Question

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

Answer 1

${\log }_{1/4}\left(\frac{35-x^2}{x}\right)\ge \left(\frac{-1}{2}\right)$

$\Rightarrow \frac{-1}{2}{\log }_2\left(\frac{35-x^2}{x}\right)\ge \left(\frac{-1}{2}\right)$

$\Rightarrow {\log }_2\left(\frac{35-x^2}{x}\right)\le 1$

$\Rightarrow \left(\frac{35-x^2}{x}\right)\le 2$

$\Rightarrow 35-x^2\le 2x$

$\Rightarrow x^2+2x-35\ge 0$

$\Rightarrow (x+7)(x-5)\ge 0$

$\Rightarrow xϵ(-\infty ,-7)\bigcup (5,\infty )$

And $\left(\frac{35-x^2}{x}\right)>0$

$\Rightarrow x(35-x^2)>0$

$\Rightarrow x(x^2-35)<0$

$\Rightarrow xϵ(-\infty ,-\sqrt{35})\bigcup (0,\sqrt{35})$

Taking intersection we get,

$xϵ(-\infty ,-7)\bigcup (5,\sqrt{35})$