Inequality Properties — Definition, Formula & Examples

Inequality properties are the rules that govern how you can manipulate inequalities while keeping them true. The most critical rule is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.

For real numbers $a$ , $b$ , and $c$ : if $a < b$ , then $a + c < b + c$ (addition property); if $a < b$ and $c > 0$ , then $ac < bc$ (multiplication by a positive); if $a < b$ and $c < 0$ , then $ac > bc$ (multiplication by a negative, reversing the inequality). The transitive property states that if $a < b$ and $b < c$ , then $a < c$ . These properties hold analogously for $>$ , $\leq$ , and $\geq$ .

How It Works

You apply inequality properties the same way you solve equations — isolate the variable using inverse operations — with one extra rule. Adding or subtracting any number on both sides keeps the inequality direction unchanged. Multiplying or dividing both sides by a positive number also preserves the direction. However, multiplying or dividing both sides by a negative number flips the inequality sign. This sign-flip is the single most important detail when solving inequalities.

Worked Example

Problem: Solve the inequality

-3x + 7 \leq 1

Subtract 7 from both sides: Use the subtraction property. The direction stays the same.

-3x + 7 - 7 \leq 1 - 7 \implies -3x \leq -6

Divide both sides by −3: You are dividing by a negative number, so flip the inequality sign from ≤ to ≥.

\frac{-3x}{-3} \geq \frac{-6}{-3} \implies x \geq 2

Answer:

x \geq 2

. Any value of

x

that is 2 or greater satisfies the original inequality.

Why It Matters

These properties appear constantly in Algebra 1, Algebra 2, and Precalculus whenever you solve inequalities, define domains, or work with absolute value expressions. In fields like economics and engineering, constraints are modeled as inequalities, and applying these rules correctly is essential to finding feasible solutions.

Common Mistakes

Mistake: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.

Correction: Every time you multiply or divide both sides of an inequality by a negative value, reverse the direction of the sign. A quick check: substitute a number from your solution back into the original inequality to verify.