arithmetic of algebraic fractions

8/13/2019 Arithmetic of Algebraic Fractions

1/16

Mrdtoètdb af

Mljenrmdb Frmbtdacs

=.7Dctragubtdac

Iust ms ace woale cu`ner gdvdgeg ny mcatoer ds bmlleg m cuèrdbml 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, ùltdpldegmcg gdvdgeg.

Xrerequdsdtes

Nefare stmrtdcj tods Rebtdac yau soaulg . . .

ne fm`dldmr wdto toe mrdtoètdb af cuèrdbmlfrmbtdacs

Lemrcdcj Autbaès

Ac ba`pletdac yau soaulg ne mnle ta . . .

mgg, suntrmbt, ùltdply mcg gdvdge mljenrmdbfrmbtdacs

6> OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm
http://www.ebookcraze.blogspot.com/


2/16

=. Bmcbelldcj ba`àc fmbtars

Bacsdger toe frmbtdac=8

25. Wa sd`pldfy dt we bmc fmbtardse toe cuèrmtar mcg toe geca`dcmtar mcg toec

bmcbel mcy ba`àc fmbtars. Ba`àc fmbtars mre toase fmbtars wodbo abbur dc nato toe cuèrmtarmcg toe geca`dcmtar. Wous

=8

2505 >

: 5 0

>

:

Cate tomt toe ba`àc fmbtar 5 oms neec bmcbelleg. Dt ds d`partmct ta reè`ner tomt acly ba`àc

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 smè 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`àc ta nato cuèrmtar mcg geca dcmtar. Wodsba`àc fmbtar bmc ne bmcbelleg ta jdve

y x

> x0

y

>

(n) Cate tomt x

xy bmc ne wrdttec

=x

xy. Woe ba`àc fmbtar afx bmc ne bmcbelleg ta jdve

= x

xy 0

=

y

(b) Dc toe expressdac xx+y

catdbe tomt mc x mppemrs dc nato cuèrmtar mcg geca`dcmtar.

Oawever x ds cat m ba`àc fmbtar. êbmll tomt fmbtars af mc expressdac mre ùltd-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


3/16


4/16


5/16

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`àc ta nato cuèrmtar mcg geca`dcmtar mcg bmcbel tods ba`àc 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 cuèrmtar mcg fmbtardse toe geca`dcmtar9

Taur salutdacx> + >x =5

>x>

5x 2

0

66 OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm


6/16

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.


6:


7/16

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 ùltdply tajetoer twa frmbtdacs (cuèrdbml ar mljenrmdb) we ùltdply toedr cuèrmtars tajetoermcg toec ùltdply toedr geca`dcmtars tajetoer. Womt ds

Key Xadct =4@ultdpldbmtdac af frmbtdacs

m

n

b

g0

mb

ng

Mcy fmbtars ba`àc ta nato cuèrmtar mcg geca`dcmtar bmc ne bmcbelleg. Wods bmcbellmtdac bmc neperfarèg nefare ar mfter toe ùltdpldbmtdac.

Wa gdvdge ace frmbtdac ny mcatoer (cuèrdbml ar mljenrmdb) we dcvert toe sebacg frmbtdac mcg toecùltdply.

6? OEL@ (>885)9Zarknaak =9 Nmsdb Mljenrm


8/16

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 =


64


9/16

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


10/16

Exm`ple 56

Rd`pldfy 7x+ >

x> + 7x+ 2

x+ 2

:x+ 5

Ralutdac

Fmbtardsdcj toe cuèrmtar 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`àc fmbtars nefare ùltdplydcj. Gact reàve

mcy nrmbkets uccebessmrdly atoerwdse ba`àc fmbtars wdll ne gdflbult ta spat.

WmskskRd`pldfy

=5

2x =

2

>x+ =

Taur salutdac

McswerWa gdvdge we dcvert toe sebacg frmbtdac mcg ùltdply9

=52x =

2>x+ =

0 =52x =

>x+ =2

0 (5)(2)(>x+ =)2(2x =)

05(>x+ =)2x =


:=


11/16

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


12/16

2. Mggdtdac mcg suntrmbtdac af mljenrmdb frmbtdacsWa mgg twa mljenrmdb frmbtdacs toe lawest ba`àc geca`dcmtar ùst ne faucg rst. Wods ds toesd`plest mljenrmdb expressdac tomt oms toe jdvec geca`dcmtars ms dts fmbtars. Mll frmbtdacs ùst newrdttec wdto tods lawest ba`àc geca`dcmtar. Woedr su` ds faucg ny mggdcj toe cuèrmtars mcg

gdvdgdcj toe result ny toe lawest ba`àc geca`dcmtar.Wa suntrmbt twa frmbtdacs toe prabess ds sd`dlmr. Woe frmbtdacs mre wrdttec wdto toe lawest ba`àcgeca`dcmtar. Woe gderecbe ds faucg ny suntrmbtdcj toe cuèrmtars mcg gdvdgdcj toe result ny toelawest ba`àc 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 =)> ùst ne m fmbtar af tods expressdac. Mlsa,nebmuse we bmc wrdte (x =)> 0 (x =)(x =) dt fallaws tomt x = ds m fmbtar taa.


:2


13/16

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`àc geca`dcmtar. Nato frmbtdacs ùst 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 smè 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 cuèrmtars mcg gdvdgdcj toe result ny toe lawest ba`àc 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`àc geca`dcmtar

Rtep >9 Express embo frmbtdac wdto tods geca`dcmtar

Rtep 29 Mgg toe cuèrmtars mcg gdvdge toe result ny toe lawest ba`àc 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 =)>



14/16

WmskskExpress

2

x+ :+

5

x+ >ms m sdcjle frmbtdac.

Fdrst cg toe lawest ba`àc geca`dcmtar9

Taur salutdac

Mcswer

(x+ :)(x+ >)

ê-wrdte nato frmbtdacs usdcj tods lawest ba`àc geca`dcmtar9

Taur salutdac

2x+ :

+ 5x+ >

0

Mcswer2(x+ >)

(x+ :)(x+ >)+

5(x+ :)

(x+ :)(x+ >)

Fdcmlly, mgg toe cuèrmtars 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`àc 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


:5


15/16

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.



16/16

? 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

::

arithmetic of algebraic fractions

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