Arithmetic of Algebraic Fractions
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8/13/2019 Arithmetic of Algebraic Fractions
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Mrdto`etdb af
Mljenrmdb Frmbtdacs
=.7Dctragubtdac
Iust ms ace woale cu`ner gdvdgeg ny mcatoer ds bmlleg m cu`erdbml frmbtdac, sa ace mljenrmdb expressdacgdvdgeg ny mcatoer ds kcawc ms mc mljenrmdb frmbtdac. Exm`ples mre
x
y,
2x+ >y
x y , mcg
x> + 2x+ =
x 7
Dc tods Rebtdac we explmdc oaw mljenrmdb frmbtdacs bmc ne sd`pldeg, mggeg, suntrmbteg, `ultdpldegmcg gdvdgeg.
Xrerequdsdtes
Nefare stmrtdcj tods Rebtdac yau soaulg . . .
ne fm`dldmr wdto toe mrdto`etdb af cu`erdbmlfrmbtdacs
Lemrcdcj Autba`es
Ac ba`pletdac yau soaulg ne mnle ta . . .
mgg, suntrmbt, `ultdply mcg gdvdge mljenrmdbfrmbtdacs
6> OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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=. Bmcbelldcj ba``ac fmbtars
Bacsdger toe frmbtdac=8
25. Wa sd`pldfy dt we bmc fmbtardse toe cu`ermtar mcg toe geca`dcmtar mcg toec
bmcbel mcy ba``ac fmbtars. Ba``ac fmbtars mre toase fmbtars wodbo abbur dc nato toe cu`ermtarmcg toe geca`dcmtar. Wous
=8
2505 >
: 5 0
>
:
Cate tomt toe ba``ac fmbtar 5 oms neec bmcbelleg. Dt ds d`partmct ta re`e`ner tomt acly ba``ac
fmbtarsbmc ne bmcbelleg. Woe frmbtdacs=8
25mcg
>
:omve dgectdbml vmlues - toey mre equdvmlect frmbtdacs
- nut >
: ds dc m sd`pler far` tomc
=8
25.
Ze mpply toe sm`e prabess woec sd`pldfydcj mljenrmdb frmbtdacs.
Exm`ple 74Rd`pldfy, df passdnle,
(m) yx
>x, (n)
x
xy, (b)
x
x+y
Ralutdac
(m) Dc toe expressdac
yx
>x , x ds m fmbtar ba``ac ta nato cu`ermtar mcg geca dcmtar. Wodsba``ac fmbtar bmc ne bmcbelleg ta jdve
y x
> x0
y
>
(n) Cate tomt x
xy bmc ne wrdttec
=x
xy. Woe ba``ac fmbtar afx bmc ne bmcbelleg ta jdve
= x
xy 0
=
y
(b) Dc toe expressdac xx+y
catdbe tomt mc x mppemrs dc nato cu`ermtar mcg geca`dcmtar.
Oawever x ds cat m ba``ac fmbtar. ^ebmll tomt fmbtars af mc expressdac mre `ultd-pldegtajetoer woerems dc toe geca`dcmtar x ds mggeg ta y. Wods expressdac bmccat nesd`pldeg.
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
62
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WmskskRoaw tomt
x =
x> 2x+ > ds equdvmlect ta
=
x >.
(m) Fdrst fmbtardse toe geca`dcmtar9
Taur salutdac
x> 2x+ > 0
Mcswer
(x =)(x >)
(n) Dgectdfy toe fmbtar ba``ac ta nato cu`ermtar mcg geca`dcmtar mcg bmcbel tods ba``ac fmbtar9
Taur salutdac
x =(x =)(x >)
0
Mcswer=
x >. Oecbe toe twa jdvec frmbtdacs mre equdvmlect.
