Question

How do you solve `2/(x-1) - 2/3 =4/(x+1)`?

Answer 1

In an equation with fractions, we can get rid of the denominators by multiplying each term by the LCD (LCM of denominators).
IN this case the LCM is `color(red)(3(x-1)(x+1))`

`(color(red)(3(cancel(x-1))(x+1))xx2)/cancel(x-1) - (color(red)(cancel3(x-1)(x+1))xx2)/cancel3 =(color(red)(3(x-1)cancel(x+1))xx4)/(cancel(x+1)`

After cancelling the denominators, this leads to a simpler equation.

`6(x+1) -2color(blue)((x-1)(x+1))=12(x-1)" "color(blue)(DOTS)`

`6x+6 -2(x^2-1) = 12x-12`

`6x+6 -2x^2+2=12x-12" make =0"`

`0 = 2x^2+6x-20" "div 2`

`x^2+3x-10= 0" factorise"`

`(x+5)(x-2)= 0`

`x = -5 or x = 2`