If |4X + 7| >= 5, then the values of X (in interval notation) are: A. (-inf, -3] only B. (-inf, -3] U [-1/2, +inf) C. [-3, -1/2] D. [-1/2, +inf) only

GATE 2013 · General Aptitude · Absolute Value · medium

Answer: The solution set is (-inf, -3] U [-1/2, +inf). Answer: B.

1. Apply the absolute value inequality rule: |4X + 7| >= 5 splits into two cases: (1) 4X + 7 >= 5, and (2) 4X + 7 <= -5.
2. Solve Case 1: 4X + 7 >= 5: Subtract 7 from both sides: 4X >= -2. Divide by 4 (positive, no flip): X >= -1/2. Solution: [-1/2, +inf).
3. Solve Case 2: 4X + 7 <= -5: Subtract 7 from both sides: 4X <= -12. Divide by 4: X <= -3. Solution: (-inf, -3].
4. Combine the two cases: The full solution is the union of both intervals: X in (-inf, -3] OR X in [-1/2, +inf), written as (-inf, -3] U [-1/2, +inf). The middle region (-3, -1/2) does NOT satisfy the inequality.