Let A = { x ∈ R:|x + 1| < 2} and B = { x ∈ R:|x - 1| ≥ 2}. Then which one of the following statements is NOT true?

Solution

Maths sir · 4 steps

Turn each modulus inequality into intervals on the number line, then test every option against them.

1. Find A. |x + 1| < 2 means -2 < x + 1 < 2, so -3 < x < 1. Hence A = (-3, 1), with both ends open.
2. Find B. |x - 1| ≥ 2 means x - 1 ≤ -2 or x - 1 ≥ 2, so x ≤ -1 or x ≥ 3. Hence B = (-∞, -1] ∪ [3, ∞), with both ends closed.

   Option	Actual set	Claimed set	Verdict
   (a) A - B	(-1, 1)	(-1, 1)	true
   (b) B - A	(-∞, -3] ∪ [3, ∞)	(-∞, -3] ∪ [1, ∞)	not true
   (c) A ∩ B	(-3, -1]	(-3, -1]	true
   (d) A ∪ B	(-∞, 1) ∪ [3, ∞)	ℝ - [1, 3)	true

3. Where (b) fails. ℝ - (-3, 1) contains every number in [1, 3), for example x = 2. But |2 - 1| = 1 < 2, so 2 is not in B, and it cannot be in B - A.
4. So. The statement that is NOT true is (b). The correct set is B - A = ℝ - (-3, 3).

The idea to keep. Write |x - a| < r as one open interval and |x - a| ≥ r as two closed rays. Then sketch both on one number line and watch which endpoints are included.

Answer from the NTA final answer key. Solution by PrepWiser.