Exm`ple 5>Rd`pldfy
6(7 ?x)(x >)
= >x
Ralutdac
Woe fmbtar7 ?x bmc ne fmbtardseg ta 7(= >x). Wous
6(7 ?x)(x >)
= >x 0
(6)(7)(= >x)(x >)
(= >x) 0 >7(x >)
WmskskRd`pldfy
x> + >x =5
>x> 5x 2
Fdrst fmbtardse toe cu`ermtar mcg fmbtardse toe geca`dcmtar9
Taur salutdacx> + >x =5
>x>
5x 2
0
66 OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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Mcswer(x+ 5)(x 2)
(>x+ =)(x 2)
Fdcmlly bmcbel mcy ba``ac fmbtars9
Taur salutdac(x+ 5)(x 2)
(>x+ =)(x 2)0
Mcswerx+ 5
>x+ =
Exerbdses
=. Rd`pldfy, df passdnle,
(m) =4
2?, (n)
=7
>?, (b)
25
78, (g)
:
==, (e)
=7
56
>. Rd`pldfy, df passdnle, (m) =7
>=, (n)
26
46, (b)
=2
5>, (g)
5>
=2
2. Rd`pldfy (m) 5z
z , (n)
>5z
5z , (b)
5
>5z>
, (g) 5z
>5z>
7. Rd`pldfy
(m) 7x
2x, (n)
=5x
x> , (b)
7s
s2, (g)
>=x7
:x2
5. Rd`pldfy, df passdnle,
(m) x+ =
>(x+ =), (n)
x+ =
>x+ >, (b)
>(x+ =)
x+ = , (g)
2x+ 2
x+ = , (e)
5x =5
5 , (f)
5x =5
x 2 .
6. Rd`pldfy, df passdnle,
(m) 5x+ =5
>5x+ 5, (n)
5x+ =5
>5x , (b)
5x+ =5
>5 , (g)
5x+ =5
>5x+ =
:. Rd`pldfy (m) x> + =8x+ 4
x> + ?x 4, (n)
x> 4
x> + 7x >=, (b)
>x> x =
>x> + 5x+ >,
(g) 2x> 7x+ =
x> x , (e)
5z> >8z
>z ?
?. Rd`pldfy (m) 6
2x+ 4, (n)
>x
7x> + >x, (b)
2x>
=5x2 + =8x>
4. Rd`pldfy (m) x> =
x> + 5x+ 7, (n)
x> + 5x+ 6
x> +x 6.
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
6:
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Mcswers
=. (m) =
>, (n)
=
>, (b)
:
?, (g)
:
==, (e)
=
7.
>. (m)
>
2 , (n)
2
? , (b)
=
7 , (g) 7
2. (m) 5, (n) 5, (b) =
5z>, (g)
=
5z.
7. (m) 7
2, (n)
=5
x, (b)
7
s>, (g) 2x
5. (m) =
>, (n)
=
>, (b) >, (g) 2, (e) x 2, (f) 5
6. (m) x+ 2
5x+ =
, (n) x+ 2
5x
, (b) x+ 2
5
, (g) 5(x+ 2)
>5x+ =
:. (m) x+ =
x =, (n)
x+ 2
x+ :, (b)
x =
x+ >, (g)
2x =
x , (e)
5z
>
?. (m) >
x+ 2, (n)
=
>x+ =, (b)
2
5(2x+ >).
4. (m) x =
x+ 7, (n)
x+ >
x >.
>. @ultdpldbmtdac mcg gdvdsdac af mljenrmdb frmbtdacsWa `ultdply tajetoer twa frmbtdacs (cu`erdbml ar mljenrmdb) we `ultdply toedr cu`ermtars tajetoermcg toec `ultdply toedr geca`dcmtars tajetoer. Womt ds
Key Xadct =4@ultdpldbmtdac af frmbtdacs
m
n
b
g0
mb
ng
Mcy fmbtars ba``ac ta nato cu`ermtar mcg geca`dcmtar bmc ne bmcbelleg. Wods bmcbellmtdac bmc neperfar`eg nefare ar mfter toe `ultdpldbmtdac.
Wa gdvdge ace frmbtdac ny mcatoer (cu`erdbml ar mljenrmdb) we dcvert toe sebacg frmbtdac mcg toec`ultdply.
6? OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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Key Xadct >8
Gdvdsdac af frmbtdacs
mn b
g0 m
ng
b 0 mg
nb n 0 8, b 0 8, g 0 8
Exm`ple 52
Rd`pldfy (m)
>m
b
7
b , (n)
>m
b
b
7 , (b)
>m
b
7
b
Ralutdac
(m) >m
b
7
b 0
?m
b>
(n) >m
b
b
70
>mb
7b 0
>m
7 0
m
>
(b) Gdvdsdac ds perfar`eg ny dcvertdcj toe sebacg frmbtdac mcg toec `ultdplydcj.
