11

8
MELCs:
GENERALMATHEMATI

M11GM-lb-
1,
CS
QUARTER1–MODULE2
M11GM-l
b-2,
M11GM-
lb
-
3,M11GM-lb-
4,M11GM-lb-
5

LET’
SBERATI
ONAL!
PARTI .
A.Introduct i
on
Welcomet oanot herinter
est i
ngandchal lengingGener alMat hemat i
cs
Self-
Lear ni
ngModul e!Enjoyawonder f
ul week-long l earning!
Thismoduleisall aboutrationalfunct i
ons, equations, and i nequalit
ies.
Aftergoingt hroughthi
smodul e,youshoul dbeabl eto r epr esent real-
li
fe
sit
uationsusi ngrati
onalfunctions;disti
nguish r at
ionalf unct ion,rati
onalequat i
on,
andr ati
onal i
nequali
ty;solv
e r at
ional equat ions and i nequal i
ti
es; represent a
rat
ional functi
on t hroughi ts:(
a)t ableofv alues,( b)graph,and( c)equat i
on;and
fi
nd t he domai nandr angeofar ationalfunction.
B.Pr etest
Directions:Chooset helet teroft hecor rectanswer .Wr it
ey ouranswer sinyour
answersheet .
1.Int hesetofor deredpai rs(1,2),(2,3), (
3,4) ,(4,5),(
5,6),thedomai nD=_ _____.
A.{ 1,
2,3,4,5}B.{ 2,3,
4, 5,6} C.{ 6,5,
4,3, 2} D.{ 1,
2,3,4,5,
6}
2.Findt hedomai nandr angeoft hel inearf unct i
onf (
x)=2x+4.
A.D: {xϵR/ x≠- 4};R: {yϵ R/y≠4} C.D:{ xϵ R/ x≠0};R:{yϵR/ y≠0}
B.D: {x/xϵ R} ;R: {
y /yϵR} D.D:{ x/xϵN};R: {
y/ yϵN}
3.Whi choft hef ol l
owi ngi sar ational funct ion?
x2- 9 x+3 x-
6
A.f (x)= B.x=4x+ 5 C.x+ 2≥ D. =10
x+3 x-2 2x
4.A t ruckt hatdel iversessent i
alsi nr emot ear east r
av els75ki l
omet ers.
Whi choft hef ollowingexpr essest hev eloci tyvasaf unct ionoft r
av eltmeti
i n
hour s?
75 t 75 v
A.v ()=
t B.v (t)= C.t (v)= D.t(
v)=
t 75 v 75
5.I ft het r
ucki npr oblem number4wasdel ayedby5hour sduet ot he
checkpoi ntst hati tpassedt hrough,whatwi llbet het metasaf
i uncti
onof
velocityvi nkm/ hr ?
75 t 75 v
A.v ()= +5
t B.v( t)= +5 C.t(v)= +5 D.t (v)= +5
t 75 v 75
x2-4
6.Howwi l
lyoucl assifyy= ?
x+2
A.Rat ional Equat ion C.Rat i
onal Funct ion
B.Rat ional I
nequal i
ty D.Rat i
onal Expr ession
x 1 x
7.Whati st heLCDoft het ermsi nt heequat ion + = ?
3 4 2
A.3 B.6 C.8 D.12
8.Ifwhensol vingar ationalequat i
ony ouobt ainedanumbert hatmadean
expr essioni nt heequat ionundef i
ned, whatwi llyoudo?
A.Donotr ejectsi ncei twi llsatisfyt heequat i
oninal ongr un.
B.Cont i
nuet hesol utionev eni fitwi ll
gi veanundef i
nedanswer .
C.Acceptev eni fitisanunt ruev alue.
D.Itisnotar eal solution.Di scar di t.
x-1
9.Compl etet het abl
eusi ngt heequat i
onf (x)= .
x+1
x 1 2 3 4
f(x) 0 1/ 3 ½ ?

1
 3 5
A. B. C.3 D.5
5 3
x
10.Whi
choft
hef
oll
owi
ngi
sthegr
aphoff
(x)= usi
ngt
hev
aluesofx=-
2,-
x-
2
1,0,
1and2?

A. B. C. D.

C.Pr
esent
ati
on/
Discussi
on
MELC1(
M11GM-
lb-
1):Repr
esent
sreal
-l
if
esi
tuat
ionsusi
ngr
ati
onal f
unct
ions
Rat
ionalf
unctionsarequoti
entsofpolynomialsfuncti
onsthatcan be
p(x)
expressedasf x)=
( wher
ep(x)andq(x)arepolynomialfunct
ions.The domain
q(x)
off(x)i
sthesetofal l
number sexceptthex-v
aluesthatwill maket hedenominat
or
zero.

