Question

Solve for x : `(x+1)/(x-1)+(x-1)/(x+2)=4-(2x+3)/(x-2);x!=1,-2,2`

Answer 1

`(x+1)/(x-1)+(x-2)/(x+2)=4-(2x+3)/(x-2)`

`=>((x+1)(x+2)+(x-1)(x-2))/((x-1)(x+2))=(4(x-2)-(2x+3))/(x-2)`

`=>((x^2+2x+x+2)+(x^2-2x-x+2))/(x^2+2x-x-2)=(4x-8-2x-3)/(x-2)`

 `=>(x^2+3x+2+x^2-3x+2)/(x^2+x-2)=(2x-11)/(x-2)`

 `=>(2x^2+4)/(x^2+x-2)=(2x-11)/(x-2)`

 ⇒(2x²+4)(x−2)=(2x−11)(x²+x−2)

⇒2x³−4x²+4x−8=2x³+2x²−4x−11x²−11x+22

⇒2x³−4x²+4x−8=2x³−9x²−15x+22

⇒2x³−2x³−4x²+9x²+4x+15x−8−22=0

⇒5x²+19x−30=0

⇒5x²+25x−6x−30=0

⇒5x(x+5)−6(x+5)=0

⇒(5x−6)(x+5)=0

⇒5x−6=0, x+5=0

⇒x=`6/5`, x=−5