Understanding Absolute Value Inequalities
Absolute value inequalities compare |ax + b| to a constant c using <, ≤, >, or ≥. The solution method depends on whether it is a "less than" or "greater than" type.
The Two Fundamental Rules
"Less Than" = AND
|ax + b| < c becomes:
-c < ax + b < c
Solution is a single interval (between two values).
"Greater Than" = OR
|ax + b| > c becomes:
ax + b < -c OR ax + b > c
Solution is two rays (outside two values).
Graphical Interpretation
y = |ax + b| is a V-shape. |ax + b| < c asks: where is the V below y = c? Answer: between the two intersection points. |ax + b| > c asks: where is the V above y = c? Answer: outside the two intersection points.
Worked Examples
|x - 3| < 5
-5 < x - 3 < 5
-2 < x < 8
Interval: (-2, 8)
|2x + 1| ≥ 7
2x+1 ≤ -7 OR 2x+1 ≥ 7
x ≤ -4 OR x ≥ 3
(-inf, -4] U [3, inf)
Special Cases
- |...| < negative: No solution (absolute value is never negative)
- |...| > negative: All real numbers (always true)
- |...| > 0: All reals except where expression = 0
- |...| ≤ 0: Only the single point where expression = 0
Common Mistakes
- Forgetting to flip inequality when dividing by negative a
- Confusing AND (less than) vs OR (greater than)
- Not checking special cases when c ≤ 0
- Writing intervals in wrong order