Inequalities in Two Variables with Absolute Value
📐 What Are Absolute Value Inequalities in Two Variables?
An absolute value inequality in two variables is an inequality that contains an absolute value expression involving two variables (typically $x$ and $y$). These inequalities define regions in the coordinate plane rather than simple lines or curves.
Recall: The absolute value $|a|$ represents the distance of $a$ from zero on the number line. So $|a| = a$ if $a \geq 0$, and $|a| = -a$ if $a < 0$.
📋 Standard Forms
The most common forms of absolute value inequalities in two variables are:
| Form | Meaning | Shape of Region |
|---|---|---|
| $|x| + |y| \leq c$ | Sum of absolute values is at most $c$ | Diamond (square rotated 45°) |
| $|x| + |y| \geq c$ | Sum of absolute values is at least $c$ | Exterior of a diamond |
| $y < |x - h| + k$ | $y$ is below a V-shaped graph | Region below a V-shape |
| $y > |x - h| + k$ | $y$ is above a V-shaped graph | Region above a V-shape |
| $|ax + by| \leq c$ | Linear expression bounded | Strip between two parallel lines |