Key to Algebra Book 5 Rational Numbers

Key to

&ebra Rstional Numbers

By fulie Kingand PeterRasmussen

Name

Class

TABLEOF CONTENTS R a t i o n aNl u m b e r s D i v i d i nIgn t e g e r s Equations withRational Solutions............ N u m b eLr i n e s Graphing lntegers Graphing Rational Numbers. I n e q u a l i t i.e. .s. . . . . . . . . . A b s o l u tVea l u e G r a p h i nIgn e q u a l i t i.e. .s. . . . . . . . . . S o l v i n lgn e q u a l i t i e s . . . . . . . . . . . . . Relations Functions W r i t t eW n ork Practice Test........

............1 ..............2 ........4 ...................5 ............6 ............7 . . . . . .1. .1 ...............14 . . . . . .1. .5 ...........18 ........27 ........29 ..................35 ..........36

Systemsof Equations The graphof a linearequationis a lineif the equationhastwovariables and a plane if the equationhas three variables. A systemof linear equations is a setofsuchequations considered atthe sametime. Thehistoryof systemsof linearequationshada ratherunusualbeginning-itstartedon the backof a turtle.Accordingto ancientChinesetraditiona turtle carrieda specialsquarefromthe riverLo to a man. Hereis thesquare. Sucha squareiscalleda magicsquarebecause thethreenumbersin everyrow,columnanddiagonal addup to 15. TheChinesewereespecially fondof patterns,so it is notsurprising that they wouldbe intriguedby magicsquares.About250 e.c.a book calledNineChapterson theMathematicalArf devotedoneentiresection to constructing them. lt involvedthreelinearequations, markingthefirst timein historythata systemof linearequationswas everencountered. Chinesemathematicians continuedto developand refine techniquesfor solvingsystemsof linearequations.The peakof this development occurredin 1303r0. withthepublication of a mathematics book havingthe unlikelylitle PreciousMirrorof the FourElements.lt describeda methodfor solvingsystemsof four equationswhose unknownswerecalledheaven,earth,manand matter. The resultsof theseChineseadvancesremainedunknownin the West.Duringthe earlypanofthe19thcenturytheGerman mathematician KarlGauss('1777-1855) introduced an effectivemethodfor solvingsuch systems.lt was modifiedslightlyby Jordan,andtodaythe procedure is calledGauss-Jordan elimination. Whois Jordan? Forovera centurythe nameJordanwasassumedto be a tributeto theFrenchmathematician CamilleJordan(1838-1922). it was However discoveredin 1986that the methodwas actuallydue to the German geodesist WilhelmJordan(1841-1899). On the coverof this book you see the legendaryChineseturtle emergingfromthe riverLo witha magicsquareon its back. Thepattern on the turtle'sbackrepresents the magicsquareshownabove.

Historicalnoteby DavidZitarelli lllustration by Jay Flom

IMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a duplicatang master. No part of this book may be reproducedin any form without the prior written permissaonof the publisher. Copyrightinfringementis a violationof FederalLaw. Copyright @1990by KeyCurriculum 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 MetricMeasurement arc registered trademarks of KeyCuniculumPress. Published by KeyCurriculum Press,115065thStreet,Emeryville, CA 94608 Printedin the UnitedStatesof America lsBN 1-55953-005-7 23 22 21 08 07 06 0s

R a t i o n aN l umbers wholenumbersand0). andnegative In Books1 to 4 we workedwithintegers(positive integers, butwhenwe gotto division We hadno troubleadding,subtracting andmultiplying youcan problems we ranintodifficulties. Division like 9 + 0 haveno answer,because neverdivideby 0. Otherproblems, like 10- 3, do nothaveanswerswhichareintegers. To solveproblems like 10 + 3 we needa newclassof numberscalledrationalnumbers. Rationalnumbersare numberswhichcan be writtenas fractions.The numerator(top number)anddenominator(bottomnumber)of a fractionmustbe integersandthe maynotbe 0. denominator

IL

r J . 9 -h- -3_-1

(-

numerators

(-

denominators

T

l - 7 7

B

I

l

3

Everyintegeris a rationalnumberbecause it can be writtenas a fractionwith a denominator of 1. Rewriteeachintegeras a fraction.

