Lesson 8-6: Write and Solve One-Step Multiplication and Division 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

To solve an inequality using multiplication or division, multiply or divide both sides by the same positive or negative number. If you multiply or divide by a negative number, you must reverse the inequality sign. If you multiply or divide by a positive number, the inequality sign stays the same.

Examples

Property

When graphing inequalities on a number line, we use different symbols to show whether the boundary point is included in the solution set.
For strict inequalities like $x > 3$ or $x < 3$, we use an open circle at the boundary point to show it is not included.
For non-strict inequalities like $x \geq 3$ or $x \leq 3$, we use a closed circle (or filled dot) at the boundary point to show it is included.
We then shade the number line in the direction that contains all the solutions.

Examples