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Question
Solve for x, the inequality given below.
`1/(|x| - 3) ≤ 1/2`
Solution
`1/(|x| - 3) ≤ 1/2`
⇒ `1/(|x| - 3) - 1/2 ≤ 0`
⇒ `(2 - |x| + 3)/(2(|x| - 3)) ≤ 0`
⇒ `(5 - |x|)/(|x| - 3) ≤ 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 ∈ (–`oo` , –5] or [5, `oo`) or x ∈ (–3, 3)
⇒ x ∈ (`-oo`, –5] ∪ (–3, 3) ∪ [5, `oo`)
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