. -Multiplytheentireequationbytheleastcommonmultiple(alwaysa multipleof10),thereforeeliminatingalldecimals.

-Solvetheequationandcheck.

Section 6Solving Equations with Decimal Coefficients

39

Decimal Coefficient Equation

Decimal Coefficient Word

Problem

WORKWORDPROBLEMCHART:

PracticeProblems

Person Rate (1/# alone)

# Hours/Minutes Together

Part of Job

Person 1

Person 2

Section 7Work Word Problems

40

**Rateisalwayswrittenas1/#alone.Asin“1jobcompletedper3hours.”**

**The3rdcolumnisALWAYSthecombinedtimeandmustbethesameforeachperson.**

ToSolve:

-Writealetstatement.

-FillintheChart,multiplyingacrosstogetthePartofJobcolumn.

-Setupanequationusingthefollowingformula:

PartofJob1+PartofJob2=1

-Solve.

-Check.

-Answerthequestionintheproblem.

Work Word Problem

SYSTEMS OF EQUATIONS

Chapter 5

5.1 Addition Method5.2 Substitution Method5.3 Business Word Problems5.4 Two-Variable Word Problems

**PossibleOutcomes**

PracticeProblems

ToSolve:

-Putbothequationsintheform“x+y=#”.

-Getthecoef;icientsofONEvariabletobeopposites.

-Multiplyoneorbothequationstoreachopposites.

-Onceyouhaveopposites,ADDthetwoequationstogetherbycombining liketerms.

-Thisshouldeliminateoneofyourvariablesandleaveyouwitha singlevariableequation.

-Solvefortheonevariableyouhaveleft.

-Substitutethisvalueintoeitheroriginalequationtosolvefor2ndvariable.

-Checkbothvariablesintobothequations.

Section 1Addition Method

42

1.Onesolution:theanswerwillbeasinglepoint.

2. Nosolution(inconsistent):theequationsareparallellines.Thelineswillnevercrosseachothersothereisnoanswersincetheyhavenopointsincommon.

-Answersuchas0=5

3. InGinitesolutions(Dependent):theequationsarethesameline.Thelineswillhaveanin;initenumberofsolutionsbecauseeverypointthatisononelinewillalsobeonthesecondline.

-Answersuchas0=0

Addition Method 1

Addition Method 2

Addition Method-

Inconsistent

Addition Method-

Dependent

PracticeProblems

ToSolve:

-Isolateonevariableononesideoftheequation.**Doesnotmatterwhichequationorvariableyouisolate**

-ex:x=ory=

-SubstituteinthevalueofthatvariableintotheOTHERequation.

-Solvetheequation.

-Plugthevalueintoeitherequationandsolvefortheothervariable.

-Checkbothvariablesintobothequations.

Section 2Substitution Method

43

**Rememberthereare3possibleoutcomes**

Substitution Method-Easy

Substitution Method-Harder

Substitution Method-

Inconsistent

PracticeProblems

ToSolve:

-LetStatements.(1foreachvariable)

-Setuptwoequations.

-Solveusingeithertheadditionmethodorthesubstitutionmethod.

-Checkbothvariablesinbothequations.

Section 3Business Word Problems

44

Business Problem 1

Business Problem 2

PracticeProblems1

PracticeProblems2

ToSolve:

-LetStatements.(oneforeachvariable)

-Setup2equations.

-Solveusingeithertheadditionmethodorthesubstitutionmethod.

-Checkbothvariablesintobothequations.

Section 4Two-Variable Word Problems

45

Two-Variable Age Word Problem

Two-Variable Word Problem

GRAPHINGChapter 6

6.1 Plotting Points, Basics6.2 Area6.3 Slope6.4 Finding the Equation of a Line6.5 Slope-Intercept Method of Graphing6.6 X- and Y-Intercept Method6.7 Graphing Systems of Equations6.8 Graphing Inequalities6.9 Graphing Systems of Inequalities

GraphingPoints

Onacoordinateaxis,allpointscanbegraphedasanorderedpair(x,y)wherexisthehorizontalvalueandyistheverticalvalue.Thecenterofthegraphistheintersectionofthexandyaxisandiscalledtheorigin(0,0).

PracticeProblems1

SolutionstoaLine

Tocheckifapointisasolutiontothegivenline,pluginthevalueforxandthevaluefory.Iftheequationchecksout,thepointfallsontheline,andtherefore,isasolution.Ifthepointdoesnotcheckout,itisnotontheline,andtherefore,notasolution.

PracticeProblems2

Section 1Plotting Points, Basics

47

Example: y = 2x - 4, (6, 8) 8 = 2(6) - 4 8=12 - 4 8 = 8 ✔️ (6,8) is a solution for the line y = 2x - 4

Example: 2y = 3x - 5, (1, -3) 2(-3)= 3(1) - 5 -6 = 3 - 5 -6 = -2 ❌ (1, -3) is not a solution for the line 2y = 3x - 5

Example: plot the point (4,5).

AreaofShapes:

PracticeProblems

Section 2Area

48

To;indtheareaofashapeonacoordinateplane:

-Plotthepoints.

-Determinetheareaformulaforthegivenshape.

-Pluginthegivenvalues.

-Solve.

