Question

The complete solution set of the inequation $\left|{\log }_{1/2}\left|x\right|\right|>1$ is

• A. $(-2,-\frac{1}{2})\cup (\frac{1}{2},1)\cup (1,2)$
• B. $(-\infty ,0)\cup (0,\infty )$
• C. $(-2,-1)\cup (-1,-\frac{1}{2})\cup (\frac{1}{2},1)\cup (1,2)$
• D. None of these

Answer 1

The correct option is D None of these

$\left|{\log }_{1/2}\left|x\right|\right|>1$

$\Rightarrow {\left({\log }_{1/2}\left|x\right|\right)}^2>(1{)}^2$

$\Rightarrow {\left({\log }_{1/2}\left|x\right|\right)}^2-(1{)}^2>0$

$\Rightarrow ({\log }_{1/2}\left|x\right|-1)({\log }_{1/2}\left|x\right|+1)>0$

$\Rightarrow (-{\log }_2\left|x\right|-1)(-{\log }_2\left|x\right|+1)>0$

$\Rightarrow ({\log }_2\left|x\right|-1)({\log }_2\left|x\right|+1)>0$

$\Rightarrow {\log }_2\left|x\right|ϵ(-\infty ,-1)\cup (1,\infty )$

$\Rightarrow \left|x\right|ϵ(2^{-\infty },2^{-1})\cup (2^1,2^{\infty })$

$\Rightarrow \left|x\right|ϵ(0,\frac{1}{2})\cup (2,\infty )$

$\Rightarrow xϵ(-\infty ,-2)\cup (\frac{-1}{2},0)\cup (0,\frac{1}{2})\cup (2,\infty )$