2.1SolvingEquationsUsingPropertiesofEqualityTrueandFalseEquationsEx:Determineifthefollowingaretrueorfalse.1)3+4=72)3+4=83)x+6=10Ex:Determinewhether4isasolutionofx+6=10PropertiesofEqualityLet’sassumethatThorandLokiarethesameheight:Iftheystandtogetheronabox,theyarestillthesameheight:So:(AdditionPropertyofEquality)Iftheystandtogetherinaditch,theyarestillthesameheight:So:(SubtractionPropertyofEquality)

True

False

Sometimes TrueSometimes False

x= 4 : 4+6=10 ✓

X +6=10 ⇒ 4+6=10 ✓T

substitute 4 for ×

Enjoy#

§§ etnatgus

A = B

k¥ Ffa-

At C = B + c

IE ¥1 Ter

A- c = B - c

IfyoucloneThorandLoki,thentheyarestillthesameheight:So:(MultiplicationPropertyofEquality)Hardtodrawapictureof:DivisionPropertyofEqualitySolvingEquations:1)Locatethevariable(s)anddecidewhichsidetokeepthemon.2)UseAdditionandSubtractionto“move”termssothatallthevariablesareononeside,andalltheconstantsareontheotherside.3)UseMultiplicationorDivisiontoisolatethevariable.Ex:Solve1)5t=402)-80=-5w

k¥¥Ithe #¥¥w 0¥

a ^Ac = Bc ⇐ 0 ← 0=0

¥ = Be to

where they

c- are happier

=L get it all by itself

5. t = 40'

s¥÷4¥t = 8

- 8o÷;5¥to:# Este

F - 5 +80 ; +80

16 = W 5=80

who / 5¥ = 8¥or W = 16

3)7.7=-3.2+2s4)2a+4=a+85)!" # − 1 =

'() # +

("()

+3i÷:t=10.9÷ ZS

101 =2522

5. 45 = S

:= 5.45

substitute or

.

plugin: Check :

TUYET 2 at 4 = at 8

- 4 ;- 4

2C 4) +4 E4+8- 8+4 = 12

a = 4 = 12 #✓

Clear the fractions

Denominators : 5,

1

, 10,10

LCD = 10 ← The smallest

big # they all

zgo into

;,

F.¥t . 10.1 =,#Pot +F. to, , a

2. 4t - 10 =3++1.5

8t - 10 =3 t + 15= =

- 3t - 3t

5¥15

+105T = 25

5¥ = 2¥ ⇒ t=5DCheck :

2.2MoreAboutSolvingEquationsEx:SolveandCheck1)9(t+2)=-6(t–3)+15tNote:Thesetypesofequationsarecalledidentities.2)5(x+3)–3x=2(x+8)Note:Thesetypeofequationsarecalledcontradictions.

pikoytt's

a aCheck

:t=3<9++92=-6++6. }+ ,st

9 (3+2) 't -6 (3-3)+15.3

9++18=-6++18+15 't

¥¥siI 9¥ 's.IE?sIID18=18 True

Solutions : All |9( ot 2) I -6 (0-3)+156)Real Numbers

or R | 982-6+3+0

c- 180

~~ At

5×+5.3-3×-2×+28

5×+15=-2×+165×-3×+15=2×+16

2×+15 = 2×+16

- ZX- Zx

l5#FalseNo Solution

No check( for you )

ClearingFractionsandDecimalsEx:SolveandCheck1)0.105x+0.06(20,000–x)=17402)"+ , + 3, = −

(' , −

((/

AtCheck :

0.1-05×+1200-0.061=1740 0.1056304+0.06/20,000-12,000) £17400.045×+1200=1740

- -1200 - izoo 1260 + 0.06 ( 8,000 )÷lsx = 540 1260+480

%fs¥.to#nyo&X = 12,000

Or Clear decimals

-

6,3 ,9⇒LCD=l8

¥5#+ is.3nHF¥nt¥t⇒3. 5h +18 .3n= 6th ) -241 )

15h + 54n= - 6h - 22

69N = - 6h -22

+ 6h + 6h-

75N = - 22

*75=3,1 -

n=¥

0.105×+0.0640000 -D= 1740w -

3 decimal 2 decimal

PlacesPlaces

Worst =3 decimal places⇒ Use

10003Zeros

10006.105Gt 1000 . 0.0640000 - × ) =

10%0

105×+60 ( 20000 - D= 1,740,000

105×+1,200,000-60 X= 1,740,000=

45×+1,200,000=1,740,000

- 1,200,000-

1,200,00045=540,000

45×145=51*0 X.