Example:

Theslopeofalineis:

PracticeProblems1

PracticeProblems2

Section 3Slope

49

!"#$%&!!"!!− !""#$%&'()!!"#$%&!"#$%&!!"!!− !""#$%&'()!!"#$%& !!!"!!

∆!∆!!

To;indtheslopebetween2pointsusetheslopeformula:

! = !!! − !!!! − !! !

To;indtheslopeofanexistingline:

-Find2pointsonthelineandcountupordownfromthe;irstpointtothe secondpoint.Placethatnumberoverthedistanceyoumoveleftorright fromthe;irsttothesecondpoint.Determineifyourslopeispositiveor negative.

• Positiveslopesincreasefromlefttoright.

• Negativeslopesdecreasefromlefttoright.

Example: Find the slope of the line passing through the points (3,4) and (−1, 3).

m = y₂ − y₁ x₂ − x₁ m = 3 − 4 = −1 = 1 −1 −3 −4 4

Example: Graph y = 4 Every point on the line y = 4 has a y-coordinate of 4.

Example: Graph x = −2 Every point on the line x = −2 has an x-coordinate of −2.

Example: Graph y = 4 Every point on the line y = 4 has a y-coordinate of 4. Horizontal Lines

• Horizontallineshaveaslopeof0.

• Verticalslopeshavenoslope(unde;ined).

• Parallellineshavethesameslope.

• Perpendicularlineshaveslopesthatarenegativereciprocalsofeachother.

FindingtheEquationofaLine:

PracticeProblems1

PracticeProblems2PracticeProblems3

To;indtheequationofalinegivenaslopeandpoint:

-Pluginm,xandyintotheequationofaline.

-Solveforb.

-Writetheequationofaline;illingintheslopeandy-intercept.

Section 4Finding the Equation of a Line

50

To;indtheequationofalinegivenalineandapoint:

-Determinetheslopeofthelinegiven.

-Isthenewlineparallelorperpendiculartothegivenline?

-Parallel:Usethesameslope.

-Perpendicular:Usethenegativereciprocal.

-Pluginyournewm,xandyintotheequationofaline.

-Solveforb.

-Writetheequationofaline;illingintheslopeandy-intercept.

To;indtheequationofalinegivenaslopeandy-intercept:

-Pluginm(slope)andb(y-intercept)intotheequationofaline.

Finding the Equation of a Line Given a Parallel Line

Finding the Equation of a Line

Given a Perpendicular Line

Example: Write the equation of a line that has a slope of −6 and a y-intercept of 4. The equation of a line is y = mx + b m is the slope, b is the y-intercept Plug in the slope and y-intercept into the equation of a line:

y = mx + b ➡️ y = −6x + 4

Answer: y = −6x + 4

Example: Write the equation of a line that has a slope of 2 and passing through the point (6, −4).

1) Plug slope and point into the equation of a line. y = mx + b

-4 = 2(6) + b -4 = 12 + b -12 -12 -16 = b 2) Plug slope and y-intercept into the equation of a line.

y = mx + b ➡️ y = 2x - 16

Answer: y = 2x - 16

PracticeProblems4

To;indtheequationofalinegiventwopoints:

-Determinetheslopeofthelinegivenusingtheslopeformula.

-Pluginmandthexandy-valuesofONEofthepointsintotheequationof aline.

-Solveforb.

-Writetheequationofaline;illingintheslopeandy-intercept.

51

Finding the Equation of a

Line Given Two Points

Slope-InterceptFormofaLine

PracticeProblems1

PracticeProblems2

Alllinescanbewrittenasy=mx+b

Whentheequationissolvedfory,thenthex-coef;icientistheslopeofthelineandtheconstantisthey-intercept.

Section 5Slope-Intercept Method of Graphing

52

Example:

Example: Find the slope and y-intercept the line: 3x + 2y = 8 - 3x - 3x 2y = 8 -3x 2 2 y = 4 - 3x 2 The slope is -3/2 and the y-intercept is (0,4)

X-andY-InterceptMethod

PracticeProblems

ToSolve:

-To;indthex-interceptofalinesetyintheequationequalto0andsolve forx.

-To;indthey-interceptofalinesetxintheequationequalto0andsolve fory.

Section 6X- and Y-Intercept Method

53

Example: Find the x and y intercepts of: 4x + y = 8

PracticeProblems

ToSolve:

-Graphthe;irstlineusingslope-interceptformorx-andy-intercepts method.

-Onthesamegrid,graphthesecondline.

-Ifthelinesintersect,determinethecoordinatesofthepointandcheck thatthepointisasolutiontobothequations.

-Ifthelinesareparallel(havethesameslope),thenthereisnosolution.

-Ifthelinesarethesamelinethentherearein;initelymanysolutions.

Section 7Graphing Systems of Equations

54

Graph the system of equations: y = 2x +1 and y = x + 6

Example:

Graph the system of equations: y = 2x - 1 and 2y = 4x - 2 2y = 4x - 2 2 2

y = 2x - 1

Example:

Graph the system of equations: y = 4x - 4 and y = 4x + 3

Example:

PracticeProblems

ToSolve:

-Useeitherslope-interceptformorx-andy-interceptsmethodto;indtwo points.

-Connectthepointswithasolidlineiftheinequalityis≤or≥.

