Let U = ℝ, A = {x ∈ ℝ : (x - 2)(x + 5) ≤ 0}, and B = {x ∈ ℝ : |x - 1| < 3}. What is (A′ ∩ B)′?
Answer and explanation
Correct answer: (-∞, 2] ∪ [4, ∞)
The inequality (x - 2)(x + 5) ≤ 0 holds between the roots, including both roots, so A = [-5, 2]. Also, |x - 1| < 3 gives -3 < x - 1 < 3, hence -2 < x < 4 and B = (-2, 4). Therefore A′ = (-∞, -5) ∪ (2, ∞), and A′ ∩ B = (2, 4). Taking the complement in ℝ gives (A′ ∩ B)′ = (-∞, 2] ∪ [4, ∞), which is option A.
Frequently asked questions
What is the correct answer to this question?
(-∞, 2] ∪ [4, ∞)
Why is this the correct answer?
The inequality (x - 2)(x + 5) ≤ 0 holds between the roots, including both roots, so A = [-5, 2]. Also, |x - 1| < 3 gives -3 < x - 1 < 3, hence -2 < x < 4 and B = (-2, 4). Therefore A′ = (-∞, -5) ∪ (2, ∞), and A′ ∩ B = (2, 4). Taking the complement in ℝ gives (A′ ∩ B)′ = (-∞, 2] ∪ [4, ∞), which is option A.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
