Question

Solve the following equation:
$\sqrt{\frac{x+1}{x-1}}-\sqrt{\frac{x-1}{x+1}}=\frac{3}{2}$

Answer 1

$\sqrt{\frac{x+1}{x-1}}-\sqrt{\frac{x-1}{x+1}}=\frac{3}{2}$

let $\sqrt{\frac{x+1}{x-1}}$ be t

$t-\frac{1}{t}=\frac{3}{2}2(t^2-1)=3t2t^2-3t-2=02t^2-4t+t-2=0(2t+1)(t-2)=0t=2,\frac{-1}{2}\sqrt{\frac{x+1}{x-1}}=2\frac{x+1}{x-1}=4x+1=4x-4x=\frac{5}{3}\sqrt{\frac{x+1}{x-1}}=\frac{-1}{2}\frac{x+1}{x-1}=\frac{1}{4}4x+4=x-1x=-\frac{5}{3}$