Summary

A linear inequality in one variable $x$ is of the form $ax + b > 0$, $ax + b < 0$, $ax + b \ge 0$, or $ax + b \le 0$, where $a \neq 0$. The solution set is an interval on the number line.

For two variables $x$ and $y$, a linear inequality is $ax + by + c > 0$ (or $<, \ge, \le$). Its solution set is a half-plane (open or closed) bounded by the line $ax + by + c = 0$.

Rules for solving: Adding or subtracting the same number on both sides does not change the inequality. Multiplying or dividing by a positive number preserves the inequality sign; multiplying or dividing by a negative number reverses the inequality sign.

To graph a linear inequality in two variables:

1. Draw the boundary line (dashed for $>$ or $<$, solid for $\ge$ or $\le$).
2. Test a point not on the line (usually $(0,0)$) to decide which half-plane to shade.
3. Shade the region containing the test point if it satisfies the inequality; otherwise, shade the opposite side.

The solution of a system of linear inequalities is the intersection of the half-planes of each inequality — the region common to all shaded areas. …