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

Do NOT reverse the inequality symbol when:

• Adding or subtracting any number (positive or negative): If $a > b$, then $a + c > b + c$ and $a - c > b - c$
• Multiplying or dividing by a positive number: If $a > b$ and $c > 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$
• Working with negative numbers that are NOT being multiplied or divided

Examples

Property

When solving inequalities, use estimation and compatible numbers to verify that your solution makes sense in the context of the problem. Replace variables with test values from your solution set to check if the original inequality holds true.

Examples