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  • Solve for x the inequalities in 1/ |x|-3 <1/2

Solve for x the inequalities in \frac{1}{\left | x \right |-3}\leq \frac{1}{2}

Answers (1)

Given: \frac{1}{\left | x \right |-3}\leq \frac{1}{2}

From this, we get,

\frac{1}{\left | x \right |-3}- \frac{1}{2}\leq 0

\frac{1}{\left | x \right |-3}- \frac{1}{2}\leq 0

\\\frac{2-|x|+3}{2(|x|-3)} \leq 0 \\\\ \frac{5-|x|}{|x|-3} \leq 0

Thus, we get,

 5-\left | x \right |\leq 0  &  \left | x \right |-3\geq 0      or  5-\left | x \right |\geq 0 & \left | x \right |-3< 0

Thus \left | x \right |\geq 5 & \left | x \right | > 3  or \left | x \right | \leq 5 & \left | x \right | < 3

Thus,

x \epsilon \left ( -\infty , -5 \right ] or \left [ 5, -\infty \right )  or x \epsilon \left ( -3, 3 \right )

Therefore,

x \epsilon \left ( -\infty , -5 \right ] \upsilon \left ( -3,3 \right ) \upsilon \left [5,\infty \right )

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