B=+

- ? =3 \-,,

I

-21=

L+=

o=

-15=

25=

6=

Everymixednumberis a rationalnumberbecause it canbewrittenas a fraction. Rewrite eachmixednumberas a fraction. 1 3

r +

oo

7

1 6+ 3

3 I

lg

3?=

6 l J 2 -

+8=

loa= = l

l7=

23 J 7 -

Everydecimalis alsoa rationalnumber(unlessit goes on foreverwithoutrepeating). A terminatingdecimal(onethatcomesto an end)is a rationalnumberbecauseit equalsa fractionwitha denominator of 10 or 100or 1000,etc. Rewriteeachterminating decimalas a fraction.

o6=# o06=# 0006= u.u)b = n

n

t-,

@1990by Ksy Curiculum Proioct,Inc. Do not duplicale wilhout p€rmission.

O.1= 0.01= 0.1= 9 0 t t?=

l 3 = l #19, lo= 2 . 1= 5.Ol= 3.27=

DividingIntegers Nowwe candivideanyintegerby any otherintegerexcept0. All we haveto do is write (top)and a fractionwiththe dividend(thenumberwe aredividinginto)as the numerator (bottom). the divisor(thenumberwe aredividingby)as the denominator

rl\-/n \., a -- l o 3 Do eachdivisionproblem.lf the divisorgoesevenlyintothe dividend,writeyouranswer writeit as a fraction. as an integer.Otherwise,

1 2* - 2 = - 6 4O 4 O= 7 = +I i -3.5=

15=3= -7=2=

5 =4 =

-$+-3 =

= b =

54*7=

-Ll5=-1 =

-7 = 6=

- / + 1 9=

A fractioncan be positiveor negative.To findthe sign,justfollowthe rulesfor division. Whendivisionis writtenusinga fractionbar,the ruleslooklikethis: POSITIVE

NEGATIVE =

NEGATIVE

NEGATIVE

POStlvE

POffi

=

POSITIVE

ffi

POSITIVE

=

NEGATIVE

iir

Nffiriw

=

we willwriteit lf a fractionis positive, we willwriteit withno signs. lf a fractionis negative, withthe negativesignon top.

sowewillwrite|. S is positive,

so wewillwritef . f is negative,

fraction. problem.Writeyouransweras a positive or negative Doeachdivision

=+

-{+-7=

15--2 =

l B* - 5 =

!+lO=

l.-5=

- l= 9 =

40*21=

- l + - l O=

9 :-5 =

1 2+ - 7=

- J + - l O 0=

12*ll=

4=5=

-f+-14=

-20=l=

1 3* - 3

2

0199 by Ksy CurriculumProjsl, Inc. 0o not duplicats without pormission.

Divide.Writeyouransweras an integeror as a positive or negative mixednumber.

50 *-5 = -fO 2B*-t+= -80=-lO=

- 1 2= 1=- l i

- l B+ - 7=

30=?= -lB=5=

- J = - 6=

2 5 + - 7=

4 5 +- 2 = % 3 + - q=

O+-6= - ll = 4 =

20=3= -25 =-4 =

%0=4=

64*-3= -100 =3= - l ++ - 5= -37=lO=

Divide.Thistimewriteyouransweras a positiveor negativedecimal.

- J + l O = B = - O . 3 lo7+too =is =l# =t.o7 -r+1 3q + lO =

.75

olgeo by KeyCuriqrlum p|Dlect,Inc. Do nd dwlicate wltl|outpermi$bn.

-253

Equationswith RationalSolutions is notan NowwecanusetheDivision Principle evenwhentheanswer to solveequations integer.Solveeachequation.Writeyouransweras a fraction or as a mixednumber.

-4x = 17

9x=40

&=-25 Y . 7

x =-3+ 2x- 5 = ltf

3x* 7=-4

-5x*l=15

x-3=8x*5

-2h -5) = 7

x-5x+7=-8

Solvetheseequations, writeit as a decimal. too. Thistimeif youransweris notan integer,

.fA

Itx

= - lO

{

1=

I rlio7 J oo\-\--#

F -2,5

-5x=

l8

7 x - 7= 3 x + 2 0

x - ? =6 x * J

3 ( x- 5 ) = x - 2 0

4(x+6)=23

-2c^ 5

- -7 I

2(x-3)*x=9

0199 by Key CurriculumProiecl,Inc. Do not duplicatewithoutpermission.

NumberLines In Book1 we usednumberlinesto helpus thinkaboutaddingand multiplying integers. Thefootballfieldwasa kindof numberline. Rulersandthescaleson thermometers are alsonumberlines. To makea numberlinewe drawa lineanddivideit intosections of equallengthcalled units. Thenwe numberthe pointswhichseparate the units.Herearethreenumberlines:

-60

-59 -58

-57 -56 -55 -54 -53

-52 -51

-50

Thearrowson the endsof eachnumberlineshowthatthe numberlinekeepsgoing. We canstartwithanynumberas longas we numberthe pointsin order(usually fromleftto right).Sometimes we do notshoweveryunit. Thisnumberlineonlyshowseveryfifthunit: -25

-20

-15

-10

-5

0

5

10

Hereare some numberlinesfor you to finishnumbering:

-18

- 15

Makea numberlineshowingall the integersfrom -5 to 5.

