Lesson 5: Solving Two-Step Inequalities

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

A two-step linear inequality has the form $ax + b < c$, $ax + b \leq c$, $ax + b > c$, or $ax + b \geq c$, where $a \neq 0$. To solve a two-step inequality, use the same steps as solving a two-step equation, but reverse the inequality sign when multiplying or dividing both sides by a negative number.

Examples

Property

When solving inequalities containing parentheses, first apply the distributive property: $a(b + c) = ab + ac$, then solve using the standard two-step process. Remember to reverse the inequality symbol when multiplying or dividing by a negative number.

Examples