Lesson 2: Solving Inequalities Using Addition or Subtraction

Property

Equivalent inequalities are inequalities that have exactly the same solution set. If two inequalities have identical solutions, they are equivalent regardless of how they appear.

Examples

Property

Subtraction Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a - c < b - c$.
if $a > b$, then $a - c > b - c$.

Addition Property of Inequality
For any numbers $a$, $b$, and $c$,
if $a < b$, then $a + c < b + c$.
if $a > b$, then $a + c > b + c$.

Examples

• To solve $x + 7 \leq 15$, subtract 7 from both sides. This gives $x \leq 8$. The solution is all numbers less than or equal to 8, or $(-\infty, 8]$.

Property

To solve an inequality using addition and subtraction:

1. We can add or subtract the same quantity on both sides.
2. The direction of the inequality sign remains unchanged when adding or subtracting.

Examples