Multiplication and Division Properties for Inequalities

Property If $c < 0$ and $a < b$, then $ac bc$ and $\frac{a}{c} \frac{b}{c}$. If $c 0$ and $a < b$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$.

Solve $ 4x 20$. Dividing by $ 4$ flips the sign: $\frac{ 4x}{ 4} < \frac{20}{ 4}$, so $x < 5$. Solve $\frac{y}{ 3} \leq 2$. Multiplying by $ 3$ flips the sign: $(\frac{y}{ 3}) \cdot ( 3) \geq 2 \cdot ( 3)$, so $y \geq 6$. Solve $6a < 18$. Dividing by a positive 6 means no flip is needed: $\frac{6a}{6} < \frac{18}{6}$, so $a < 3$.

Think of an inequality like a balanced seesaw. Multiplying by a positive number keeps it level. But multiplying or dividing by a negative number is like swapping the kids to opposite ends—the seesaw flips! To keep the comparison true, you must flip the inequality sign. It's the one tricky rule you must always remember to check for.