Section 11.2: Solving Inequalities Using Addition or Subtraction

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 verify an inequality solution, substitute test values from the solution set into the original inequality. The inequality should remain true for values in the solution set and false for values outside it.

Examples

Property

Real-world problems can be modeled using inequalities of the form $x + a \geq b$ or $x - a \leq b$, where $x$ represents an unknown quantity and the inequality describes a constraint or condition that must be satisfied.

Examples