Il
lustr
ati
vePr oblems:
1.Becauseoft hepandemi cin ourcount r
y,oneoft he localbar angaysin
Pangasinanr ecei vedanaddi ti
onalbudgetofPhp200,000t opr ovi
demedi cal
checkupsf ort hechildreninthebarangay.Theamountistobeal l
ott
edequally
amongal lthechi l
dreni nthebarangay.Wr i
teanequat i
onr epresenti
ngthe
rel
ati
onshi poft heal l
ottedamountperchi l
d(y-
var
iabl
e)tothet otalnumberof
chil
dren(x-variable).
200, 000
Answer :y=
x
Fi
llupt het ablebelowwi ththeall
otmentamountcorrespondingt o the
gi
vennumberofchi ldren:
(
No.of
chil
dren) 10 20 50 100 200 300 500 1,
000
x
(Al
lotted
amount ) 20,000 10,
000 4,
000 2,
000 1,
000 666.
67 400 200
y
2.Anobjecttr
avel
sadi st
anceof20meter
s.Expr
essvel
oci
tyvasafunct
ionv
(t)
oft
ravelti
met,i
nseconds.
Sol
uti
on:Thefoll
owingtabl
eofval
uesshowsvforv
ari
ousval
uesoft.
t(seconds) 1 2 4 5 10
v(meterpersecond) 20 10 5 4 2
20
Thef
unct
ionv(
t)= canr
epr
esentvasaf
unct
ionoft
.
t
MELC 2 ( M11GM- l
b-2)
:Di st
ingui
shes r
ati
onalfunct
ion,r
ati
onalequati
on,and
rat
ionali
nequali
ty
Arati
onalexpr
essionisanexpressi
onthatcanbewritt
enasar at
iooftwo
pol
y nomi
als.

2
Exampl
es: , ,

Ar
ati
onalequat
ioni
sanequat
ioni
nvol
vi
ngr
ati
onal
expr
essi
ons.
Exampl
es: , ,
Ar
ati
onali
nequal
i
tyi
sani
nequal
i
tyi
nvol
vi
ngr
ati
onal
expr
essi
ons.
Exampl
es: , ,

Ar
ati
onalf
unct
ioni
saf
unct
ionoft
hef
orm wher
e and

ar
epol
ynomi
alf
unct
ions, .

Exampl
es: or , ,
MELC3(
M11GM-
lb-
3):Sol
vesr
ati
onalequat
ionsandi
nequal
it
ies

Thefoll
owingar
ethestepsinsolvi
ngar at
ionalequation:
1.El
iminat
edenominat
orsbymul ti
ply
ingeacht erm oftheequat
ionbythei
r
l
eastcommondenomi nator(
LCD) .
2.Checkthesol
uti
onsofthetransf
ormedequat ionswi t
htheori
ginal
equati
on.

El
i
mi nat
ingdenominat
orsmayi ntr
oduceextr
aneoussolut
ions,ther
efor
e,i
tis
i
mport
anttocheckthesol
uti
onsobt
ainedagai
nsttheor
igi
nal
equation.
I
ll
ust
rat
iveExampl
es:
1.Sol
vef
ort
hev
alueofx: .

Step1:
Mult
ipl
yi
ngbyt
he Since6=6i sat
rue
St
ep2:
In
LCDwhichi
s5, equation,t
hen
subst
it
uti
ngxwi
th10, x=10 isthe
solut
ion

of .
Di
vi
dingbothsi
desoft
he
equat
ionby3,
x=10

2.Sol
vef
ort
hev
alueofx: .

Step1:
Mult
ipl
yi
ngbyt
he SinceinSt
ep2,
St
ep2:
In
LCDwhichi
sx, a.2=2,and
a.subst
it
uti
ngxwi
th2, b.3=3arebot h
trueequat
ions,
thenx=2andx=3
arethesol
utions
Adding-5x+6tobot
hsi
des 5-3=2
oftheequati
on, 2=2 of .
b.subst
it
uti
ngxwi
th3,
Appl
yfact
ori
ngmet
hodt
o
3
 sol
vef
ort
hev
alueofx,

