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

Aug 24, 2017

There are no valid solutions to this equation

#### Explanation:

Note that $\frac{x - 1}{x - 2} - \frac{x + 1}{x + 2} = \frac{4}{{x}^{2} - 4}$ is only defined if $x \ne \pm 2$

If we attempt to solve this equation by converting all terms to the common denominator of $\left(x - 2\right) \left(x + 2\right) = {x}^{2} - 4$
we get
$\frac{\left(x - 1\right) \left(x + 2\right)}{{x}^{2} - 4} - \frac{\left(x + 1\right) \left(x - 2\right)}{{x}^{2} - 4} = \frac{4}{{x}^{2} - 4}$

$\rightarrow \left(x - 1\right) \left(x + 2\right) - \left(x + 1\right) \left(x - 2\right) = 4$

$\rightarrow \left({x}^{2} + x - 2\right) - \left({x}^{2} - x - 2\right) = 4$

$\rightarrow \cancel{{x}^{2}} + x \cancel{- 2} \cancel{- {x}^{2}} + x \cancel{+ 2} = 4$

$\rightarrow 2 x = 4$

$\rightarrow x = 2$

BUT the original equation is not defined if $x = 2$

Therefore there is no valid solution.