Ol 990by KsyCurriculum Proiscl,Inc. Oonol duplicatswithoutg€rmission

5

Graphinglntegers Wecanusea number lineto picture a setof numbers. linewe makea dot Onthenumber to showeachnumberin theset. Thisis calleda graphoftheset. A graphcanhelpyousee a pattern or answera question.lf a pattern continues forever to theleftor right,wefillin the arrowthatpointsin thatdirection. Grapheachset of integersbelow. Oddintegers: Evenintegers: Integers lessthan4: lntegersgreaterthan-3: Integersbetween-3 and-4: Integers notequalto 2:

-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

6

-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 6

Integers divisible by 2: 0

10

20

0

10

20

0

10

20

0

10

0

10

lntegers divisible by 3: Integers divisible by 2 and3: Thesquaresof integers: Integerswithsquares whicharelessthanl0:

-10

-10

Didyou noticeany interesting patternsin the graphsyou made? @199 by Key CurriculumProj6cl,Inc Do not duplicalo without p€rmissbn.

G r a p h i n gR a t i o n aN l umbers Integers are notthe onlypointson a numberline. On the numberlinebelowwe have also labeledthe pointshalfwaybetweeneachintegerandthe next.

3 -2+ -;

-i

-i

o

+

b z t

ln fact,thereis a pointon the numberlinefor eachrationalnumber.To findthispoint, firstwritethe rationalnumberas a fraction.Thedenominator of the fractiontellshow manypafisto divideeachunitof the numberlineinto. The numerator tellshowmany partsto countoffto the rightof 0 (ifthe numberis positive) o_r to the leftof 0 (ifthe number is negative) point. find to the Here'showto find f ,'f , anOf . 6 Pafts

I3,o

7

i 2 + 2 + 2+ 2+ 2?

On eachnumberline,firstfinishlabeling the points.Thengraphthe rationalnumberat the left. ^ J

T -2 3

z5 lr

3

0

1

Labeleach rationalnumbershownon the numberline below. '2 O1990by K€y CurriculumProjocl,Inc Do not duplicats wilhout p€fmissbn.

7

Eachdecimalis a rationalnumber(unlessit goeson foreverwithoutrepeating), so it also place hasa on the numberline. To findthe pointfor a decimal, thinkof it as a fractionor mixednumber. 0.4is thesameas # . Thisnumberis between0 and1 so we dividethatunitintoten parts andcountfourto the rightof 0.

-3.7is equalto-3fr . Thisnumberis between -3 and-4 so we dividethatunitintoten parts andcountsevenunitsto the leftof -3. -4

-3.7

-3

-2

Forhundredths we coulddividethe unitintoa hundredparts,butto savetimeit makes senseto divideit intotenthsfirstandthento divideonlyoneof thetenthsintoten parts. To graph0.32thiswaywe firstnoticethatit is between0.3and0.4. Thenwe dividethe sectionbetween0.3 and0.4 intoten pails. Eachof thesepartsis a hundredth of the unit.

Grapheachdecimalbelow.

0.7

-o.2 5.8 -t+.I 325 -1.71+ 0.75 -0 08 0

I

@19S by Key CurriculumProjscl,Inc Oo nol duplicalBwithoul p€rmi$ion.

Finda decimalnamefor eachpointgraphedon the numberlinesbelow.

-0.1

0

0.1

lmaginemakinga graphof all the rationalnumbersbetween2 and 3.

Firstwe wouldgraphthe halves,

2+

thenthethirds,

2l

2+

zt

z)

zt zi

thenthefourths,

zt zi thenthe fifths,

2

z l z i z i z t z2+I zztt z3 z S2eo z2t lzt

3

andsoon... We wouldneverbe finished!Soonthe linewouldbe so crowdedwithdotsthatyoucouldn't tellonefromanother.So whenwe wantto show alltherationalnumbersbetween2 and3 we justshadethe wholesectionof the linebetween thosenumbers. 2 "between" wheneverwe say we willmean"notincluding the endpoints." We haveusedhollowdotsat 2 and3 to showthatthosenumbersare notincluded. Yougraphallthe rationalnumberswhichare: -1 and4 between between-3 and0

-4 -3

-2 -1

-4

-2

-3

between2 and3.5 O1990by K€y CutriculumProjecl,Inc. Oo not duplicatswilhoulpsrmission.

