Lesson 5: Solve Inequalities Using Multiplication or Division

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 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.