Lesson 3: Solving Inequalities Using Multiplication or Division

Property

For any real numbers $a$, $b$, $c$
if $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
if $a > b$ and $c > 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a < b$ and $c < 0$, then $\frac{a}{c} > \frac{b}{c}$ and $ac > bc$.
if $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$ and $ac < bc$.
When we divide or multiply an inequality by a:
• positive number, the inequality stays the same.
• negative number, the inequality reverses.

Examples

• To solve $6x > 48$, divide both sides by 6. Since 6 is positive, the inequality stays the same: $x > 8$.

• To solve $-4y \geq 20$, divide both sides by -4. Since -4 is negative, the inequality reverses: $y \leq -5$.

Property

Common errors in two-step inequalities include: forgetting to reverse the inequality symbol when multiplying or dividing by negative numbers, applying operations to only part of an expression instead of both sides, and misinterpreting inequality symbols in word problems.

Examples