-Connectthepointswithadottedlineiftheinequalityis<or>.

-Shadethegraphonthesideofthelinethatmakesallthepointstheretrue fortheinequality.

-Chooseapointononesideoftheline:abovethelineif>or≥,below if<or≤,andcheckintheinequalitytotestifitisatruestatement.

Section 8Graphing Inequalities

55

Example: Graph y > x - 3 m = 1, b = (0, 3) dotted line Check (0,0)

y > x - 3 0 > 0 - 3 0 > -3

Example: Graph: y < 1x + 2 3 Slope: 1 y-int: (0, 2) 3 Solid line Check: (0, 0) y < 1x + 2 3 0 < 1(0) + 2 3 0 < 0 + 2 0 < 2

PracticeProblems

Tosolve:

-Graphbothinequalitiesandshadeeach(usedifferentshadingforeach).

-Chooseapointwherebothshadingsoverlapandcheckthepointineach inequalitytoseeifitcreatestruestatements.Iftrue,labelthearea“S”for solutionset.

Section 9Graphing Systems of Inequalities

56

Graph the following system of inequalities: y > 2x + 1 and 2y < −x + 6

Example:

POLYNOMIALSChapter 7

7.1 Addition and Subtraction7.2 Properties of Exponents7.3 Scientific Notation7.4 Multiplication7.5 Division

OrderingPolynomials

Polynomialsshouldbeplacedindescendingdegreeorder.To;indthedegreeofapolynomialaddtogetheralloftheexponentsineachterm.Iftwotermshavethesamedegree,placetheminorderalphabetically.

PracticeProblems1AddingPolynomials

Whenaddingpolynomials,addtogetherthosepolynomialsthatareliketerms.Remember,liketermshavethesamevariablesandthesameexponents.Toaddthem,addthecoef;icientsandkeepthebaseandexponentsthesame.Placeinthecorrectpolynomialorder.

SubtractingPolynomials

Whensubtractingpolynomials,subtractthosepolynomialsthatareliketerms.Ifthereisasubtractionsignoutsideoftheparenthesesmakesuretodistributethesubtractionsigntoeachtermbeforecombiningtheliketerms.Placeinthecorrectpolynomialorder.

PracticeProblems2

PracticeProblems3

Section 1Addition and Subtraction

58

Example: Place the polynomial in the correct order: 5y² + 7xy − 2x² + 3y − 8 Degree of two: (Place alphabetically) − 2x² + 7xy + 5y² Degree of one: 3y Constants: − 8 Correct Order: − 2x² + 7xy + 5y² + 3y − 8

Example: Add: (4x² + 6x − 7) + (−5x² −6x)

4x² + 6x − 7 + −5x² −6x −1x² + 0x − 7 Answer: −x² − 7

Example: Subtract: (6x² + 2x − 8) − (−4x² + 3x −4)

6x² + 2x − 8 + 4x² − 3x + 4 10x² − 1x − 4 Answer: 10x² − x − 4

MultiplicationProperties

•MultiplyingExponents-whenmultiplyingexponentswiththesamebase,addthe exponentsandkeepthebasethesame.

• Example:

€

x 2 ⋅ x 4 = x 6

•ZeroExponents-anynumberortermraisedtoanexponentofzeroisalways equalto1.

• Example:

•PowerofaPowerMultiplication-whenoneexponentisraisedtoanother exponent,multiplytheexponentsandkeepthebasethesame.

• Example:

DivisionProperties

•DividingExponents-whendividingexponentswiththesamebase,subtractthe exponentsandkeepthebasethesame.

• Example:

•PowerofaPowerDivision-whenafractionwithexponentsisraisedtoanother exponent,multiplytheoutsideexponenttobothoftheexponentsinthe numeratoranddenominatorandkeepthebasesthesame.

•

PracticeProblems1

NegativeExponents

• Sincetherearenonegativeexponents,allnegativeexponentsmustbeturnedintopositiveexponents.Todothis,movethenegativeexponenttothedenominatorofafractionandmaketheexponentpositive.Ifthereisanegativeexponentinthedenominatorofafraction,moveittothenumeratorandmakeitpositive.

•

Example:

PracticeProblems2

PracticeProblems3

!! = 1!

Section 2Properties of Exponents

59

(!!)! = !!!

€

x 6 ÷ x 4 = x 2

€

a2

b−2= a2b2

Example:

€

x 3

y 4

"

# $

%

& '

2

=x 6

y 8

Example:

€

4 −2 =116

€

142

=116

StandardNotationtoScientiGicNotation

• Scienti;icnotationisusedtoexpressverylargeorverysmallnumbers.Whenwritinganumberinscienti;icnotation,movethedecimalpointtomakeanumberthatisgreaterthanorequalto1butlessthan10.Multiplythenumberbyapoweroften.Todeterminethepoweroftencounthowmanyplacesthedecimalpointwasmoved.Ifthedecimalwasmovedtotheleft,theexponentwillbepositive.Ifitwasmovedtotheright,theexponentwillbenegative.

• Example:4,621,000canbewritteninscienti;icnotationas:

• Example:0.00462canbewritteninscienti;icnotationas:

ScientiGicNotationtoStandardNotation

• Whenwritinganumberinstandardnotation,movethedecimalpointaccordingtotheexponentinthepowerof10.Iftheexponentispositive,movethedecimalpointtotheright.Iftheexponentisnegative,movethedecimalpointtotheleft.

