Question

Solve the following inequality:
${\log }_{1/4}\left(\frac{35-x^2}{x}\right)⩾-\frac{1}{2}$

Answer 1

$\frac{-\log \left(\frac{35-x^2}{x}\right)}{\log 4}\ge \frac{-1}{2}\left(\frac{35-x^2}{x}\right)\le 4^{1/2}35-x^2\le 2x{x}^2+2x-35\ge 0(x+7)(x-5)\ge 0x\in (-\infty ,-7]\bigcup [5,\infty )$

and

$\frac{35-x^2}{x}>0(35-x^2)x>0x\ne 0(x^2-35)x<0x\in (-\infty ,-\sqrt{35})\cup (0,\sqrt{35})$

Taking intersection, we get

$x\in (-\infty ,-7]\cup [5,\sqrt{35})$