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What is the set of all x for which 1/(x – 1)(3 – x) ≤ 1?(a) (-∞, 1) ∪ (3, ∞)(b) (-∞, 1) ∪ (3, ∞) ∪ {2}(c) (-∞, 1) ∪ {2}(d) (3, ∞) ∪ {2}The question was asked in an international level competition.Question is taken from Sets and their Representations topic in portion Sets of Mathematics – Class 11

Answer»

The CORRECT answer is (b) (-∞, 1) ∪ (3, ∞) ∪ {2}

The explanation is: 1/(x – 1)(3 – x) ≤ 1

Now, on solving the equation further we get,

 1/(x – 1)(3 – x) -1 ≤ 0

This ALSO implies,

1- 1/(x – 1)(3 – x) ≥ 0

(x – 1)(3 – x) – 1/(x – 1)(3 – x) ≥ 0

So, (x – 2)^2/(x – 1)(3 – x) ≥ 0.

This implies, (-∞, 1) ∪ (3, ∞) ∪ {2}.


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