9

Canyoutellwhatsetshavebeengraphed?

-

2

-

1

0

1

2

3

4

5

6

7

8

9

Thefirstgraphshowsall rationalnumberswhichare greaterthan.3. The hollowdot shows that3 is notincluded. Thesecondgraphshowall rationalnumbers whicharelessthanor equalto 3. Thistime3 is included, so we haveuseda soliddot. On bothgraphsthe arrowshavebeenfilledin to showthatthe graphscontinue. Graphallthe rationalnumberswhichare: lessthan6

greaterthan1

greaterthanor equalto 4

lessthanor equalto 0 greaterthanor equalto -1.4

lessthanor equalto 0.5

between8 and8.5 7,8

8.0

8.2

8.4

8.8

8.6

notequalto 1 4

10

5

6

7

@19$ by K€y CurriculumProjscl,Inc Do not duplicalswilhoutpermission.

Inequalities In Book3 we workedwithequations.Rememberthat an equationis a sentenceabout numbersbeingequal,liker + -4 = 5. Anotherkindof sentenceis an inequality- a sentenceaboutnumbersbeingunequal. Herearetwo examplesof inequalities: utr+ -4 is less x + -4 3

x+8=l0

x 4 0

x+BlO

x
l0

X
10

x >

x +B { l0

x
o +3 14

l x l) x l x ls 2 Ol 99 by Key Cuiriculum Proiecl, Inc. Oo not duplicale wilhoul permission.

GraphingInequalities Lookat thelastequation page.Thefiveintegers ontheprevious whicharesolutions are -2,-1,0, 1 and2. we could easilymakea graphofthissetofsorutions.

lxls 2

- 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5

Someinequalities, like r ) -3, havean infinitenumberof solutions.lt woutdbe impossible to listallthe integerswhicharesolutions, butwe couldshowthe solutionset by startinga list andthenusingthreedotsto showthatit continues on andon.

x >3

o ,1 ,z , g ., . . I {-2,-1

We couldalsographtheset of solutions usinga darkened arrowon the rightto showthat the dotscontinue to the right.

x > 3

{ - 2 , - 11 , 0, 2 , ,g,...1

-5-4 -3-2 -1 0

1

2

3

4

5

Foreachinequality, showthe integers whicharesolutions in twoways:by makinga listand by graphing. lisf Graph

x 5

t

x s-3

t

x >-+ t x >2lot xso

t

l x l( T

t

l x l> 2 t x + f > 5 t Ol99O by Key Cuilicutum proisct, Inc. Do not duplicate without p€amissbn

15

ofthe numbers whicharesolutions It wouldbeimpossible to listalltherational -3. -3 greater than-3 is. number inequality is nota solution, r > buteveryrational Wecanshowthesolutionsetveryclearly by graphing. -

5

-

4

-

3

-

2

-

1

0

1

2

3

4

For each equationor inequalitybelow,graphthe set of all rationalnumberswhich

aresolutions.

x >

-5

-4

-3

-2

x ( 2

-5

-4

-3

-2

xs2

-5

-4

-3

-2

-1

x >-Ll

-5

-4

-3

-2

-1

-4

-3

-2

x +-3 x >

-5

-4

-3

-2

x

x-4

x - 5 x +t(

l >x

x < lo -+* 5x+ |

ff I ir biqqli thonr, th'e?r r is fessthon f.

7x-1
0

x >7 5 6 7

x-g

( x * 2 ) ( x - 2 )< x z+ x x e - & +( x t + x

(x * A(x-ll

5x-6
-tf

proioct.Inc. 01990by KeyCurriqrlum Do notduplbatewithoutp€rmjssion.

