Question

Solve for $x:\frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6},x\ne 1,-1$

Answer 1

$\frac{x+1}{x-1}-\frac{x-1}{x+1}=\frac{5}{6}$

$\Rightarrow \frac{{(x+1)}^2-{(x-1)}^2}{x^2-1}=\frac{5}{6}$

$\Rightarrow \frac{(x+1+x-1)(x+1-x+1)}{x^2-1}=\frac{5}{6}$

$\Rightarrow \frac{4x}{x^2-1}=\frac{5}{6}$

$\Rightarrow 24x=5x^2-5$

$\Rightarrow 5x^2-24x-5=0$

$\Rightarrow 5x^2-25x+x-5=0$

$\Rightarrow 5x(x-5)+(x-5)=0$

$\Rightarrow (5x+1)(x-5)=0$

$\therefore x=\frac{-1}{5},5$