Key to
&ebra Adding snd Subtracting Rations,I Expressions
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By JulieKingand PeterRasmussen
Name
Class
.at,!
TABLEOF CONTENTS Review .........1 ... AddingFractions witha CommonDenominator............ ...................2 The Oppositeof a Fraction Subtracting Fractions witha Common Denominator.......... ..............7 Addingand Subtracting Fractions withDitferent Denominators..... ................11 Combining Integers and Rational Expressions ..............14 Combining Polynomials andRational Expressions............ ............ 15 MoreAddingandSubtracting Fractions withDifferent Denominators...........................16 LeastCommonDenominators ......... ..............18 UsingCommonDenominators to SolveEquations .......23 Proportions.............. . . . . .2. 6 RatioProbfems. ............. 28 PercentProblems. .........30 TimeProblems .............. 32 Problems aboutRational Numbers ............... 34 Written Work ................. 35 Practice Test........ ......... 36 Youngand Giftedll Manymathematicians triedto solvethegenerallitthdegreeequation from 1545until1820,whentheteenagerNielsHenrikAbelsucceeded where theyhadfailed.He showedthatit is impossible to constructa formulato solveeveryequationof degreefiveor higher. Meanwhilein Franceanotherchildprodigymeta similarfate. gabWAH;l8l11832)leda rebellious EvaristeGalois(pronounced life. Hedidn'tattendschooluntilhewas12 yearsold;hismotherprovided him witha classicaleducationuntilthen. Evaristewasonlyan average studentuntil he took a.tnath coursewhoset€xtbookby the famous mathematician Legendiecaughthisfancy.Intwodayshehadfinishedthe bookdesignedfor two yearsof study. 'l knowit." Galoisthenhandedin histextbookwithtre explanation, He was right. WhenGaloissaidhe knewsomething,he meanthe had masteredit. Histeacher,LouisPaulEmile Richard,playedan impofiantrolein hislife,particularly sinceGaloisdidn'tgetalongwithmanyotherstudents at the boardingschool. Mr. Richardarrangedfor Galoisto take the examinationfor entranceinto France'smost famousuniversity,the Polytechnic Institute.Unfortunately theyoungGaloisfailedtheoralpaft of theexamwhenhe refusedto answera question,he feltwasimproperly posed. DoyouthinkthatGalois'attitudewasimpertinent?Shouldhe have challengedthe experts? Most mathematicians today agreethat his opinionwas correct,evenif his behaviorwasn'tright. So Galoisretumedto highschoolandworkedind€p€ndently under Mr. Richard.Soon,however,he becameinvolvedin radicalpoliticsand at onepointmadea threaton lhe king'slife. HEwassentto prisonforthe threat,and he remainedthere until ill healthforcedthe authoritiesto translerhimto a sanitorium. Whiletherehe waschallengedto a duel. He accepted. Thatwas verystupid. Galoishadno experience withpistols. The nelt morninghe was killedin the duel. Hewasonly20 yearsold.
Historicalnoteby DavidZitarelli lllustration by Jay Flom
:l,f
