Question

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

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

Answer 1

The correct option is A

$(-1,1]\cup [5,7)$

Explanation for the correct option

Given, $2\le \left|x-3\right|<4$

A function which always gives a non negative real value irrespective of the input values is called a modulus function. It is given as,

$f\left(x\right)=\left|x\right|=\left\{\begin{array}{ll}x;& x\ge 0\\ -x;& x<0\end{array}\right.$

Case 1:

$\begin{array}{ccccc}2& \le & x-3& <& 4\\ 2+3& \le & x-3+3& <& 4+3\\ 5& \le & x& <& 7\end{array}$

$\Rightarrow x\in [5,7)$

Case 2:

$\begin{array}{ccccc}2& \le & -\left(x-3\right)& <& 4\\ -2& \ge & x-3& >& -4\\ -2+3& \ge & x-3+3& >& -4+3\\ 1& \ge & x& >& -1\end{array}$

$\Rightarrow x\in (-1,1]$

Thus, $x\in (1,-1]\cup [5,7)$.

Hence, the correct option is option(A) .