Question

Write the set of values of x satisfying |x − 1| ≤ 3 and |x − 1| ≥ 1.

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \left|x-1\right|\le 3\mathrm{and}\left|x-1\right|\ge 1 \\ \mathrm{We}\mathrm{know}: \\ \left|x-a\right|\le r\Rightarrow a-r\le x\le a+r \\ A\mathrm{nd},\left|x-a\right|\ge r\Rightarrow x\le a-r\mathrm{or}x\ge a+r \\ \therefore 1-3\le x\le 1+3\mathrm{and}x\le 1-1\mathrm{or}x\ge 1+1 \\ \Rightarrow -2\le x\le 4\mathrm{and}\mathrm{x}\le 0orx\ge 2 \\ \Rightarrow x\in [-2,4]\mathrm{and}x\in (-\infty ,0]\cup [2,\infty ) \\ \Rightarrow x\in \left[-2,0\right]\mathrm{U}\left[2,4\right] \end{gathered}$$