Linear inequalities

Description

A linear inequality is like a linear equation but uses $<$, $\leq$, $>$, or $\geq$ instead of $=$. Solve using the same operations as equations, with one important extra rule.

Rule: when you multiply or divide by a negative number, the inequality sign reverses.

$$-2x > 6 \implies x < -3$$

Example: Solve $3x - 4 \leq 14$ $$3x \leq 18 \implies x \leq 6$$

Number lines: show solutions using:

• Open circle (○) for $<$ or $>$ (not included)
• Closed circle (●) for $\leq$ or $\geq$ (included)

Integer solutions: list all integers satisfying the inequality. For $-2 < x \leq 4$: integers are $-1, 0, 1, 2, 3, 4$.

Graphical representation: shade regions on a graph. Use a dashed line for strict inequalities and a solid line for $\leq$ or $\geq$.

Common error: forgetting to reverse the inequality sign when dividing by a negative number.

Links

Solving linear equations Quadratic inequalities