Question

Solve the inequality $|1-\frac{\left|x\right|}{1+\left|x\right|}|\ge \frac{1}{2}$.

• A. $x\in [-1,1]$
• B. $x\in [0,1]$
• C. $x\in [-1,0]$
• D. $x\in [-2,1]$

Answer 1

The correct option is A $x\in [-1,1]$

$|1-\frac{\left|x\right|}{1+\left|x\right|}|\ge \frac{1}{2}$
$\Rightarrow \left|\frac{1}{1+\left|x\right|}\right|\ge \frac{1}{2}$
$\Rightarrow 1+\left|x\right|\le 2$
$\Rightarrow \left|x\right|\le 1$
Therefore, $x\in [-1,1]$
Ans: A