5-
2=3
x=2, x=3 3=3

x 4
3.Sol
vef
ort
hev
alueofx: =3- .
x+4 x+4
Step1: Multi
plyi
ngbyt heLCD Step2: SinceinStep2,
whichi s(x+4), x 4 x=-4makest he
In =3- ,
(x
x
+4) =(
x+4
x+4)[3- ( )
4
x+4
]
x+4
substi
tuti
x+4
ngxwi th -
origi
nal
undefi
equat
ned,
ion

x=3(x+4) -4 4wi l
lmaket he x=- 4isan
Dist
ributecarefull
ythensimpl i
fy. denomi natorzero.I
t extraneous
x=3x+12- 4 willmaket he soluti
on.
x=3x+8 functi
onundef ined.
Adding- 3xtobot hsidesoft he
equation,
-2x=8
Dividi
ngbot hsidesoft he
equationby- 2,
x= -4

Thef ollowi ngar et hestepsi nsol vingr ationalinequalit

ies:
1.Rewr it
et hei nequalityasasi ngl er ationalexpressi onononesi deoft he
i
nequal itysy mbol and0ont heot hersi de.
2.Det ermi neov erwhati nterv alst her ationalexpressiont akesonposi t
iveand
negativev alues.
a.Locat et he xv aluesf orwhi ch t her at
ionalexpr ession i szer o or
undef i
ned( factoringthenumer atoranddenomi natori susef ulstrategy).
b.Mar kt henumber sf oundi n( a)onanumberl ine.Useashadedci rcl
et o
i
ndicat et hatt hev al
uei si ncl udedi nt hesol uti
onset ,andahol lowci r
clet o
i
ndicat et hatt hev aluei sex cluded.Thesenumber spar t
iti
ont henumberl ine
i
ntoint erv als.
c.Sel ectat estpoi ntwi thi nt hei nteriorofeachi nt ervali
n( b).Thesi gnof
ther ationalexpr essionatt hist estpoi ntisalsot hesi gnoft herat i
onal
expressi onateachi nteriorpoi ntint heaf orementi
onedi nterval.
d.Summar izet heinterv alscont ainingt hesoluti
ons.

War ni
ng!Mult
ipl
yi
ngbothsidesofaninequal
i
tybyanumberr equi
resthatt
hesign
(posit
iveornegati
ve)ofthenumberi sknown.Sincethesi gnofav ar
iabl
eis
unknown,iti
snotvali
dtomulti
plybot
hsidesofani
nequali
tybyav ar
iabl
e.

I
ll
ust
rat
iveExampl
es:
3x
1.Sol
vet
hei
nequal
i
ty ≥2.
x+2

4
1)Rewr
( it
ethei
nequal
i
tyasa 3x 3x-
2x-
4
≥0
si
ngl
efr
acti
onononesi
de, ≥2 x+2
x+2
and0ontheot
hersi
de.
3x x-4
-2≥0 ≥0
x+2 x+2
3x-
2(x+2)
≥0
x+2

(2.
a)Ther ati
onal (2.
b)Mar
ktheseont henumberl
i
ne.Useashadedcir
cle
expressionwil
lbezero f
orx=4( asol
uti
on)andanunshaded(hol
l
ow)cir
clefor
forx=4andundef i
ned x=- 2(
notasolut
ion)
.
forx=–2.Thev al
uex=
4isincludedwhilex=–2
i
snot .

(
2.c)Chooseconv
eni
ent Const
ructat
abl
eofsi
gnsasshownbel
ow.
t
estpoi
ntsi
nthe
i
nter
val
sdeter
minedby Int
erval x<-2 -
2<x<4 x˃4
–2and4todeter
mine TestPoint x=-3 x=0 x=5
x-
4 x-4 - - +
t
hesignof i
nthese x+2 - + +
x+2
i
nter
val
s. x-4 + - +
x+2
(2.
d)Sincewearelookingfort
hei nt
erv
alswheretherat
ionalexpr
essioni
sposi
tiv
e
orzero,wedeter
minethesoluti
ont obetheset{
x /x<-2orx≥4}.I
ncanalsobe
wri
ttenusingt
heinter
valnotat
ion:(-
∞,-2)∪ [
4,∞).Pl
otthissetonthenumberli
ne.

3 2
2.
Sol
vet
hei
nequal
i
ty < .
x-
5 x
1)Rewr
( it
ethei
nequal
i
ty 3 2 3x-
2x+10
<0
asasingl
efr
acti
ononone < x(x-
5)
x-
5 x
si
de,
and0ont heot
her x+10
3 2 <0
si
de. - <0 x(x-5)
x-
5x

2(
3x- x5)
-
<0
x(
x-5)

(2.
a)Ther ati
onal (2.b)Plott hepoint
sonanumberl ine.Useholl
ow ci
rcl
es
expressionwi l
lbezero sincethesev al
uesar enotpartofthesolut
ion.
forx=- 10and
undefinedf or
x=0andx=5.
( c)Const
2. ructatabl
e I
nterval x<-
10 -10<x<0 0<x<5 x˃5
ofsignst odetermine TestPoi nt x=- 12 x=-6 x=1 x=6
thesignoft hefuncti
on x+10 - + + +
i
neachi nterv
al x - - + +
x- 5 5 - - - +
x+10 - + - +
x(
x-5)
det
ermi
nedby-
10,
0
and5.

