Question

Solve the following inequality:
${\log }_{1/4}(2-x)>{\log }_{1/4}\left(\frac{2}{x+1}\right)$

Answer 1

${\log }_{1/4}(2-x)>{\log }_{1/4}(\frac{2}{x+1})$

$(2-x)<\frac{2}{x+1}⟹\frac{(2-x)(x+1)-2}{x+1}<0$

$⟹\frac{-x^2+x}{x+1}<0⟹\frac{\left(x\right)(x-1)}{x+1}>0⟹x(x-1)(x+1)>0⟹x\in (-1,0)\cup (1,\infty )$