If U = ℝ and A = {x : x² − 2x − 8 ≤ 0}, what is A'?
Answer and explanation
Correct answer: (-∞,-2) ∪ (4,∞)
Factor the quadratic: x² − 2x − 8 = (x − 4)(x + 2). Since the parabola opens upward, the inequality (x − 4)(x + 2) ≤ 0 holds between and at the roots, so A = [−2,4]. The complement in ℝ contains numbers strictly outside this closed interval. Hence A' = (−∞,−2) ∪ (4,∞). The roots are excluded from the complement because they belong to A.
Frequently asked questions
What is the correct answer to this question?
(-∞,-2) ∪ (4,∞)
Why is this the correct answer?
Factor the quadratic: x² − 2x − 8 = (x − 4)(x + 2). Since the parabola opens upward, the inequality (x − 4)(x + 2) ≤ 0 holds between and at the roots, so A = [−2,4]. The complement in ℝ contains numbers strictly outside this closed interval. Hence A' = (−∞,−2) ∪ (4,∞). The roots are excluded from the complement because they belong to 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.
