Question

Write the solution set of the inequation $\left|\frac{1}{x}-2\right|>4$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}: \\ \left|\frac{1}{x}-2\right|>4 \\ \mathrm{Here},\mathrm{two}\mathrm{cases}\mathrm{arise}. \\ \mathrm{CASE}1:\mathrm{When}\frac{1}{x}-2>0,\mathrm{then}\left|\frac{1}{x}-2\right|=\frac{1}{x}-2 \\ \therefore \frac{1}{x}-2>4 \\ \Rightarrow \frac{1}{x}-2-4>0 \\ \Rightarrow \frac{1}{x}>6 \\ \Rightarrow x\in \left(0,\frac{1}{6}\right)...\left(\mathrm{i}\right) \\ \mathrm{CASE}2:\mathrm{When}\frac{1}{x}-2<0,\mathrm{then}\left|\frac{1}{x}-2\right|=-\left(\frac{1}{x}-2\right) \\ \therefore -\frac{1}{x}+2>4 \\ \Rightarrow -\frac{1}{x}>2 \\ \Rightarrow \frac{1}{x}<-2 \\ \Rightarrow x\in \left(-\infty ,\frac{-1}{2}\right)...\left(\mathrm{ii}\right) \\ \mathrm{Hence},\mathrm{the}\mathrm{solution}\mathrm{set}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{inequation}\mathrm{is}\mathrm{the}\mathrm{union}\mathrm{of}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right). \\ \therefore x\in \left(-\infty ,\frac{-1}{2}\right)\cup \left(0,\frac{1}{6}\right) \end{gathered}$$