Question

The set of values of $x$ satisfying $2\le |x-3|<4$ is

• A. $(-1,1]\cup [5,7)$
• B. $-4\le x\le 2$
• C. $-1<x<7$ or $x\ge 5$
• D. $x<7$ or $x\ge 5$

Answer 1

Answer: (A)

We have, $2\le |x-3|<4$
Case I If $x<3$, then
$2\le |x-3|<4$
$\Rightarrow 2\le -(x-3)<4$
$\Rightarrow 2\le -x+3<4$
Subtracting 3 from both sides,
$-1\le -x<1$
Multiplying $(-1)$ on both sides,
$-1<x\le 1$
$\Rightarrow x\in (-1,1]$
Case II If $x>3$, then
$2\le |x-3|<4$
$\Rightarrow 2\le x-3<4$
Adding 3 on both sides,
$\Rightarrow 5\le x<7$
Hence, the solution set of given inequality is $x\in (-1,1]\cup [5,7)$