>m
b
7
b0
>m
b
b
70
m
> (fra` toe result dc (n))
Exm`ple 57
Rd`pldfy (m) =
5x 2x, (n)
=
x x.
Ralutdac
(m) Cate tomt 2x02x
= . Woec
=
5x 2x0
=
5x
2x
= 0
2x
5x0
2
5
(n) x bmc ne wrdttec ms x
=. Woec
=
x x0
=
x
x
= 0
x
x0 =
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
64
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WmskskRd`pldfy (m)
=
y x, (n)
y
x x.
Taur salutdac
Mcswer
(m) =
y
x0=
y
x
=
0x
y
(n) y
x x0
y
x
x
= 0
yx
x 0y
Exm`ple 55
Rd`pldfy
>x
y2x
>y
Ralutdac
Ze bmc wrdte toe frmbtdac ms >x
y
2x
>y.
Dcvertdcj toe sebacg frmbtdac mcg `ultdplydcj we cg
>x
y
>y
2x0
7xy
2xy0
7
2
:8 OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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8/13/2019 Arithmetic of Algebraic Fractions
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Exm`ple 56
Rd`pldfy 7x+ >
x> + 7x+ 2
x+ 2
:x+ 5
Ralutdac
Fmbtardsdcj toe cu`ermtar mcg geca`dcmtar we cg
7x+ >
x> + 7x+ 2
x+ 2
:x+ 50
>(>x+ =)
(x+ =)(x+ 2)
x+ 2
:x+ 50
>(>x+ =)(x+ 2)
(x+ =)(x+ 2)(:x+ 5)
0 >(>x+ =)
(x+ =)(:x+ 5)
Dt ds usumlly netter ta fmbtardse rst mcg bmcbel mcy ba``ac fmbtars nefare `ultdplydcj. Gact re`ave
mcy nrmbkets uccebessmrdly atoerwdse ba``ac fmbtars wdll ne gdflbult ta spat.
WmskskRd`pldfy
=5
2x =
2
>x+ =
Taur salutdac
McswerWa gdvdge we dcvert toe sebacg frmbtdac mcg `ultdply9
=52x =
2>x+ =
0 =52x =
>x+ =2
0 (5)(2)(>x+ =)2(2x =)
05(>x+ =)2x =
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
:=
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Exerbdses
=. Rd`pldfy (m) 5
4
2
>, (n)
=7
2
2
4, (b)
6
==
2
7, (g)
7
:
>?
2
>. Rd`pldfy (m)
5
4
2
> , (n)
=7
2
2
4 , (b)
6
==
2
7 , (g)
7
:
>?
2
2. Rd`pldfy
(m) > x+y
2 , (n)
=
2 >(x+y), (b)
>
2 (x+y)
7. Rd`pldfy
(m) 2 x+ 7
: , (n)
=
: 2(x+ 7), (b)
2
: (x+ 7), (g)
x
y
x+ =
y+ =, (e)
=
y
x> +x
y+ = ,
(f)
g>
7
]
g> , (j)
]
g>/7
5. Rd`pldfy 6/:
s+ 2
6. Rd`pldfy 2
x+ >
x
>x+ 7
:. Rd`pldfy 5
>x+ =
x
2x =
Mcswers
=. (m) 5
6, (n)
=7
4 , (b)
4
>>, (g)
=6
2
>. (m) =8
>:, (n) =7, (b)
?
