Linear Inequations-14

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Solve -1/{|x| - 2} ≥ 1, where x ∈ R, x ≠ ± 2

Solution: We have, -1/{|x| - 2} ≥ 1

=>-1/{|x| - 2} – 1 ≥ 0

=>[-1 – (|x| - 2)]/[|x| - 2] ≥ 0

=> [1 - |x|]/[|x| - 2] ≥ 0

[|x| - 1]/[|x| - 2] ≤ 0

=>(y - 1)/(y - 2) ≤ 0, where y = |x|

=> 1 ≤ y < 2

=> 1 ≤ |x| < 2 [since, y = |x|]

=> x ∈ (-2, -1] U [1, 2) [since, a < |x| < b <=> x ∈ (-b, -a) U (a, b)]

Hence, the solution set of the given inequation is (-2, -1] U [1, 2)

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