Question

Solve $1\le \left|x-2\right|\le 3$ [NCERT EXEMPLAR]

Answer 1

$$\begin{gathered} \mathrm{As},1\le \left|x-2\right|\le 3 \\ \Rightarrow \left|x-2\right|\ge 1\mathrm{and}\left|x-2\right|\le 3 \\ \Rightarrow \left(\left(x-2\right)\le -1\mathrm{or}\left(x-2\right)\ge 1\right)\mathrm{and}\left(-3\le \left(x-2\right)\le 3\right)\left(\mathrm{As},\left|x\right|\ge a\Rightarrow x\le -aorx\ge a;\mathrm{and}\left|x\right|\le a\Rightarrow -a\le x\le a\right) \\ \Rightarrow \left(x\le 1\mathrm{or}x\ge 3\right)\mathrm{and}\left(-3+2\le x\le 3+2\right) \\ \Rightarrow \left(x\le 1\mathrm{or}x\ge 3\right)\mathrm{and}\left(-1\le x\le 5\right) \\ \Rightarrow x\in (-\infty ,1]\cup [3,\infty )\mathrm{and}x\in \left[-1,5\right] \\ \therefore x\in \left[-1,1\right]\cup \left[3,5\right] \end{gathered}$$