Lesson 7: Solving Inequalities Using Multiplication or Division

Property

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

Examples

• To solve $3x > 21$, divide by 3 (a positive number): $\frac{3x}{3} > \frac{21}{3}$, so $x > 7$. The sign stays the same.
• To solve $-4y \geq 20$, divide by -4 (a negative number): $\frac{-4y}{-4} \leq \frac{20}{-4}$, so $y \leq -5$. The sign reverses.
• To solve $\frac{z}{-2} < 5$, multiply by -2 (a negative number): $-2(\frac{z}{-2}) > -2(5)$, so $z > -10$. The sign reverses.

Explanation

The game-changing rule! When you multiply or divide both sides by a positive number, nothing changes. But if you use a negative number, you MUST flip the inequality sign. For example, > becomes <.

Property

When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.

• If $a < b$, then $-a > -b$.
• The solution set of $E < F$ is the same as the solution set of $-E > -F$.

Examples

• To solve $-x < 8$, multiply by $-1$ and reverse the inequality sign to get $x > -8$.
• For $-5w \geq 30$, divide by $-5$ and reverse the inequality sign to get $w \leq -6$.
• To solve $12 - 3x > 6$, first subtract 12 to get $-3x > -6$. Then, divide by $-3$ and reverse the sign to get $x < 2$.

Explanation

Multiplying by a negative number flips everything to the opposite side of zero on the number line. What was smaller (more to the left) becomes larger (more to the right), so you must flip the inequality sign.