• Example:canbewritteninstandardnotationas:4,621,000

• Example:canbewritteninstandardnotationas:0.00462

PracticeProblems1

MultiplyingwithScientiGicNotation

• Whenmultiplyingwithscienti;icnotation,multiplythefactors,andthenmultiplythepowersoften.Tomultiplythepowersoften,addtheexponents.Ifthefactorsmultiplytoanumbergreaterthanorequaltoten,adjustthefactorandexponentaccordinglytoproperscienti;icnotation.

DividingwithScientiGicNotation

• Whendividingwithscienti;icnotation,dividethefactors,andthendividethepowersoften.Todividethepowersoften,subtracttheexponents.Ifthefactorsdividetoanumberlessthanorequaltoten,adjustthefactorandexponentaccordinglytoproperscienti;icnotation.

PracticeProblems2

4.621!×!10!!

Section 3Scientific Notation

60

4.62!×!10!!!

4.621!×!10!!

4.62!×!10!!!

Multiplying with Scientific Notation

Dividing with Scientific Notation

MultiplyingMonomials

• Whenmultiplyingmonomials,multiplythecoef;icients,keepthevariablethesame,andaddtheexponents.Ifatermisoutsideofparenthesesdistributeittoeachtermintheparentheses.

PracticeProblems1MultiplyingBinomials

• Whenmultiplyingbinomials,multiplyeachterminthe;irstparenthesesbyeachterminthesecondparentheses.UsethetermsFOILtohelprememberthesteps.MultiplytheFIRSTterms,thentheOUTERterms,nexttheINNERterm,nexttheLASTterms.Combineanyliketermsattheend.

PracticeProblems2

MultiplyingPolynomials

• Whenmultiplyingpolynomials,multiplyeachterminthe;irstparenthesesbyeachterminthesecondparentheses.Combineanyliketermsattheend.

PracticeProblems3

AreaWordProblems

PracticeProblems4

Section 4Multiplication

61

Multiplying Polynomials

Tosolveanareawordproblem:

-Setuptheoriginalshape’sletstatement(s)andpicture.

-Setupthenewshape’sletstatement(s)andpicture.

-Identifytheformula(s)neededtosolvetheproblem.

-Writeanequationthatrepresentsthegiveninformation.

AreaofaSquare:

!! = ! !!!

AreaofaRectangle:

!! = !!"! Area Word Problems without

Factoring

Example: Multiply: (x + 6)(x − 2) F: x(x) = x² O: x(-2) = -2x I: 6(x) = 6x L: 6(-2) = -12 Answer: x² + 4x - 12

Example:

Example: Multiply: (−4x²)(6x³y) −4(6) = −24 x²(x³) 3+2=5 x⁵ y Answer: −24x⁵y

DividingaMonomialbyaMonomial

• Whendividingmonomials,dividethecoef;icients,keepthevariablethesame,andsubtracttheexponents.

PracticeProblems1DividingaPolynomialbyaMonomial

• Whendividingapolynomialbyamonomial,eachterminthenumeratorshouldbedividedbythemonomialinthedenominator.Dividethecoef;icients,keepthevariablethesame,andsubtracttheexponents.

PracticeProblems2

DividingPolynomialsUsingLongDivision

• Whendividingapolynomialbyabinomialortrinomial,uselongdivisioninordertosolve.Thestepsforlongdivisionofpolynomialsarethesameasregularlongdivision.Thereareafewthingstokeepinmindwhensettingupthedividend(thetermbeingdivided).Thedividendshouldhavedescendingexponents.Ifadegreeismissing,additinwithazeroasthecoef;icient.Iftherearetwovariablesinthedividendmakesureitisincorrectpolynomialorder.

PracticeProblems3

Section 5Division

62

Dividing Polynomials 1

Dividing Polynomials 2

Dividing Polynomials with

2 Variables

Dividing Polynomials with

Remainders

Example: Divide: −16x²y²z 8xyz −16 = -2 x² = x y² = y z = 1 8 x y z Answer: -2xy

Example: Divide: 12x³y² − 8xyz − 4x 12x³y² = −3x²y² − 8xyz = 2yz − 4x − 4x Answer: −3x²y² + 2yz

FACTORING POLYNOMIALS

Chapter 8

8.1 Greatest Common Factor8.2 Difference of Two Squares and Grouping8.3 Trinomials8.4 Trinomials with Leading Coefficients8.5 Factoring Completely8.6 Quadratic Equations8.7 Quadratic Equation Word Problems8.8 Graphing Parabolas

GreatestCommonFactor

• Whenitcomestofactoringpolynomials,thereareseveralmethodstouse.The;irstmethodthatyoushouldALWAYSlookforisaGreatestCommonFactor.

PracticeProblems

ToFindtheGCF:

-Lookateachterminthepolynomial.

-FindtheGREATESTfactorofthecoef;icientsthattheyALLhavein common.

-FindtheHIGHESTexponentofeachvariablethattheyALLhavein common.

-Thesecommonfactorsshouldallgooutsideoftheparentheses.

