← Algebra
Absolute Value & Compound Inequalities
Absolute value equations/inequalities split into two cases; compound inequalities require care with direction, especially when multiplying or dividing by a negative number.
Splitting into cases
|x| = k (for k>0) means x=k OR x=-k. |x| < k means -k < x < k. |x| > k means x > k OR x < -k. Always isolate the absolute value expression before splitting into cases.
When multiplying or dividing an inequality by a negative number, flip the inequality sign — this is the single most common inequality mistake.
Worked Example
Solve |2x - 3| = 7.
- Case 1: 2x-3 = 7 → 2x=10 → x=5.
- Case 2: 2x-3 = -7 → 2x=-4 → x=-2.
Answer: x = 5 or x = -2
Key terms (1)
- Absolute value
- The distance of a number from 0 on the number line; always non-negative.
Practice Quiz
Question 1 of 2Score so far: 0/0
Solve -3x + 6 > 15.