How do you solve (x-1)/(x-2) - (x+1)/(x+2) = 4/(x^2-4)x1x2x+1x+2=4x24?

1 Answer
Aug 24, 2017

There are no valid solutions to this equation

Explanation:

Note that (x-1)/(x-2)-(x+1)/(x+2)=4/(x^2-4)x1x2x+1x+2=4x24 is only defined if x!=+-2x±2

If we attempt to solve this equation by converting all terms to the common denominator of (x-2)(x+2)=x^2-4(x2)(x+2)=x24
we get
((x-1)(x+2))/(x^2-4)-((x+1)(x-2))/(x^2-4)=4/(x^2-4)(x1)(x+2)x24(x+1)(x2)x24=4x24

rarr (x-1)(x+2)-(x+1)(x-2)=4(x1)(x+2)(x+1)(x2)=4

rarr (x^2+x-2)-(x^2-x-2)=4(x2+x2)(x2x2)=4

rarr cancel(x^2)+xcancel(-2)cancel(-x^2)+xcancel(+2)=4

rarr 2x=4

rarr x=2

BUT the original equation is not defined if x=2

Therefore there is no valid solution.