Question

The set of all the values of ${}^{\prime }{x}^{\prime }$ satisfying the inequation $\left(\frac{|x|-1}{|x|-2}\right)\ge 0$ are

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

Answer 1

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

Given $\frac{|x|-1}{|x|-2}\ge 0$
Critical points are -2, -1,+1,+2
Using, the number line rule for $|x|$ we can say the given inequation is true for
$\Rightarrow |x|\le 1$ or $|x|>2$

$\Rightarrow -1\le x\le 1$ or $x<-2,x>2$
ie, $x\in [-1,1]\cup (-\infty ,-2)\cup (2,\infty )$