-Anyremainingpartsofthetermsremaininsideoftheparentheses.

Section 1Greatest Common Factor

64

Example: Factor: 12x⁴ − 3x³ + 9x² Greatest common factor is 3x² Factoring: 3x²(4x² − x + 3)

Example: Factor: a²b³ + ab² Greatest common factor is ab² Factoring: ab²(ab + 1)

Example: Factor: 4x + 16 Greatest common factor is 4 Factoring: 4(x + 4)

DifferenceofTwoSquares

• Whenanexpressionisintheforma²–b²,thewayyoufactoriscalled“DifferenceofTwoSquares”(DOTS).ThekeyhereistheDIFFERENCE.ThereMUSTbeasubtractionsignbetweentheperfectsquares.

PracticeProblems1

Grouping

• IfyouareunabletofactorusingtheGreatestCommonFactorforalloftheterms,youmaybeabletogroupwithinthepolynomialinordertouseaGCF.Youneed4termstousegrouping.

PracticeProblems2

**Remember,eachpartoftheexpressionmustbeaperfectsquare--variablesareaperfectsquareiftheyhaveanevenexponent**

Section 2Difference of Two Squares and Grouping

65

To;indtheDifferenceofTwoSquares:

-Setuptwopairsofparentheses.

-EachsquarerootgoesintoBOTHparentheses.

-Putoppositesignsineachpair.

Totakethesquarerootofvariables,dividetheexponent

Totakethesquarerootofafraction,takethesquarerootofthe

numeratorandthedenominator.

DOTS 1 DOTS with Fractions

ToFactorUsingGrouping:

-GroupthetermsintopairssoeachpairhasaGCF.

-Factoreachsetofterms.

-TheGCFsoutsidebecomeonesetofparenthesesandtheremaining partsofthetermsbecometheotherparentheses.

-Thetermsinsideoftheparenthesesoriginallymustbethe same.Factorouta-1ifyouneedtochangethesignsinoneset ofparentheses.

Grouping

Example: Factor: a² − 81 √a² = α √81 = 9 Factoring: (a + 9)(a − 9)

Example: Factor: x² + xy + 4x + 4y

x² + xy 4x + 4y x(x + y) 4(x + y) Answer: (x + y)(x + 4) Check: (x + y)(x + 4) = x² + xy + 4x + 4y x² + 4x + xy + 4y = x² + xy + 4x + 4y ✔️

TrinomialswithoutLeadingCoefGicients

• Trinomialsarewrittenintheform“ax²+bx+c”where“a,”“b,”and“c”arenumbers.Inthiscase,“a”isalways1.

PracticeProblems1

PracticeProblems2

ToFactor:

-Setuptwopairsofparentheses(x+/-_____)(x+/-_____).

-Lookatthesigninfrontof“c”.

-Ifitispositive,thesignsinparenthesesaresameas“b”.

-Ifitisnegative,thesignsareopposite,with“b”tellingusthe signofthelargerfactor.

-Findfactorsof“c”thatadduptothe“b”value.

-UseFOILtocheck.

Section 3Trinomials

66

Trinomials 1 Trinomials 2 Trinomials 3

FactoringTrinomialswithaLeadingCoefGicient

PracticeProblems

ToFactor:

TheEyeglassMethod

-Takeoutanygreatestcommonfactors.

-Multiply“a”and“c”together.

-Findfactorsofthatnumberthataddupto“b”.

-Useyoursignstohelpyou:apositive“ac”meansusethesame signsinyourfactors,whileanegative“ac”meansusedifferent signsinyourfactors.

-Replaceyour“bx”withthevaluesthatadduptoit.

-Usethegroupingmethodto;inishfactoring.

-CheckbyFOILtomakesureyouransweriscorrect.

Section 4Trinomials with Leading Coefficients

67

Trinomials with Leading

Coefficient 1

Trinomials with Leading

Coefficient 2

Trinomials with a Leading

Coefficient 3

FactoringCompletely

PracticeProblems

ToFactorCompletely:

-StartbylookingforaGCF.

-TaketheGCFoutifpossible.

-Factorwhatremains.

-Checkyournewpolynomialstoseeiftheycanbefactoredanyfurther.

Section 5Factoring Completely

68

Youmayhavetofactorseveraltimesbeforeapolynomialisfactoredcompletely.

**UseGCF,DOTS,Grouping,TrinomialFactoring,andEyeglassFactoring**

Factoring Completely 1

Factoring Completely 2

Example: Factor: 4x² - 16

4(x² - 4) 4(x - 2)(x + 2)

Check: 4(x - 2)(x + 2) = 4x² - 16 (4x - 8)(x + 2) = 4x² - 16 4x² + 8x - 8x - 16 = 4x² - 16

4x² - 16 = 4x² - 16 ✔️

Example: Factor: 2x² - 4x - 30

2(x² - 2x -15) 2(x - 5)(x + 3)

Check: 2(x - 5)(x + 3) = 2x² - 4x - 30

(2x - 10)(x + 3) = 2x² - 4x - 30 2x² + 6x - 10x - 30 = 2x² - 4x - 30

2x² - 4x - 30 = 2x² - 4x - 30 ✔️

SolvingQuadraticEquations

PracticeProblems

Tosolve:

-Combineanyliketerms.

