Question

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

Answer 1

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