Section 11.3: 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

To solve an inequality using multiplication or division:

1. We can multiply or divide both sides by the same positive number without changing the inequality sign.
2. If we multiply or divide both sides by a negative number, we must reverse the direction of the inequality sign.

Examples

Property

To solve an inequality using multiplication or division, multiply or divide both sides by the same positive or negative number. If you multiply or divide by a negative number, you must reverse the inequality sign. If you multiply or divide by a positive number, the inequality sign stays the same.

Examples