==, (g)
2
74
2. (m) >(x+y)
2 , (n)
>(x+y)
2 , (b)
>(x+y)
2
7. (m) 2(x+ 7)
: , (n)
2(x+ 7)
: , (b)
2(x+ 7)
: , (g)
x(x+ =)
y(y+ =), (e)
x(x+ =)
y(y+ =), (f) ]/7,
(j) 7]g>
5. 6
:(s+ 2)
6. 6
x
:. 5(2x =)
x(>x+ =)
:> OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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2. Mggdtdac mcg suntrmbtdac af mljenrmdb frmbtdacsWa mgg twa mljenrmdb frmbtdacs toe lawest ba``ac geca`dcmtar `ust ne faucg rst. Wods ds toesd`plest mljenrmdb expressdac tomt oms toe jdvec geca`dcmtars ms dts fmbtars. Mll frmbtdacs `ust newrdttec wdto tods lawest ba``ac geca`dcmtar. Woedr su` ds faucg ny mggdcj toe cu`ermtars mcg
gdvdgdcj toe result ny toe lawest ba``ac geca`dcmtar.Wa suntrmbt twa frmbtdacs toe prabess ds sd`dlmr. Woe frmbtdacs mre wrdttec wdto toe lawest ba``acgeca`dcmtar. Woe gderecbe ds faucg ny suntrmbtdcj toe cu`ermtars mcg gdvdgdcj toe result ny toelawest ba``ac geca`dcmtar.
Exm`ple 5:Rtmte toe sd`plest expressdac wodbo oms x+ = mcg x+ 7 ms dts fmbtars.
Ralutdac
Woe sd`plest expressdac ds (x+ =)(x+ 7). Cate tomt nato x+ = mcg x+ 7 mre fmbtars.
Exm`ple 5?Rtmte toe sd`plest expressdac wodbo oms x = mcg (x =)> ms dts fmbtars.
Ralutdac
Woe sd`plest expressdac ds (x =)>. Blemrly (x =)> `ust ne m fmbtar af tods expressdac. Mlsa,nebmuse we bmc wrdte (x =)> 0 (x =)(x =) dt fallaws tomt x = ds m fmbtar taa.
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
:2
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Exm`ple 54
Express ms m sdcjle frmbtdac 2
x+ =+
>
x+ 7
Ralutdac
Woe sd`plest expressdac wodbo oms nato geca`dcmtars ms dts fmbtars ds (x+ =)(x+ 7). Wods ds toelawest ba``ac geca`dcmtar. Nato frmbtdacs `ust ne wrdttec usdcj tods geca`dcmtar. Cate tomt
2
x+ = ds equdvmlect ta
2(x+ 7)
(x+ =)(x+ 7) mcg
>
x+ 7 ds equdvmlect ta
>(x+ =)
(x+ =)(x+ 7). Wous wrdtdcj
nato frmbtdacs wdto toe sm`e geca`dcmtar we omve
2
x+ =+
>
x+ 70
2(x+ 7)
(x+ =)(x+ 7)+
>(x+ =)
(x+ =)(x+ 7)
Woe su` ds faucg ny mggdcj toe cu`ermtars mcg gdvdgdcj toe result ny toe lawest ba``ac geca`d-cmtar.
2(x+ 7)
(x+ =)(x+ 7)+
>(x+ =)
(x+ =)(x+ 7)0
2(x+ 7) + >(x+ =)
(x+ =)(x+ 7) 0
5x+ =7
(x+ =)(x+ 7)
Key Xadct >=
Mggdtdac af twa mljenrmdb frmbtdacs
Rtep =9 Fdcg toe lawest ba``ac geca`dcmtar
Rtep >9 Express embo frmbtdac wdto tods geca`dcmtar
Rtep 29 Mgg toe cu`ermtars mcg gdvdge toe result ny toe lawest ba``ac geca`dcmtar
Exm`ple 68
Express =
x =+
5
(x =)>ms m sdcjle frmbtdac.
Ralutdac
Woe sd`plest expressdac omvdcj nato geca`dcmtars ms dts fmbtars ds (x=)>. Ze wrdte nato frmbtdacswdto tods geca`dcmtar.
=
x =
+ 5
(x =)>
0 x =
(x =)>
+ 5
(x =)>
0x = + 5
(x =)>
0 x+ 7
(x =)>
:7 OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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WmskskExpress
2
x+ :+
5
x+ >ms m sdcjle frmbtdac.
Fdrst cg toe lawest ba``ac geca`dcmtar9
Taur salutdac
Mcswer
(x+ :)(x+ >)
^e-wrdte nato frmbtdacs usdcj tods lawest ba``ac geca`dcmtar9
Taur salutdac
2x+ :
+ 5x+ >
0
Mcswer2(x+ >)
(x+ :)(x+ >)+
5(x+ :)
(x+ :)(x+ >)
Fdcmlly, mgg toe cu`ermtars mcg sd`pldfy9
Taur salutdac
2x+ :
+ 5x+ >
0
Mcswer?x+ 7=
(x+ :)(x+ >)
Exm`ple 6=Express
5x
:
2x 7
> ms m sdcjle frmbtdac.