- 12.000

Check for # 2

It

¥n+3n=- In - ¥

Et⇒+3t⇒e÷t⇒a±¥t¥¥¥#¥¥÷t⇒¥

25

¥ . ¥

E* ¥

9.5 5.5 ! ( CD = 225

LCD=9 . 5.5 .

= 225 . 11.25275¥I*i .¥÷÷ie :

Fires

-55 198 ; 2¥ }÷FIs

'

IF

'}¥_=szs¥

2.3ApplicationsofPercentConvertingfrompercentstodecimalsandbackagainEx:Converteachfrom%todecimal,orfromdecimalto%1)31%=2)130%=3)0.08=4)50.456=DirectTranslationProblemsEx:Whatnumberis5%of96?Ex:102is21.3%ofwhatnumber?Ex:31iswhatpercentof500?

O . 3 I why 31 %3 I

. = 31W Too2 '

1. 30

8% 0.

08W

5045.6%

50.456ur ÷

x t }oF. 96

× = 4.8

at tote .int

OFodds naeis ÷

n = 478 .8732394 . . .

§,± Tf !

soo

31 = × .

50/0Foo ¥ . not3 '

×det

Foo=

f

X = ¥ = 0 . 062

= 6 . 2 %

AppliedPercentProblemsEx:(#32)AguestattheSanAntonioHiltonAirportHotelpaid$180foraroomplus9%cityroomtax,a1¾%countyroomtax,anda6%stateroomtax.Findthetotalamountoftaxthattheguestpaidontheroom.Ex:(#44)ApearlnecklaceofformerFirstLadyJacquelineKennedyOnassis,originallyvaluedat$700,wassoldatauctionin1996for$211,500.Findthepercentincreaseinthevalueofthenecklace.Ex:Apairofshoesthatusuallysellsfor$110wasmarkeddownto$88.Findthepercentdecreaseinthepriceoftheshoes.

Tax =Total . Tax % rate

180 ( o.O a) + 180 (0-0175)+180/0.06)or

180 (0.09+0.0175+0.06) = 180(0-1675)= $30.15

Percent Increase / Decrease

Slept : find the change / difference

211,500 - 700 = 210,800

step 2 : Calculate the % usingthe original value

.

÷gear"

# =÷

,¥÷ = 0.2 =

Zfeohease

Original : $700

New : $211,500Change # ference :$210,800

Formal method :

The changekiffuena is what % of the original?

210,800 =

p . 700

21%5 = PEE'

r.ae#=E*toiIitI.

= 30,114.285..%Round to nearest whole % : 30,114%

Round to one decimal place : 30,114.3%' ' ' '

2 ' ' ' i

: 30,114.29%

2.4FormulasWhataresomeexamplesofformulas?Formula:Anequationthatstatesamathematicalrelationshipbetweentwoormorevariables.I=Prt Interest=Principal*interestrate*timeP=R–C Profit=Revenue–Costr=c+m Retailprice=cost+markupD=RT Distancetravelled=Rate(speed)*TimeC=5/9(F–32) Temperaturein°Celcius=5/9(Temperaturein°Fahrenheit–32)P=2l+2w Theperimeterofarectangle=two*thelength+two*thewidthA=pr2 Theareaofacircle=(pi)r2V=pr2h Thevolumeofacylinder=pr2hEx:(#14)Findthemarkuponadozenrosesifafloristbuysthemwholesalefor$12.95andsellsthemfor$47.50.

- Monthan

A=w.lone variable

P=l+w+l+w

L =

hftd

:- m

47.50=12.95 's m

-12.95 - 12.95.

34.55 = m

m=$34.55@

Ex:(#20)Threeyearsafteropeninganaccountthatpaidsimpleinterestof6.45%annually,adepositorwithdrewthe$3,483ininterestearned.Howmuchmoneywasleftintheaccount?Ex:(#22)RoseParadefloatstraveldownthe5.5-mile-longparaderouteatarateof2.5mph.Howlongwillittakeafloattocompletetherouteiftherearenodelays?Ex:(#28)Convert2,212°C,thetemperatureatwhichsilverboils,todegreesFahrenheit.Roundtothenearestdegree.

