Answer
$x=-2$
Work Step by Step
Expressing the terms of the given equation, $
\dfrac{x-1}{x+1}+\dfrac{x+7}{x-1}=\dfrac{4}{x^2-1}
,$ in factored form results to
\begin{array}{l}\require{cancel}
\dfrac{x-1}{x+1}+\dfrac{x+7}{x-1}=\dfrac{4}{(x+1)(x-1)}
.\end{array}
Multiplying both sides by the $LCD=
(x+1)(x-1)
$, then the solution to the equation above is
\begin{array}{l}
(x-1)(x-1)+(x+1)(x+7)=1(4)
\\\\
x^2-2x+1+x^2+8x+7=4
\\\\
(x^2+x^2)+(-2x+8x)+(1+7-4)=0
\\\\
2x^2+6x+4=0
\\\\
x^2+3x+2=0
\\\\
(x+1)(x+2)=0
\\\\
x=\{-2,-1\}
.\end{array}
Upon checking, only $
x=-2
$ satisfies the original equation.