(2.
d)Summar izethei
nterv
alssati
sfy
ingt
heinequal
i
ty.Thesoluti
onsetofthe
i
nequalit
yistheset{x / x<-
10or0<x<5}.I
ncanalsobewr i
ttenusi
ngtheint
erv
al
notat
ion:(-
∞, -10)∪(0,5)
.Pl
otthi
ssetonthenumberline.

MELC 4( M11GM-lb-
4):Representsar
ati
onalf
unct
iont
hroughi
ts:(
a)t
abl
eof
val
ues,
(b)graph,
and(c)equation
I
llustr
ati
v eExampl e:
12
Given:f(x)= .
x-4
(a)Const ructatableofval
uesusingnumber sf
rom -2to8.
(b)Plott hepoi ntsintheCartesi
anplaneanddet er
minewhethert
hepoi
nts
form asmoot hcur v
eorast rai
ghtl
ine.
Soluti
on:
(a) x -2 -1 0 1 2 3 4 5 6 7 8
f(x) - 2 - 2.
4 - 3 -4 -
6 -
12 und. 12 6 4 3
(b)Pl
otti
ngandconnect ingthepoi
nts,wegett hefol
lowinggraphwhichf
ormst
wo
diff
erentsmoot hcur ves.

I
llust
rati
vePr oblem:
Ahy potheticalfunct
ionrepresent
ingtheconcentrati
onofadr uginapati
ent’
s
5t
bloodstr
eam ov ertime(i
nhour s)isgi
venasc(t)= 2 .
t+1
(a)Constructat abl
eofv al
ues.
(b)PlotthepointsinaCar t
esianplaneandconnectthem.
(c)Whatcany ousayaboutt hefuncti
on?
Soluti
on.
(a)Sinceti sint i
me,wecanonl yusenon-negat
ivevaluesfori
t.Usi
ngthef
ir
stten
wholenumber s,weget
t 0 1 2 3 4 5 6 7 8 9
c(t
) 0 2. 5 2 1.
5 1. 18 0. 96 0. 81 0. 7 0. 62 0. 55
(b)Pl
otti
ngt hepointsandconnect i
ngthem:

6
(
c)Att=0,t
heconcent
rat
ioniszerosincethedr
ughasnotent
eredt
hebl
oodst
ream
y
et.I
tshoot
supatt=1butitst
artsdecreasi
ngaf
tert
hat
.

MELC5(
M11GM-
lb-
5):Fi
ndst
hedomai
nandr
angeofar
ati
onalf
unct
ions
Thedomai
nofar
ati
onal
funct
ion i
sal
lval
uesofxt lnotmakeq(
hatwi
l x)

equalt
ozero.Ther
angeofarat
ionalfunct
ionisthesetofallt
hepossibl
eresul
ti
ng
val
uesoft
hedependentv
ari
abl
eafterwehav esubst
it
utedthedomain.
FindingtheDomai nandRangeusi ngaGr aphofaRat
ionalFunct
ion
Il
lustrat
iveExampl es:
1.Findt hedomai nandr angeoft hegraphthatisshownbel
ow.
Tof indtheDomai n:
•Thedomai nofagr aphconsist sofallt
hev al
uesof
xasshowni nthehor i
zontalli
neort hex-axis.
•Obser vet hatwhenweext endourgr aphfrom lef
t
toright,itwi l
lNOTt ouchthey -axiswithequation
x=0.Thus, thiswil
lbeourr estricti
onforthedomai n.
•Ther efore,thedomai nofthef uncti
onisthesetof
allrealvaluesofxEXCEPTx=0.
•Innot ationitcanbewr it
tenas
Tof i
ndt heRange:
•Ther angeofagr aphconsi stsofal lthevaluesof
yasshowni nthev ert
icallineort hey-axi
s.
•Obser vet hatwhenweext endourgr aphfrom the
bottom t othetop, i
twillonlyincl udethoseabov e
y=0.Thus, t
hiswillbeourr estrictionfortherange.
•Theref ore,therangeoft hef unct ionisthesetof
allr
eal valuesofyt hatismor et hany =0orinot her
word,y >0.
•Innotat i
oni tcanbewr i
ttenas ˃0}
.

