Question

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

Answer 1

`x=-4`

Explanation:

Note that `x^2-1 = (x-1)(x+1)`

Multiply the given equation through by `x^2-1 = (x-1)(x+1)` to get:

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

Multiply out to get:

`2x-2+x^2+x = 2`

Simplify to get:

`x^2+3x-2 = 2`

Subtract `2` from both sides and transpose to get:

`0 = x^2+3x-4 = (x+4)(x-1)`

From this we find `x=1` or `x=-4`

Note that `x=1` is a solution of the derived equation, but is not a solution of the original equation since it causes two of the denominators to be `0`.

So the only valid solution is `x=-4`