Question

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 1

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}.