Lesson 50: Graphing Inequalities

New Concept

A linear inequality in one variable is an inequality that can be written as $ax < b$, $ax > b$, $ax \le b$, $ax \ge b$, or $ax \neq b$, where $a$ and $b$ are real numbers.

What’s next

This is your introduction to inequalities. Soon, we'll dive into worked examples on graphing solutions and writing inequalities from visual representations.

Property

A solution to an inequality is any value that makes the statement true. To check, substitute the value for the variable and see if the resulting inequality holds.

Examples

• Is 6 a solution for $4y - 5 \ge 11$? Yes, because $19 \ge 11$ is true.
• Is 2 a solution for $4y - 5 \ge 11$? No, because $3 \ge 11$ is false.

Explanation

Think of it as trying a key in a lock. You plug the number in, and if the inequality “unlocks” by making a true statement, it's a solution! If not, that number isn't in the solution club.

Property

Use a number line to graph solutions. A closed circle includes the endpoint ($\ge, \le$), while an open circle excludes it (>, <).

Examples

• To graph $g > 5.8$: Place an open circle at 5.8 and draw an arrow pointing right.
• To graph $r \le -2$: Place a closed circle at -2 and draw an arrow pointing left.

Explanation

A closed circle means "you're invited" ($\ge$). An open circle means the party starts after that number ($>$). The arrow shows where all the fun is happening!