Let A = x ∈ R : |x| < 1 , B = x ∈ R : |x − 1| ≥ 1 and A ∪ B = R \ D. Then the set D is

Question

Let A = x ∈ R : |x| < 1 , B = x ∈ R : |x − 1| ≥ 1 and A ∪ B = R \ D. Then the set D is

1. A. x ∈ R : 1 ≤ x < 2 (Correct answer)
2. B. x ∈ R : 1 < x ≤ 2
3. C. x ∈ R : 0 ≤ x ≤ 2
4. D. x ∈ R : 1 ≤ x ≤ 2

Correct Answer

Option A — x ∈ R : 1 ≤ x < 2

Detailed Solution & Explanation

The correct answer is { x ∈ R : 1 ≤ x < 2 }.

Key Points

• A={(−1,1)}, B=(−∞,0]∪[2,∞). A∪B=(−∞,1)∪[2,∞).
• D = R\(A∪B) = {x: 1≤x<2}.
• The complement of the union excludes (−∞,1) and [2,∞), leaving [1,2).

Additional Information

• Modulus inequalities unpack directly: |x| < 1 means −1 < x < 1, and |x − 1| ≥ 1 means x − 1 ≤ −1 or x − 1 ≥ 1, i.e. x ≤ 0 or x ≥ 2.
• Sketching each set on a number line makes the union and its complement obvious and avoids sign slips.
• R \ D denotes the complement of D in the reals, so D is exactly what the union fails to cover.
• Interval notation: (a, b) excludes the endpoints, [a, b] includes them — the difference decides whether 1 and 2 belong to D.