Section 11.1: Writing and Graphing Inequalities

Property

An inequality is used in algebra to compare two quantities that may have different values. We use four main inequality symbols:
$a < b$ is read $a$ is less than $b$
$a > b$ is read $a$ is greater than $b$
$a \leq b$ is read $a$ is less than or equal to $b$
$a \geq b$ is read $a$ is greater than or equal to $b$

Examples

Property

A statement that uses one of the symbols $>$ or $<$ is called an inequality. An inequality that uses the symbol for less than, $<$, or greater than, $>$, is called a strict inequality. A nonstrict inequality uses one of the following symbols: $\geq$ means "greater than or equal to"; $\leq$ means "less than or equal to".

Examples

• The inequality $x > 5$ represents all numbers strictly greater than 5. On a number line, this is shown with an open circle at 5 and an arrow pointing to the right.
• The inequality $y \leq -2$ represents -2 and all numbers less than it. On a number line, this is shown with a solid dot at -2 and an arrow pointing to the left.
• The values 8, 9.5, and 200 all satisfy the inequality $x \geq 8$, but 7.9 does not.

Explanation

Inequalities describe a range of possible values, not just a single answer. A strict inequality ($<$ or $>$) uses an open circle on a number line, while a non-strict one ($\leq$ or $\geq$) uses a solid dot to show the endpoint is included.

Property

To check if a value is a solution to an inequality, substitute the value for the variable and evaluate both sides. If the resulting statement is true, the value is a solution. If false, it is not a solution.

Examples