Thatis verysad. Fortunatsly Galoisstay€dup theentirenightbeforetheduelwriting his resultsin algebraand a letterto his bestfriend. In it he summarized outlinedth6 directiontrat his researchwouldlead. Wittroutknowingof Abel'swork Galoisprovedexacdythe same result_it is impossibleto constructa formulato solve fifth degree could equations.Buthewentevenfurther.Heknewthatsomeequations be solvedand otherscould not. He introducedgrorps of oblectsto whetheragivenequationof anydegrEecouldbesolvedor not. determine Galoiswas aheadof his time. Hisworkswerenot understoodfor some 25 yearsafter his death.
[email protected] 'abstract "Galois algebra'beginby studyinggrouptheoryand end by str.rdying theory.' On the coverof this bookGaloisprepareslor his duel. Unlikethe "duels'oftheRenaissance, pistols mathematical thisduelwasfoughtwith and broughtthe endto Galois'shortlife.
IMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a duplicating master. No part of this book may be reproducedin any form without the prior written permissionof the publisher. Gopyrightinfringementis a violationof FederalLaw.
[email protected] KeyGurriculum Project,Inc.All rightsreserved. @Key to Fractions,Key to Decimals,Key to Percents,Key to Algebra,Key to Geometry,Key to Measurement,and Key to MetricMeasurementareregisteredtrademarksof Key CurriculumPress. Publishedby KeyCurriculumPress,115065thStreet,Emeryville, CA 94608 lsBN 1559s30073 Printedin the UnitedStatesof Ameri:a 23 22 21 08 07 06 0s
Review In thisbookyou willlearnhowto add andsubtractfractions(bothrationalnumbersand rational multiply expressions). To do thisyouwillhaveto remember howto add,subtract, andfactorpolynomials, howto simplifyrationalexpressions and howto findequivalent expressions in higherterms.Testyourmemoryby doingtheseproblems. Simplify. 3a
3b
3a+3b __
=
a+b
2x* I
xtl = x'lb
x"+7x*12
Find an equivalentfraction.
1 = vl 5 l v
x =_ 12 lzxy
3
^4
=1*ffi11
Add.
(x"*4x6)*(2x,+x8)= ( x " * 5 ") ( x  l ) = Writethe oppositeof the expression in parentheses.
(x'"8)=
(3x'5x*2)=
Subtract.
3 x  5 x =
( 3 x ' * 5 x )  ( x ' * 4 x= )
2a' 
( x . * l x * B )  ( x ' 2 x  5 ) =
6a' =
Multiply.Makesureyouransweris in simplest form.
(x*7)(xl)= Bx' *, =
2x'(x 5)=
5 y ' .( e i )=
Factorcompletely.
1 0 5= l 5 g= 01990 by Ksy CutriculumProiecl.Inc Do not duplicats withoul pormission.
3 x ' 2 7= xL + 3x  BB=
AddingFractionswith a CommonDenominator Whentwo fractionshavethe samedenominator we say thattheyhavea common denominator.Addingfractions witha commondenominator is easy. Denominators tell whatkinds of numbersor expressions are beingadded. Numerators tell howmany, so to addfractionswe just addthe numerators and keepthe denominators the same.
3 T
+
2 +
5=
+
t
+
+
1 a
a
=
3 a
Add.
_ 1 
5
2
r
B
B
5
7
4
1 6+  2
+
6
6
=
5 q
5_+
7y 1
5
l
 f
ty
5
+
3=
t
+ 
l
xl
= + _ x+3
n
5
X
3
=
4 x +. 3I v=
3
3
V + + =
=
t + t +  3 =
8
x +3
+
3 1
+
8
a
+ 
2 1
3
r
o
x2
+
3=
1 2 3 + _ = al al
3
2
+ 
5b
B
3=
5b
7  4
xy
+