I=Prt 3483=170.064533483=13.1935 )p= ? t=3yeas§÷=hootedr= 6.45% I = $3483

= 0.0645↳

¥8,0000

D= RT D= RTDert

5.5=2.5'T

D= 5.5 miles s

R ' 2.5m¥hI÷= 3¥ -

T= ?hourst'

-1=2.20

C=Ia( F- 32 )2212 =¥( F- 32 )

iii.¥i¥¥ef÷E÷I¥19

,

908=5 F- 160

tl60t1#

2. 2 hours

= 2 hours + 0.460) min

= 2 hours + 12 minutes

-

2. 5 hours

2 hours 30 minutes

.5 = £ of an hour

÷days1 day ,

0.2 ( 24 ) hours

1 day , 4.8 hours →

I day ,4.8 hours

i 1 day,

4 hours, 0.8/60) min

= I day ,4 hours

.48 minutes

SolvingforaSpecifiedVariableEx:SolveP=R–CforR.Ex:Solve! = 123

4 forb.

Ex:Solve 5)" + 16- = −81forGEx:SolveB=50+r(x+y)fory

2.5ProblemSolvingEx:(#46)FirstAidAslingisintheshapeofanisoscelestrianglewithaperimeterof144inches.Thelongestsideoftheslingis18incheslongerthaneitheroftheothertwosides.Findthelengthsofeachside.Ex:(#16)Arockgroupplanstotravelforatotalof38weeks,makingthreeconcertstops.TheywillbeinJapanfor4moreweeksthantheywillbeinAustralia.TheirstayinSwedenwillbe2weeksshorterthanthatinAustralia.Howmanyweekswilltheybeineachcountry?

Ex:(#40)Thethreenumbersofthecombinationforalockareconsecutiveintegers,andtheirsumis81.Findthecombination.

2.6MoreProblemSolvingInvestmentProblemsEx:(#22)Aninvestorwantstoreceive$1,000annuallyfromtwoinvestments.Hehasput$4,500inamoneymarketaccountpaying4%annualsimpleinterest.Howmuchshouldheinvestinastockfundthatpays10%annualsimpleinteresttoachievehisgoal? I=Prt

P r t I

MoneyMarket

StockFund

UniformMotionProblems D=R*TEx:(#34)Howlongwillittakeamother,runningat4feetpersecond,tocatchupwithhertoddler,runningdownthesidewalkat2feetpersecond,ifthechildhada5-secondheadstart?“Catchup”-

R T D

Mother

Toddler

MixtureProblemsEx:(#40)Aphotographerwishestomix2litersofa5%aceticacidsolutionwitha10%solutiontogeta7%solution.Howmanylitersof10%solutionmustbeadded?

Amount Strength

Mixture

Ex:(#48)Astoresellsregularcoffeefor$8perpoundandgourmetcoffeefor$14perpound.Togetridof40poundsofthegourmetcoffee,ashopkeepermakesablendtoputonsalefor$10perpound.Howmanypoundsofregularcoffeeshouldheuse?

Number-ValueProblemsEx:(#60)Ascubadiver,hiredbyanamusementpark,collected$121innickels,dimesandquartersatthebottomofawishingwell.Therewere500nickelsand90morequartersthandimes.Howmanyquartersanddimeswerethrownintothewishingwell?

2.7SolvingInequalitiesEx:Foreachinequality,expressthesolutionseta)inset-buildernotation,b)asaninterval,andc)asagraph.1)x>52)x>53)x<54)x<55)-3<x<15

LokiandTheHulkLokiisshorterthanTheHulk:Whenthey’rebothstandingonaplatform,LokiisshorterthanTheHulk:Whenthey’rebothstandinginaditch,LokiIsshorterthanTheHulk:Whenyouclonethem,LokiisstillshorterthanTheHulk:

However,whenyouclonethemandhangthemupsidedown,whathappens?MultiplicationandDivisionPropertiesofInequality:5<10Whathappensifyoumultiplybothsidesby2?Divideby5?Whathappensifyoumultiplybothsidesby-2?Divideby-5?What’stherule?

Ex:Solveforx:1)4<-4(x–2)2)4<-4(x–2)<20

3)'9:()" ≤ ! + 4