-Getalltermsontoonesideoftheequationsothatitisequaltozero.

-Factorcompletelythesidewiththeterms.

-Seteachfactorequaltozeroandsolve.

-CheckbypluggingEACHanswerintotheoriginalequation.

Section 6Quadratic Equations

69

Theanswerstotheseproblemsarecalled“roots.”Theseareallofthevaluesofxwhenthe

equationisequalto0.

Quadratic Equation 1

Quadratic Equation 2

Example: Solve: x² - 3x - 18 = 0 (x + 3)(x - 6) = 0 x + 3 = 0 x - 6 = 0 - 3 - 3 +6 +6 x = - 3 x = 6 {-3, 6} Checks: x² - 3x - 18 = 0 x² - 3x - 18 = 0 (-3)² -3(-3) -18 = 0 (6)² -3(6) - 18 = 0 9 + 9 - 18 = 0 36 - 18 - 18 = 0 0 = 0 ✔️ 0 = 0 ✔️

QuadraticEquationWordProblems

PracticeProblems

ToSolve:

-SetupLetStatement(s)inONEvariable.

-Setupanequation.

-Movealltermstoonesidetogetequaltozero.

-Factor.

-Solve.

-Determineifanyanswerdoesnotmakesense.

Section 7Quadratic Equation Word Problems

70

Quadratic Equation Word

Problem 1

Quadratic Equation Word

Problem 2

GraphingParabolas

• Thegraphofaquadraticequationisaparabola.Thesearecurvesthatopenup(likeasmile)ordown(likeafrown).Thewaytodeterminethedirectionislookingatthe“a”coef;icient.Ifitispositive,yourparabolawillopenup,andifitisnegative,yourparabolawillopendown.

• Parabolasaresymmetricalsotheyhavewhatiscalledan“axisofsymmetry”thatrunsverticallydownthemiddle.Thisimaginarylineiswherethegraphchangesdirectionandisthereforecalledtheturningpoint.

PracticeProblems1

PracticeProblems2

FormulaforAxisofSymmetry:x=-b/2a

Section 8Graphing Parabolas

71

ToGraph:

-Getequationinstandardform(equaltozero).

-Identifyyour“a,”“b,”and“c”values.

-FindtheAxisofSymmetry.

-Setupatableofvaluestograph.

-3Columns:x-value,equation,y-value

-Yourtableshouldhave5valueswiththemiddleonebeingyour turningpoint.

-Plotthe5pointsandyourROOTSonyourgraph.Remember,your rootsarewheretheparabolacrossesthex-axisandtheanswertoa quadraticequation.

-Connectyourpoints—it’sacurve,nota“V”.

Rememberthataverticallineisx=#

Example: Graph: -x² - 3x +6 = y a = -1, b = -3, c = 6 axis of symmetry: x = -b = -(-1) = -1

2a 2(-1) 2 table of values: roots: -x² -3x +6 = 0 -1(x² +3x -6) = 0 -1(x - 2)(x + 3) = 0 -1 = 0 x - 2 = 0 x + 3 = 0 ❌ +2 +2 -3 -3 x = 2 x = -3 (2, 0) (-3, 0)

Example:

RATIONAL EXPRESSIONS

Chapter 9

9.1 Simplifying9.2 Multiply and Dividing9.3 Adding and Subtracting with a Common Denominator9.4 Adding and Subtracting with a Non-Common Denominator

SimplifyingRationalExpressions

• Arationalexpressionisafractionthatcontainsvariables.Itcanalsobecalledanalgebraicfraction.

PracticeProblems

Tosimplify:

Findthegreatestcommonfactorofboththenumeratoranddenominatorandcancelthem.

Section 1Simplifying

73

Forbinomialandtrinomialexpressionsinafraction:

a)Factorboththenumeratoranddenominatorcompletely.

b)Cancelanycommonfactors.

Example:Simplify

sincexisthecommonfactor,cancelthemandansweris

3!4!!!!!!

34!

Example:Simplify

cancelallofthecommonfactorsandtheansweris3x

6!!2! !!!!!

!

Example:!!4x!+!2!!!!factors!to!!!!!!!!2(2x!+!1)!!!!!!!!!!!!!cancel!out!the!!(2x!+!1)!!!and!the!answer!is!!!!!!!2!!!!!!!!!!!!!!!!!!!!!!!!!!!!4x2!–!1!!!!!!!!!!!!!!!!!!!!!!!!(2x!+!1)(2x!–!1)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!2x!–!1!!!!Example:!!x2!+!4x!+!3!!!=!!(x!+!3)(x!+!1)!!cancel!out!the!!(x!+!3)!!!and!the!answer!is!!!x!+!1!!!!!!!!!!!!!!!!!!!!!x2!+!5x!+!6!!!!!!!!(x!+3)(x!+!2)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!x!+!2!!

Example:

Example:

MultiplyingandDividingRationalExpressions

• Tomultiply:factorcompletelyallnumeratorsanddenominatorsandcancelallcommonfactorsfromanynumeratortoanydenominator.Thenmultiplyacrossusinganyremainingfactors.

• Todivide:First,rewritetheproblemchangingthefractionafterthedivisionsigntoitsreciprocalandthenfollowmultiplicationrules.

