Question

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

Answer 1

There are no valid solutions to this equation

Explanation:

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

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

`rarr (x-1)(x+2)-(x+1)(x-2)=4`

`rarr (x^2+x-2)-(x^2-x-2)=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.