2.
Findt
hedomai
nandr
angeof whosegr
aphi
sshownbel
ow.
Tof i
ndtheDomai n:
•Observet hatourgraphforthi
srat
ional
functi
onext endsindef
ini
tel
yonbothsides
withnor estri
cti
on.
•Therefore,thedomainofthefuncti
onisthe
setofallrealvaluesofx.I
nnotati
onitcanbe
7
writ
tenas
Tof i
ndt heRange:
•Obser v ethegr aphf rom bot tom t otop.
•Wecanseet hatt hegr aphi sf r
om upt o
touchingeachr estr
ict edl ine.
•Ther efore,ther angeoft hef unct i
onisthesetofal lreal
v aluesofyt hatis
greatert hanorequal to−3( inMat hematicssy mbol thisi
s )butequal
orbelow5( orthisis i
nsy mbol s)
.
•Innot at i
onitcanbewr ittenas
PARTI I.ACTI VITIES
A.Answerandsol vei t!
Direct i
ons:Answerand/ orsol vethegi v
enpr obl
emsbel ow.Wr it
ey oursoluti
ons
andanswer siny ouranswersheet .
1.Anobj ectt r
av elsadi stanceof30met ers.Expressv elocityvasaf unctionv (t
)
oft raveltmet
i ,i
nseconds.
2.Thel ocalbar angayr ecei vedabudgetofP300, 000t opr ovidemedi cal
checkupsf orthechi ldren.Theamounti st obeal l
ott
edequal lyamongal lt he
chil
dreni nthebar angay .Wr i
teanequat ionrepr esentingthe r elati
onshi
p of t he
all
ottedamountperchi l
d( y-var i
able)tothet otal numberofchi l
dren(x-var
iable).
Foritem number s3- 4, refertoproblem number2andf i
llupthetable wi t
h
thecor rectanswer s.

No.ofchil
dren,x 10 20 50
Allott
edamount ,
y 3. 15,000 4.
5.Atr
uckthatdeli
ver
sessent i
alsi
nremoteareastrav
els65ki
l
omet
ers.
Expresst
hevelociyvasaf
t uncti
onoftravelt
imetinhours?
B.Determinei
t!
Di
recti
ons:Det
ermi
newhethert
hegiveni
sar
ati
onalf
unct
ion,
arat
ionalequat
ion,
ar
ationali
nequal
it
yornoneofthese.
4x+3 5
1.
fx)=
( 3.
6x- ≥0 5.
5 x+3
1 x 5 2 12
2. = + 4. ≥ -
2 3 6 x 35
C.AnswerandSolv eMeNow!
Direct
ions:
Answerand/ orsolv
ethegivenpr obl
emsbelow.
51 1
Fornumber s1-2,usetheequat i
on - = .
x3 x
1.Whati st
heLCDoft hegivenequation?
2.Solvethegiv
enequat ionandf i
ndthev al
ueofx.
3 2 1
3.Solvetheequat i
on - = .
x 3x 6
3x
Fornumbers4- 5,usetheinequal
it
y ≥1.
x+1
4.Rewr i
tet
hegi veninequali
tyasasinglefracti
onononesi
de,
and0on t
he
ot
herside.
5.Solvethegiv
eni nequali
ty.

8
 6
6.Given(x)= ,constr
uctat ableofv aluesusi
ngt heint
eger
sfrom -
5 to5.
x-2
6
7.Usingthet abl eofvaluesinnumber6, graphfx)=
( inaCart
esi
an Plane.
x-2
Fornumber s8- 10, ref
ertothepr oblem below.
Ahy potheticalfuncti
onr epresenti
ngt heconcentrati
onofadrugina
4t
pat
ient
’sbl
oodst ream ov erti
me( i
nhour s)isgi v
enasc( )= 2 .
t
t+1
8.Constructat abl
eofv aluesusingt heintegersfr
om 0t o8.
9.Plotthepoi ntsinaCar tesianplaneandconnectt hem.
10.Whatcany ousayaboutt hefunction?

D.FindMe!
Dir
ecti
ons:Fi
ndt
hedomai
nandr
angeoft
her
ati
onal
funct
ionswhose gr
aphs
ar
egi venbel
ow.

1.
Domain:
____
_____
___ 3.
Domain:
____
_____
____ 5.
Domain:
____
_____
___
2.
Range:
___
_____
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__ 4.
Range:
___
_____
_____
_ 6.
Range:
___
_____
_____

References:
GeneralMat hemat icsLear ner’sMat eri
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