3/(2x-4)-2/(x^2-x-2)=9/(x+1)32x−4−2x2−x−2=9x+1
Factorising
2x-4=2(x-2)2x−4=2(x−2)
x^2-x-2=x^2-2x+x-2x2−x−2=x2−2x+x−2
=x(x-2)+(x-2)=x(x−2)+(x−2)
=(x+1)(x-2)=(x+1)(x−2)
3/(2x-4)-2/(x^2-x-2)=3/(2(x-2))-2/((x+1)(x-2))32x−4−2x2−x−2=32(x−2)−2(x+1)(x−2)
3/(2(x-2))-2/((x+1)(x-2))=9/(x+1)32(x−2)−2(x+1)(x−2)=9x+1
3/(2(x-2))-9/(x+1)=2/((x+1)(x-2))32(x−2)−9x+1=2(x+1)(x−2)
Multiplying throughout by (x+1)(x-2)(x+1)(x−2)
3/2(x+1)-9(x-2)=232(x+1)−9(x−2)=2
3/2x+3/2-9x+18=232x+32−9x+18=2
(3/2-9)x+(3/2+18-2)=0(32−9)x+(32+18−2)=0
-15/2x+35/2=0−152x+352=0
-15x+35=0−15x+35=0
15x=3515x=35
x=35/15x=3515
x=7/3x=73
Check:
lhs=
3/(2x-4)-2/(x^2-x-2)=3/(2xx7/3-4)-2/((7/3)^2-7/3-2)32x−4−2x2−x−2=32×73−4−2(73)2−73−2
=3/(14/3-4)-2/(49/9-7/3-2)=9/(14-12)-18/(49-21-18)=3143−4−2499−73−2=914−12−1849−21−18
=9/2-18/10=45/10-18/10=(45-18)/10=27/10=92−1810=4510−1810=45−1810=2710
rhs=9/(x+1)=9/(7/3+1)=27/(7+3)=27/10rhs=9x+1=973+1=277+3=2710
lhs=rhslhs=rhs