PracticeProblems

!

Example:!!!!x!–!3!!!•!!x2!+!7x!+!12!!•!!x2!–!1!!!!!!!factor:!!!!!!!!!x!–!3!!!!!!!•!!(x!+!3)(x!+!4)!!•!!(x!+!1)(x!–!1)!! !!!!!!!!x2!–!9!!!!!!!!!4x!+!16!!!!!!!!!!!!x!+!1! !!!!!!!!!!!!!!!!!(x!+!3)(x!–!3)!!!!!!!4(x!+!4)!!!!!!!!!!!!!!!!!!x!+!1!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Cancel!any!common!binomials!and!answer!!!!!!!x!–!1!!!! ! ! ! ! ! ! ! ! ! !!!!!!!!!4!!

Section 2Multiplying and Dividing

74

!

Example:!!!!x2!–!1!!÷!!2x!+2!!!!rewrite!as!!!!!x2!–!1!!!•!!x2!–!9!!!!!factor!!(x!+!1)(x!–!1)!!•!!(x!+!3)(x!–!3)!!!!!!!!!!!!!!!!!!!!!!!!x!+!3!!!!!!!x2!!–!9!!!!!!!!!!!!!!!!!!!!!!!!!!!!x!+!3!!!!!!!!2x!+!2!!!!!!!!!!!!!!!!!!!!!(x!+!3)!!!!!!!!!!!!!!!!!2(x!+!1)!!!!!!! ! ! cancel!common!factors!and!the!answer!is:!!(x!–!1)(x!–!3)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! ! ! ! !!!!!!!!!!!!!!!!!!!!!!!!!2!!

Example:

Example:

AddingandSubtractingRationalswithCommonDenominators

• Toaddorsubtractwhenthedenominatorsarethesameaddorsubtractthenumeratorscombiningliketerms.Thensimplifytheanswer.

PracticeProblems

Rememberwhensubtractingmorethanoneterm,distributethenegativesignoveralltermsinthesecond

denominator.

Section 3Adding and Subtracting with a Common Denominator

75

!Example:!!!!2x!+!3!!+!!2x!+!1!!=!!4x!+!4!!=!!4(x!+!1)!!=!!4!!!!!!!!!!!!!!!!!!!!!!!!!!!x!+!1!!!!!!!x!+!1!!!!!!!!!!x!+!1!!!!!!!!!x!+!1!!!Example:!!!!3x!+!2!!–!!x!–!4!!!=!!2x!+!6!!=!!2(x!+!3)!!!=!!x!+!3!!!!!!!!!!!!!!!!!!!!!!!!!!2x!!!!!!!!!!!!2x!!!!!!!!!!!!!!2x!!!!!!!!!!!!!!2x!!!!!!!!!!!!!!!!!x!!

Example:

Example:

AddingandSubtractingRationalswithDifferentDenominators

PracticeProblems

Toaddorsubtractwhenthedenominatorsaredifferent,

-Factoranydenominatorsthatcanbefactoredto;indwhattheyhavein common.

-Converttolikedenominators.

-Addorsubtractthenumeratorscombiningliketermsandanswerin simpli;iedform.

Section 4Adding and Subtracting with a Non-Common Denominator

76

!Example:!!!!!x!+!1!!+!!!x!+!1!!!!=!!!2x!+!2!!+!!3x!+!3!!!=!!5x!+!5!!!!or!!5(x!!+!1)!!!!!!!!!!!!!!!!!!!!!!!!!!!3x!!!!!!!!!!!2x!!!!!!!!!!!!!!!6x!!!!!!!!!!!!!6x!!!!!!!!!!!!!!6x!!!!!!!!!!!!!!!!!6x!!

Example:

Adding with Different

Denominators

ROOTS OF NUMBERS

Chapter 10

10.1 Finding the Root10.2 Simplifying10.3 Adding and Subtracting10.4 Multiplying and Dividing10.5 Solving Equations10.6 Rationalizing the Denominator10.7 Pythagorean Theorem

TerminologyofRadicals:

• Example:,2isthecoef;icient,3istheindex,125istheradicand.

SquareRoot

• Totakethesquarerootofanumber,determinewhatnumbermultipliedbyitselfgivesyouthenumberundertheradical(theradicand).Anyradicalwithoutanindex,hasanindexoftwo,meaningtakethesquareroot.

PerfectSquares

• Numbersthatyoucantakethesquarerootofarecalledperfectsquares.Theanswerforaperfectsquareisaninteger.To;indtheperfectsquareofanumber,determinewhatintegerwhenmultipliedbyitselfgivesyoutheradicand.

• Example:,4multipliedbyitselfequals16.

CubeRoot

• Takingthecuberootofanumberisrepresentedbyanindexofthreeoutsidetheradical.Totakethecuberootofanumberdeterminewhatnumbermultipliedbyitselfthreetimesgivesyoutheradicand.

• Example:,2multipliedbyitselfthreetimesequals8.

RaisingaRadicaltoanExponent

• Whenaradicalisraisedtoanexponenttheradicalandexponentcanbedroppediftheindexandtheexponentarethesame.

• Example:

Rationalvs.Irrational

• Radicandsthatyoucantakethesquarerootofarerationalnumbers.Numbersthatyoucannottakethesquarerootofareirrationalnumbers.

