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
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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
I
B
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.
loa=
1 3
r +
oo
7
1 6+ 3
3
I
lg
3?=
6 l J 2 -
+8=
= 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-,
O.1=
0.01=
0.1=
9
0 t t?=
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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
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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 =
-253
.75
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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
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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.
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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?
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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
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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
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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
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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
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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 <5
means
than5."
"x +'4 is greater
x+-4>5
means
than5."
Sometimes
we combine
twosymbols
to makea newsymbol.Thesymbol< means
"islessthanor equal
> means"isgreater
to." Andthesymbot
thanor equalto."
sentences
usingthesecombined
symbols
arealsoinequalities.
r+-4<5
means
x+-4>5
means
"tr+ -4 rs /essthan
or equallo 5."
"x,+ -4 is greater
thanor equalto 5."
To get usedto using<, >, < and>, readthe followingstatements
carefully.
Eachof theseis true:
l 0> 6
1>-2
5
5 s5
B >-B
1 2 = 1 2 -4 >-7
-7 <-+
1 2> 1 2
Eachof theseis false:
3>r+
-6<-6
-q>?
l < o
0
3 >
|>2
2>2
Markeachstatementbelowtrue (T)or false(F).
I B> 2 T
lg >
-lt+s
I
-lr+s -15
7<
t g> t g
- 3s - 3
-l+ s -lr+
7s7
- 7s 7
o >-g
l0 s24
3 <3
0 >-g
1 3> - 2
3
-8<
-Ll > -l
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n
[flfil
roadsignfor "PassingPermitted."
is the international
\./
"NoPassing."
IACD is the signfor
\.y
Canyouligureoutwhateachsignbelowmeans?
S
A
@
@
isthesisnror
isthesisnror
isthesisnror
ror
isthesign
roadsignsto mean"No."We usethesameideato makeup
A slashis usedon international
"is
newsymbolsin algebra.Inthesesymbolsthe slashmeans not."
"7 is not equalto 4."
means
7 + 4
o>
means
"0 is notgreaterthan5."
2<
means
"2 is not lessthan2."
l0 +-lZ
means
"10is notgreaterthanor equalto 12."
3 S 0
means
"3 is not lessthanor equalb 4."
Youwritethe meaningof eachsentencebelow.
B{ +
0+3
5 + 6
- 1 0# t 0
x+Ll +g
-2x
4 lO
means
means
means
means
means
means
5 +s
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A numberis a solutionof an inequality
if it makesthe inequality
truewhenyoutry it in
placeof r.
Thesenumbers
aresolutions
o fr + - 4 < 5 :
ofr+-4<5:
7
-1
because
0
because 0+-4<5
because
Thesenumbers
arenotsolutions
7 + - 4< S
-1 + -4< 5
10 because
1 0+ - 4 { 5
9
because 9+-4{5
25 because 25 + -415
Tryto findat leastfiveintegerswhicharesolutionsfor eachequationor inequality.lf there
aren'tfiveintegersolutions,
listas manyas you canfind.
x>3
x+8=l0
x 4 0
x+B<10
x + l
x+8>lO
x <
x +8 > l0
X <
x+8< lO
x + x
x+8+10
x : x
x+8>10
x >
x +B { l0
x <
x+8#lO
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AbsoluteValue
or
Whenwe multiplyor divideintegerswe get lhe amounfof the answerby multiplying
dividingandthe srgnof the answerby followingthe rulesfor signs.The amountof a
numberis oftencalleditsabsolutevalue. We putthesymbol| | arounda numberwhen
we wantto talkaboutits absolutevalue.
"Theabsolutevalueof 6 is 6."
= 6
means
l6l
l - 6 1= 6
l0l = 0
means
means
"Theabsolutevalueof -6 is 6."
"Theabsolutevalueof 0 is 0."
Findingthe absolutevalueof a numberis easy. Justget ridof itssign. Youfindeach
absolutevaluebelow.
-al=4
7 l =
- l ol =
-l9 =
|
l9| =
Itl=
| - tr l =
t+(-5
)I=
0 . l gl =
l o o 2 l7=
(-6)(-t{)
I=
3 - 1 1 l=- 6l =
Usetrialanderrorto findas manysolutionsas youcan for eachequationbelow.
xl=5
r + 2 1= 4
x l =I
xl=O
3 xI = 2 7
x - 5 1 2=
x-Bl=O
lxl=-5
Foreachinequality
whicharesolutions.
below,tryto findat leastfiveintegers
7, l0 ond12 holl,cobsolrrta
yclccs grcatcr thaa 6,
-7,-fO
ond -lZ.
o brrt lo do
o
o
lxl{z
x
lxl$lo
>o
l x l) x
l x ls 2
+3
14
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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 <|
{
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
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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
-5
-4
-3
-2
x +L+.4
-5
-4
-3
-2
x
-5
-4
-3
-2
x|z
-5
-4
-3
-2
lxl)3
-5
-4
-3
-2
-1
-1
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4
5
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Thegraphof an inequality
dependson whatkindsof numberswe ailowas solutions.
Theset of numberswe allowas solutions
is calledthe reptacement
set.
