Lesson 7: Solving Inequalities Using Multiplication or Division

Property

When solving inequalities, you can multiply or divide both sides by the same positive number to get an equivalent inequality. The inequality sign stays the same when multiplying or dividing by positive numbers.

Multiplication by a positive number: If $a < b$ and $c > 0$, then $ac < bc$.

Property

For inequalities in the form $\frac{x}{a} \leq b$ where $a > 0$: multiply both sides by $a$ to get $x \leq ab$.

For inequalities in the form $ax > b$ where $a > 0$: divide both sides by $a$ to get $x > \frac{b}{a}$.

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.