Question

Let $A=\left\{x:x\in R,|x|<2\right\}$, $B=\left\{x:x\in R,|x-2|\ge 2\right\}$ and $A\cup B=R-C$, then the set $C$ equals

• A. $\{x:-2<x\le 2\}$
• B. $\{x:-2\le x\le 4\}$
• C. $\{x:2\le x<4\}$
• D. None of these

Answer 1

Answer: (C)

Given : $A=\left\{x:x\in R,|x|<2\right\}$
and $B=\left\{x:x\in R,|x-2|\ge 2\right\}$
$\Rightarrow A=-2<x<2$
and $B=\left\{x:x\in R,x-2\le -2\text{or}x-2\ge 2\right\}$
$=\left\{x:x\in R,x\le 0\text{or}x\ge 4\right\}$
$\Rightarrow A=]-2,2[\&B=(-\infty ,0]\cup [4,\infty )$

Now $A\cup B=(-\infty ,2)\cup [4,\infty )=R-[2,4)$
$\therefore C=[2,4)$ i.e. $2\le x<4$