Question

Solve $\frac{\left|x-2\right|-1}{\left|x-2\right|-2}\le 0$ [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{As},\frac{\left|x-2\right|-1}{\left|x-2\right|-2}\le 0 \\ \mathrm{Case}\mathrm{I}:\mathrm{When}x\ge 2,\left|x-2\right|=x-2, \\ \frac{x-2-1}{x-2-2}\le 0 \\ \Rightarrow \frac{x-3}{x-4}\le 0 \\ \Rightarrow \left(x-3\le 0\mathrm{and}x-4>0\right)\mathrm{or}\left(x-3\ge 0\mathrm{and}x-4<0\right) \\ \Rightarrow \left(x\le 3\mathrm{and}x>4\right)\mathrm{or}\left(x\ge 3\mathrm{and}x<4\right) \\ \Rightarrow \varphi \mathrm{or}\left(3\le x<4\right) \\ \Rightarrow 3\le x<4 \\ \mathrm{So},x\in [3,4) \\ \mathrm{Case}\mathrm{II}:\mathrm{When}x\le 2,\left|x-2\right|=2-x, \\ \frac{2-x-1}{2-x-2}\le 0 \\ \Rightarrow \frac{1-x}{-x}\le 0 \\ \Rightarrow \frac{x-1}{x}\le 0 \\ \Rightarrow \left(x-1\le 0\mathrm{and}x>0\right)\mathrm{or}\left(x-1\ge 0\mathrm{and}x<0\right) \\ \Rightarrow \left(x\le 1\mathrm{and}x>0\right)\mathrm{or}\left(x\ge 1\mathrm{and}x<0\right) \\ \Rightarrow \left(0<x\le 1\right)\mathrm{or}\varphi \\ \Rightarrow 0<x\le 1 \\ \mathrm{So},x\in (0,1] \\ \therefore \mathrm{From}\mathrm{both}\mathrm{the}\mathrm{cases},\mathrm{we}\mathrm{get} \\ x\in (0,1]\cup [3,4) \end{gathered}$$