In the firstproblembelowwe haveshownwhatthe graphof r < 4 lookslikewhenonly
integersare allowedas solutionsandwhatit lookslikewhenall rationalnumbersare
allowedas solutions.Makea graphof eachinequality
for eachreptacement
set.
lntegers
x<+
RationalNumbers
+- l F , l5 l l .l 0l l O l 5>
-
5
0
5
x
-
5
0
5
-
5
0
5
x >2.5
-
5
0
5
-
5
0
5
x ' >+
-
5
0
5
-
5
0
5
x<35
-
5
0
5
-
5
0
5
<2
-
5
0
5
-
5
0
5
<25-
5
0
5
-
5
0
5
>r+
-
5
0
5
-
5
0
5
xl>o
-
5
0
5
-
5
0
5
x *-t
-
5
0
5
-
5
0
5
x > l
-
5
0
5
-
5
0
5
Fromnowon in thisbookwe'llatwaysuserationat
numbersas our replacement
set.
01990 by Key CurricrJlumproiecl, Inc.
Do nol duplbalo withoul pormission.
.
,
www.pdfgrip.com
t
Solvinglnequalities
thatnumber
by substituting
We cantellif a numberis a solutionof an equation
or inequality
forthevariableandseeingwhetherthe resultis trueor false.
l s S a s o l u t i o n2 ox l+ 5 < x - 1 5 ?
We cansubstitute
8 forxto findout:
2X+5
<
Z.g + S
<
t6+5
2
t
<
<
false
so 8 is nota solution.
solution.
Luckily
we do nothaveto substitute
everytimewewantto checka possible
justaswe usedit to solve
WecanusetheAddition
Principle
to helpussolvean inequality
withthesame
inequality
equations.
TheAddition
Principle
canhelpusfinda simpler
Lookatthisexample:
solution
set. Thisis calledsolvingtheinequality.
zi^n 5-\ <
x +5- (
- 1-55 '
x <
Onlynumberslessthan-20aresolutions,
so 8 couldnotbe a solution.Neithercouldany
otherpositive
number.
Solveeachinequality
belowusingthe AdditionPrinciple.
^ + ff>
3-'t
-16
x >
x-4
x+? s -6
ll+ x >
x - 5 <-3
l0<
1 2 >
-rl
5 > x +t(
l >x
x < lo
-+*
5x+ |
ff I ir biqqli
thonr, th'e?r
r
is fessthon f.
7x-1<
x-5S 2x+8
x+l
x
18
@199 by Key CurnculumPlojacl,Inc
Do nol duolicalewilhoutosrmission.
www.pdfgrip.com
Solveeachinequality
belowusingthe AdditionPrinciple.Drawa graphof eachsolutionset.
^-7>0
x >7
5 6 7
x-g<-?
6 < x + 1 2
S g f o
2 x + l ) x - 4
3(x-il ) 2x
x + 1 61 2 x + 2 0
-Lt*ll 3 26
xz+x+8>
( x * 2 ) ( x - 2 )< x z+ x
x e - & +( x t + x
(x * A(x-ll
5x-6 <
2 k - l ) < 3 ( x* 4 )
2x''*22)2x'+x-1
x"
(x-4Xx+5)(x'-19
-q(x
?( >-tf
proioct.Inc.
01990by KeyCurriqrlum
Do notduplbatewithoutp€rmjssion.
19
www.pdfgrip.com
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 <
>$)'+
>-z+
x
>
3
l
llx <
x
-5
<
-3x <
t^
019$ by l(ey Cwriclrlum Ptoiad, Inc
Do not duplicats without potmission.
20
www.pdfgrip.com
Solvingeachinequality
belowtakesmorethanonestep. Remember
to switchthe
inequality
signwhenever
you multiply
or dividebothsidesby a negativenumber.
{@
-3x+iWt
-3x >
-5^ < t1
nlL'\
-3
W%;Z
5x +l s -L+4
- Z x + 1 5<
x
ry<
f>
-4x
-t
f , " 1 +> q
2x-7x <
t
-3(x-5)
2l
o1990 by K€y Curiculumproiec,t,Inc
Do not duplicats wilhoul pormission.
www.pdfgrip.com
-t2
ro<
on page21.
Here'showSandyandTerrysolvedthe lastinequality
Terry
Sandy
q-fs
7 '*
t{xs
7-7
-z
- l x < 'z
-XS
?t s zn* t,
2 <
x >
,k
v
-2
T
x >
BothSandyandTerryendedup withthe samesotutionset. Whosemethoddo you like
better?Why?
below.
Bothmethodswork. Useeitheroneto solveeachinequality
2-x >
3 x * f 5>
25 s lo x
1 8 + x>
x+ 20 <
+-X
T
-5x*9 >
\ t.
/ t L
1 2- 7 x <
O19O by l(.y CurtiorlumPtol€cl,hc.
Oo not dupl'Eatowirhoulp.rflission'
22
www.pdfgrip.com
Solveeachinequality
andgraphthesolution
set.
3x-{x >
x + 1 2>
t-7 <
7x-2x x >
- t 6 +I x
g + r < +
>
@1990
by KsyCuricutumproiecl.Inc.
Do not dupli:atewilhoutgermissbn.
www.pdfgrip.com
