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