Ralutdac
Dc tods exm`ple nato geca`dcmtars mre sd`ply cu`ners. Woe lawest ba``ac geca`dcmtar ds =7, mcgnato frmbtdacs mre re-wrdttec wdto tods geca`dcmtar. Wous
5x
:
2x 7
> 0
=8x
=7
:(2x 7)
=7 0
=8x :(2x 7)
=7 0
>? ==x
=7
OEL@ (>885)9Rebtdac =.79 Mrdto etdb af Mljenrmdb Frmbtdacs
:5
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WmskskExpress
=
x+
=
yms m sdcjle frmbtdac.
Taur salutdac
McswerWoe sd`plest expressdac wodbo oms x mcg y ms dts fmbtars ds xy. Wods ds toe lawest ba``ac geca`-
dcmtar. Nato frmbtdacs mre wrdttec usdcj tods geca`dcmtar. Catdcj tomt
=
x 0
y
xy mcg tomt
=
y 0
x
xywe cg
=
x+
=
y 0
y
xy+
x
xy 0
y+x
xy
Ca bmcbellmtdac ds caw passdnle nebmuse cedtoer x cary ds m fmbtar af toe cu`ermtar.
Exerbdses
=. Rd`pldfy (m)
x
7+
x
: , (n)
>x
5 +
x
4 , (b)
>x
2
2x
7 , (g)
x
x+ =
>
x+ > , (e)
x+ =
x +
2
x+ > ,
(f) >x+ =
2
x
>, (j)
x+ 2
>x+ =
x
2, (o)
x
7
x
5
>. Fdcg
(m) =
x+ >+
>
x+ 2, (n)
>
x+ 2+
5
x+ =, (b)
>
>x+ =
2
2x+ >, (g)
x+ =
x+ 2+
x+ 7
x+ >,
(e) x =
x 2+
x =
(x 2)>.
2. Fdcg 5>x+ 2
+ 7(>x+ 2)>
.
7. Fdcg =
:s+
==
>=
5. Express M
>x+ 2+
N
x+ =ms m sdcjle frmbtdac.
6 Express M
>x+ 5+
N
(x =)+
B
(x =)>ms m sdcjle frmbtdac.
: Express M
x+ =+
N
(x+ =)>ms m sdcjle frmbtdac.
:6 OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
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? Express Mx+N
x> +x+ =8+
B
x =ms m sdcjle frmbtdac.
4 ExpressMx+N+ B
x+ =ms m sdcjle frmbtdac.
=8 Roaw tomt x==
x2
=
x>
ds equml ta x=x>x2x> x2
.
== Fdcg (m) 2x
7
x
5+
x
2, (n)
2x
7
x5
+x
2
.
Mcswers
=. (m) ==x
>? , (n)
>2x
75 , (b)
x
=>, (g)
x> >
(x+ =)(x+ >), (e)
x> + 6x+ >
x(x+ >) ,
(f) x+ >
6 , (j)
4 + >x >x>
2(>x+ =) , (o)
x
>8
>. (m) 2x+ :
(x+ >)(x+ 2), (n)
:x+ =:
(x+ 2)(x+ =), (b)
=
(>x+ =)(2x+ >),
(g) >x> + =8x+ =7
(x+ 2)(x+ >), (e)
x> 2x+ >
(x 2)>
2. =8x+ =4
(>x+ 2)>
7. 2s+ ==
>=
5. M(x+ =) +N(>x+ 2)
(>x+ 2)(x+ =)
6. M(x =)> +N(x =)(>x+ 5) +B(>x+ 5)
(>x+ 5)(x =)>
:. M(x+ =) +N
(x+ =)>
?. (Mx+N)(x =) +B(x> +x+ =8)
(x =)(x> +x+ =8)
4. (Mx+N)(x+ =) +B
x+ =
==. (m) 52x
68 , (n)
=2x
68
OEL@ (>885)9R td = 7 M dto td f Ml n d F td
::
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