Question

The range of values of $x$ satisfying the following inequation, $\frac{1}{2-\left|x\right|}\ge 1$ is

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

Answer 1

The correct option is D $x\in (-2,-1]\cup [1,2)$

Given,

For $\frac{1}{2-|x|}$ to be real

$x\ne 2or-\mathrm{2........}\left(1\right)$

Now in the equation,

$\frac{1}{2-|x|}\ge 1$

or, $2-|x|\le 1$

or, $|x|\ge 1$

$\Rightarrow x\le -1$ or $x\ge 1$.

$\Rightarrow x\in (-2,-1]$ or $x\in [1,2)(Consideringeqn.1)$

So $x\in (-2,-1]\cup [1,2)$.