• Example:isrational.

• Example:isirrational.

TakingtheRootofVariables

• Totaketherootofavariable,dividetheexponentbytheindexandkeepthevariablethesame.

• Example:,multipliedbyitselfequals.

2 1253

Section 1Finding the Root

78

€

16 = 4

83 = 2

7( )2= 7

25 = 5

€

7 = 2.64575131106...

x16 = x8 x8 x16

RadicalswithFractions

• Totaketherootofafraction,taketherootofthenumeratorandtherootofthedenominator.

• Example:

• Example:

PracticeProblems

8125

3!

"#

$

%&=25

79

916 =

34!

SimplifyingRadicals

Whenaradicalisnotaperfectsquare,theradicalcansometimesbesimpli;ied.Tosimplifyaradical,simplifytheradicandintofactors.Anyfactorsthatareperfectsquaresshouldbesimpli;ied.Ifafactorisnotaperfectsquare,checktoseeifitcanbefactoredfurther.Factorsareinsimplestformwhenthesquarerootsofperfectsquaresaretaken,andallremainingradicandsareprimenumbers.Toputtheanswertogether,multiplyallperfectsquaresforthecoef;icient,multiplyanyradicandstogethertobecomethe;inalradicand.

SimplifyingVariablesinaRadical

Totakethesquarerootofavariablewithanevenexponent,dividetheexponentbytwo.Totakethesquarerootofavariablewithanoddexponent,subtractonefromtheexponent(whichshouldstayundertheradicalsign),andthendividewhatisremainingbytwotogetitssquareroot.Theevenexponentthatyoucantakethesquarerootofbecomespartofthecoef;icient.Theoddexponentofonebecomespartoftheradical.

PracticeProblems

Section 2Simplifying

80

Simplifying Radicals

Simplifying Radicals 2

Example: Simplify: √a⁵b⁶c⁷ √a⁵ √b⁶ √c⁷ √a⁴ √a b³ √c⁶ √c a² c³ a²b³c³√ac

Example:

Simplify: √18 √9 √2

3 3√2

OR Simplify: √18 √6 √3 √2 √3

√2 √3 √3

3√2

AddingandSubtractingRadicals

Radicalscanonlybecombinediftheyareliketerms.Radicalsareliketermswhentheyhavethesameradicand.Inordertocombinelikeradicals,addorsubtractthecoef;icientsandkeeptheradicandthesame.Allradicalsshouldbeinsimplesttermsbeforeaddingorsubtractingthem.

PracticeProblems

Section 3Adding and Subtracting

81

Adding Radicals Subtracting Radicals

Example: Add: √6 + 4√6 + 5√6

1 + 4 + 6 = 10 Answer: 10√6

Example: Subtract: √5 - 6√5 - 7√5

1 - 6 - 7 = -12 Answer: -12√5

MultiplyingandDividingRadicals

Allradicalscanbemultipliedanddivided.Multiply/Dividethecoef;icientsandmultiply/dividetheradicands.Simplifytheradicalsattheend.

PracticeProblems

Section 4Multiplying and Dividing

82

Multiplying Radicals

Multiplying Radicals 2

Dividing Radicals

Example:

Example: Multiply: (4√2)(2√3)

Answer: 8√6

SolvingEquationswithRadicals

Tosolveanequationwitharadical,theradicalneedstobeisolatedononesideoftheequation.Thatmeansanytermsthatareoutsideoftheradicalneedtobemovedtotheothersideusinginverseoperations.Oncetheradicalisisolated,squareeachsideoftheequation(thiswilleliminatethesquareroot).Solvetheequationandcheck.

PracticeProblems

Section 5Solving Equations

83

Solving Equations with

Radicals

Example: Solve: √x + 1 = 6

(√x + 1)² = (6)² x + 1 = 36 x = 35

Check: √x + 1 = 6 √35 + 1 = 6

√36 = 6 6 = 6 ✔️

RationalizingtheDenominator

Rationalizingthedenominatorofafractionmeansremovingtheradicalfromthedenominator.Todothis;indaradicalthatthedenominatorcanbemultipliedbywhichwillresultinaperfectsquare.Whenthedenominatorismultipliedbyaradical,thenumeratormustbemultipliedbythesameradical.Simplifytheradicalinthenumeratorandsimplifythefractionalcoef;icientattheend.

PracticeProblems

Section 6Rationalizing the Denominator

84

Rationalizing the Denominator

Example: Simplify: 2 √5 2 √5 = 2√5 = 2√5 √5 √5 √25 5

PythagoreanTheorem

PythagoreanTheoremexplainstherelationshipbetweenthethreesidesofarighttriangle.Thetheoremstatesthatthesumofthesquaresofthelegsofarighttrianglearealwaysequaltothesquareofthehypotenuseoftherighttriangle.

PracticeProblems

Formula:

(aandbrepresentthelegs,whilecrepresentsthehypotenuse,whichisalwaysoppositetherightangle).

Section 7Pythagorean Theorem

85

!! + !! = !!!

Pythagorean Theorem

3 in.

4 in. x

a² + b² = c² 3² + 4² = c² 9 + 16 = c² 25 = c² √25 = √c² 5 in. = c

Example: Find the missing side.