Solve for x, the inequalities in

Question:

Solve for x, the inequalities in 

$\frac{1}{|x|-3} \leq \frac{1}{2}$

Solution:

According to the question,

$\frac{1}{|x|-3} \leq \frac{1}{2}$

$\Rightarrow \frac{1}{|x|-3}-\frac{1}{2} \leq 0$

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

$\Rightarrow \frac{5-|x|}{(|x|-3)} \leq 0$

⇒ 5 – |x| ≤ 0 and |x| – 3 > 0 or 5 – |x| ≥ 0 and |x| – 3 < 0

⇒ |x| ≥ 5 and |x| > 3 or |x| ≤ 5 and |x| < 3

⇒ |x| ≥ 5 or |x| < 3

⇒ x ∈ (- ∞ , – 5] or [5, ∞) or x ∈ ( -3 , 3)

⇒ x ∈ (- ∞ , – 5] ∪ ( -3 , 3) ∪ [5, ∞)

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