19

for and DivisionPrinciples Maybeyou are wonderingwhetherthereare Muttiplication and The answeris yes,buttheyaren'tquitethe sameas the Multiplication lnequalities. for Equations.Whenwe multiplyor dividebothsidesof an inequality Divisionprinciples inequality.Butwhenwe multiplyor by a pos1ivenumber,we do get an equivalent dividebothsidesby a negativenumber,we mustreversethe inequalitysignto get an to seewhy: inequality.Lookat thesesentences equivatent -6 Dividing tr,^e Multiplying 2: by -52 L

-6 < I

-t7 < Multiplying by -2:

lO'2.- t^re

-2O t>

l2

trae

-6 .L

true

-3

true

t z < - 2 0(

2

Dividing by -2:

folse ./

-z

false /

3

truc wilh 4

io ) swilchcd

3

>

-5

F

true wilh ( to ) switched

to switch Remember Principles. and Division usingthe Multiplication Solveeachinequality signif you multiplyor divideby a negativenumber. the directionof the inequality

oo 5x 2 ' 3 0 0

5

" )p

x >-6

o

o

-+. c c

x q

^t{r

t - --rl f

\

x

6 < >$)'+

>-z+

x

>

3

l

llx
BothSandyandTerryendedup withthe samesotutionset. Whosemethoddo you like better?Why? below. Bothmethodswork. Useeitheroneto solveeachinequality

2-x >

3 x * f 5>

25 s lo x

22

1 8 + x>

x+ 20


\ t. / t L

1 2- 7 x


x + 1 2>

t-7


- t 6 +I x

@1990 by KsyCuricutumproiecl.Inc. Do not dupli:atewilhoutgermissbn.

>

g + r < +

you shouldnof usethe Whenan absolutevaluesignappearsin an equationor inequality, insidetheabsolute to simplifyan expression or DivisionPrinciple Multiplication Addition, valuesignmeans. thinkaboutwhatthe absolute valuesign. Instead,

Ix 1 +

i l l = 6

l = 6

o r 1 + l = - 6

x = 5 or x=-7 -7 eachlorx. we cansubstitute To makesurethatboth5 and aresolutions,

C h e c k ,l $ + l l = 1 6 l =6 l - 7+ t | = l - 6 1= 6 Solveeachequation,andcheckyoursolutions.

I x - tol = Z

l x * 5 1= 7

Check:

Check:

l-3xl = Ll

Check:

[3x+41 =lO

= frzx-616

C h e c k:

24

Proiecl,Inc @19$ by KeyCurriculum withoulpormission. Do notduplicato

Solveeachequation.

lxl*f=2O-3 l x l= 1 7

lxl-6=-Z

x=17or-|'7

l x | +I = l l

Z l x l- S = 3

| + x -5 | = 7

4 l x l - s= 7

l x + 2 1 - l= 7

f x * 5 1 + 4= l Z

I z ^- 3| +I = l l

-3lxl * 5 = -r+

@1990by Key Cufticllum proioct, Inc. Do not duplicalo without D€rmission.

25

SolveandcheckeachinequalitY. so picka fewsamples' Youcan'tcheckallthe solutions,

,--,.n^,^,-.

( wtoti inridc nus!bc J

.oo(-g[Xj:)

l x l + 7


l x l *4


^ A - 2< - 3 o r > 3 X 2 l x l< x > 5 o rx < - l x < 9 a n xd > - 9 Check: Check: 6 , l 6 l+ 2 = 6 + 2 = 8 < 1 : l q - z l= - /> 3 -rl: l - 4+12 = 4 + 2 = 6 l 3 lxl s 4 l x l > Check:

Check:

lx-ll


Check:

Itzx| > 2+

lil
2

7x-1 >4-3x

3lrl-7=11

3 ( r -2 ) < 4 x + 1 6

B. Whichof theserelations arefunctions? "brother" The relation:B(x) is a brotherol x. "mother" The relation:M(x) is a motherof r. "less The than"reration:z(r) is a numberressthanr. "2 The lessthan"relation:W(x)= x - 2. The "tripling"relation:T(x) = 936. 9'

Makea tablefor eachfunctionbelow. Choosefive numbersto substitute forr in eachtable. Showthe valuesyougetwhenyousubstitute thesenumbers. f(x)=2x-1

@1990by Koy Cuflicllum proiecl, Inc. Oo not duplicats without permissior

g ( x=) l x - 2 1

h(x) = vz

35

PracticeTest numberas a fraction. Writeeachrational

3+= -2+=

-t t -

O . 1=

4.Ol=

Q=

-2.5, from-3to 3. Graphthenumbers 0.3and2 |' numbers lineshowing Drawa number

Labeleachnumberlineandgraphthe setof numbers' -2 fntegersbetween and4'. Rationalnumbersgreater thanor equalto 5: Rationalnumbersnot equalto 1: value. Findeachabsolute

le.a1=

l-sl=

l o| =

l - z * 6l =

Solve.

36

@199 by KeYCwriculumProiocl,Inc. withoutp€lmissron. Do notduplicato

Solve.

4x-7(x+6

2 0< 5 x - 2

+ -3

7 x - 3 ^ + l )x * 5 x - 6

t*a