Inequalities and Their Solutions

Solving Inequalities

By the end of this lesson, you will be able to solve one- and two-step inequalities, graph their solutions on a number line, and write solution sets.

Inequality Symbols

Symbol	Meaning	Number line marker
$<$	Less than	Open circle
$>$	Greater than	Open circle
$\leq$	Less than or equal to	Closed (filled) circle
$\geq$	Greater than or equal to	Closed (filled) circle

Solving Inequalities

Same as equations with ONE critical rule:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Example 1: $x + 5 > 12$

Subtract 5: $x > 7$ → open circle at 7, arrow pointing right

Example 2: $-3x \leq 15$

Divide by −3 (FLIP!): $x \geq -5$ → closed circle at −5, arrow pointing right

Example 3: $2x - 4 < 10$

Add 4: $2x < 14$

Divide by 2: $x < 7$ → open circle at 7, arrow pointing left

Graphing on a Number Line

Open circle: the endpoint is NOT included ($<$ or $>$)

Closed circle: the endpoint IS included ($\leq$ or $\geq$)

Arrow points in the direction of the solution.

Writing Solution Sets

$x > 7$ means all real numbers greater than 7.

$x \leq -3$ means all real numbers less than or equal to −3.

Summary

Solve inequalities like equations — same inverse operations.

Flip the sign only when multiplying or dividing by a negative number.

Graph: open circle (strict inequality), closed circle (or-equal-to), arrow pointing toward solutions.