Lesson 70: Solving Inequalities by Multiplying or Dividing

New Concept

When dividing an inequality by a negative value, the order of the inequality changes. If $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$.

What’s next

Next, you'll apply this rule to solve, graph, and check various inequalities, including those in real-world application problems.

Property

For every real number $a$ and $b$, and for $c > 0$: If $a > b$, then $ac > bc$. If $a < b$, then $ac < bc$.

Explanation

Think of this as scaling up! If you have more cookies than your friend ($a > b$) and you both get five times as many ($c=5$), you still have more cookies. Multiplying by a positive number keeps the relationship the same, so the inequality sign doesn't change. It’s a fair game!

Examples

To solve $\frac{1}{3}x \le 7$, multiply by 3: $(3)\frac{1}{3}x \le 7(3)$, so $x \le 21$.
Since $8 > 5$, then $8(2) > 5(2)$ because $16 > 10$.

Property

For every real number $a$ and $b$, and for $c < 0$: If $a > b$, then $ac < bc$. If $a < b$, then $ac > bc$.

Explanation

This is the big twist! Multiplying by a negative number flips everything across zero on the number line. A bigger positive number becomes a smaller (more negative) number. Because the order reverses, you MUST flip the inequality sign to keep the statement true. Don't forget this crucial step!

Examples

To solve $\frac{-x}{4} < 2$, multiply by -4 and flip the sign: $(-4)\frac{-x}{4} > 2(-4)$, so $x > -8$.
Since $5 > 3$, then $5(-2) < 3(-2)$ because $-10 < -6$.

A single negative sign can flip this entire problem on its head. Let's see how the Multiplication Property of Inequality for $c < 0$ works.

Example Problem

Solve, graph, and check the solution for the inequality $\frac{-x}{3} < 4$.

Step-by-Step

1. Start with the given inequality.

2. To isolate $x$, we need to multiply both sides by $-3$. Because we are multiplying by a negative number, we must reverse the inequality symbol.

3. Simplify the expression to find the solution.

4. Graph the solution on a number line. This requires an open circle at $-12$ and an arrow pointing to the right, indicating all numbers greater than $-12$.

Property

For every real number $a$ and $b$, and for $c > 0$: If $a > b$, then $\frac{a}{c} > \frac{b}{c}$. If $a < b$, then $\frac{a}{c} < \frac{b}{c}$.

Explanation

Dividing by a positive number is like sharing equally. If you have a bigger pile of candy and you split it among your friends, your share is still bigger than your sibling's smaller pile split the same way. The inequality stays the same because the relationship doesn't change. No tricks here!

Examples

To solve $5r < 30$, divide by 5: $\frac{5r}{5} < \frac{30}{5}$, which simplifies to $r < 6$.
Since $20 > 12$, then $\frac{20}{4} > \frac{12}{4}$ because $5 > 3$.