Write the solution set of the inequation |1 x -1 |<4 - Vidyadip

Question

Write the solution set of the inequation $\Big|\frac{1}{\text{x}}-1\Big|<4$

✓

Answer

$\Big|\frac{1}{\text{x}}-2\Big|>4$
$\Leftrightarrow\frac{1}{\text{x}}<2-4$or $\frac{1}{\text{x}}>2+4$
$\Leftrightarrow\frac{-1}{\text{x}}>2$ or $\frac{1}{\text{x}}>6$
$\Leftrightarrow\frac{-1}{\text{x}}>2$ or $\frac{1}{\text{x}}>6$
$\Leftrightarrow\frac{-1}{2}>\text{x}$ or $\frac{1}{6}>\text{x}$
$\Leftrightarrow\text{x}\in\Big(-\infty,\frac{-1}{2}\Big)$or $\text{x}\in\Big(\frac{1}{6},\infty\Big)$
$\Leftrightarrow\text{x}\in\Big(-\infty,\frac{-1}{2}\Big)\cup\Big(\frac{1}{6},\infty\Big)$
Hence, the solution set of the given system of inequation is $\Big(-\infty,\frac{-1}{2}\Big)\cup\Big(\frac{1}{6},\infty\Big)$

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