Solving an Absolute Value Inequality

Solving an absolute value inequality means isolating the absolute value, unpacking by the less-than or greater-than pattern, solving each piece, and writing the set in interval notation.

Solve |2x − 1| > 7. The absolute value already stands alone, and greater-than means rays: 2x − 1 > 7 or 2x − 1 < −7. Solve each: x > 4 or x < −3. Interval form: (−∞, −3) ∪ (4, ∞). Test all three regions: x = 5 gives |9| = 9 > 7, true; x = −4 gives |−9| = 9 > 7, true; x = 0 gives |−1| = 1 > 7, false — both rays admit, the middle rejects, exactly the greater-than shape. A less-than version would have produced one trapped interval instead of two rays.