Example

Example: Solving ∣4x−3∣≥5|4x - 3| \geq 5 and ∣3x−4∣≥2|3x - 4| \geq 2

To solve the absolute value inequalities ∣4x−3∣≥5|4x - 3| \geq 5 and ∣3x−4∣≥2|3x - 4| \geq 2, apply the standard procedure for 'greater than or equal to' inequalities by translating each into a compound inequality. For ∣4x−3∣≥5|4x - 3| \geq 5, the equivalent compound inequality is 4x−3≤−54x - 3 \leq -5 or 4x−3≥54x - 3 \geq 5. Solving each part for xx yields x≤−12x \leq -\frac{1}{2} or x≥2x \geq 2. In interval notation, this solution set is (−∞,−12]∪[2,∞)(-\infty, -\frac{1}{2}] \cup [2, \infty). Similarly, for ∣3x−4∣≥2|3x - 4| \geq 2, the equivalent compound inequality is 3x−4≤−23x - 4 \leq -2 or 3x−4≥23x - 4 \geq 2. Solving for xx gives x≤23x \leq \frac{2}{3} or x≥2x \geq 2, which is expressed in interval notation as (−∞,23]∪[2,∞)(-\infty, \frac{2}{3}] \cup [2, \infty).

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Updated 2026-06-27

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