xY 7
2
+ xz8
= ?('8
+ r +
5
q2
 t . B x
q
62 + 
1
=
3
+
3
=
Proiecl,hc. @19$ by KeyCurriculum Do nol duplbalewilhoutp€rmission.
Sometimes afteraddlngwe cansimplifythe answer. 5
8 . 1 9 q
I
2x
5y d
7 x=
+
5y
t
n
r E= 5 r' g 3Y
=
tzY
,
.
L
?
t2Ll
4 =
f.4
= x V I
x+z xz tl
I
t)z
=
=
l*4lh2) I

I
t2
Add. Besureto simplify eachanswer. 3
tz
5 l 6 + E e
5=
+
tz
+.t,*=  131v  + _ =
y*4
I++ l
=
+
xt+6t+8
n"+2n
=
atl

5n
+ tlr7
3 r q 4 x +1 2  4 x + 1 2
12
+ at
th
2t
2
3i*It= 5
3 +J =
2x
4
+
5
1z+gx+g
n
nt+2n
=
q b2+2b3 Ot990by KeyCufiicrlumProiect,Inc. Do not duplicatewithoutpermissbn.
b z+ 2 b  3
3
Add. Makesureeachansweris in simplestform.
3

?
T
Y
8
2

=
Y
31
Y
5v
+ + =
2x
2t
5r
x2+6x
x2+6x
)
5x,
x21
?
V
J
r+rl
x
x+{

* xllx 4= 5n
5x2 x24?"+3
lfx
xz4r+3
4x
3x6
3r 6
x
3x
Xs+tl
=
x2q
ttt
3x2

11+5t
a
x
=
2* 
d2+4
2 Oz+ 2a + 
4
6
1z
a
+
tt+4
2 At +2a+ 
=
O19S by Key Cufriqrlum Projecl, Inc 0o not duplicats wilhout p€rmission.
The numerators of theserational expressions arepolynomials. Add. Thensimplify the
answerif youcan. qdd
odd
x1 + 5
2x"4 "^3 + x  3
? ( +l o +
N +z
2x+3
t
x+ 
/ t * + * L   * Z y= Y*f Y * 2 vz 3 vt
T
v 2 + 7= 3 lv
2 x 3 . x  l ?
x+6
=
x+b
.'  2E t z  a Ja(x + Zl 3 x( r + 2 ) xz+1x
+
 x z x ( r t +l ) ( r  3 ) Ztt+5
(r+l)(r3) 0199O by Ksy Curridrlum Proiod, Inc. Oo nol dupli:ate without pormission.
7+xxL
(x+l)(x3)
5
The Oppositeof a Fraction Subtracting rationalnumbersandexpressions worksthe sameway as subtracting integers. To subtract,we addthe opposite so we needto knowhowto findthe oppositeof a fraction. The opposite of a fraction has the oppositenumeratorbut the same denominatoras the originalfraction.
3  3 = r f t 6 =6
,,Theopposite ot I is f :' ,,Theopposite of * u *:
? ( x  r + l
x  l :(ar3
= jJ
rJ
,,Theopposire ot **
j'
;s ;f,i{
Writetheoppositeof eachfraction. 5
xl 3
=
8 5

  =
8
o +3
=
o3
=
_cr = xl
r+  =
3
_g=
x z+2 x  3
x
_ 2 x +4 = x
i1 =
_ 2x4 =
3x'x++
x
x
4
?(   =
lo
x
x2
xal

=
5
^22x+l
=
Add. 5 + 
8
s=
8
 x x
z 6
?

=
2
2!:l 5
^2!:'7 5
t.tfi+3 t+f
+
Xe+tlr3 ?(+l
=
0199 by Key Cutriculum Poj6ct, Inc. Do not duplicats rvithoul permission.
SubtractingFractionswith a CommonDenominator fromanother: expression Tosubtract onerational sign. signwithan addition 1. Replacethe subtraction 2. Replacethe second fractionwithits opposite. 3. Go aheadand add. Subtract.
2x 3x +  = 5 5
t 2 5
{
E
z _tl 15 t5
x+5
x
6
6
=
'2t +'5
*T
n+I
a+r{
aZ
1 3x ? ( 5=
2x
4r
I _3^= ttr 4r
L= **y r(*y
x + 3_ I = 1 q
x+4
{ L +
x+5
6x
=
l3x
x+3
@1990by Koy CuriqJlum Prolecl, Inc. 0o not duplbate without p€rmissbn.
=
?(
3rl
=
r( f,
2a6 = aZ
Zx
 x2
T
2x+3 x+f
x2+3xl
?(
t+3
x+3
x" 5x+2

?x 7
Subtract.Simplifythe answerif you can.
3x{
2x5
3x+3
3t+3
,(+y
=
 . ? (  y
2x
2x
3x+2 a
3x
4x 12 h
3x
x. + 5x
xz +5r(
lz + 4r(
x+rl
a5 =
a + 5
qb
ab
2(
4x
t5
t5
6a 5or+a
=
al 5ar+a
t ( z + 3 n  5 _ xz 2x + lO
lo
8
lo
o l9S by KoyCuriculumPtoied,lnc. wilhoulpomissbn. Oonotduplicato
Hereare somemorerationalexpressions for youto add. Thistimeyouwill haveto multiply the factorsin eachnumerator to findout if thereare any liketermsyou can combine. =
5rt5 7
3x+18 7

B r + 3
7
{"2) tl
5 ( z