Lesson 1: Writing and Graphing Inequalities

Property

An inequality is used in algebra to compare two quantities that may have different values or to show a range of possible values. We use these four specific symbols for inequalities:

• $<$ is read "$a$ is less than $b$"
• $>$ is read "$a$ is greater than $b$"
• $\leq$ is read "$a$ is less than or equal to $b$"
• $\geq$ is read "$a$ is greater than or equal to $b$"

Examples

• The statement "15 is greater than 9" is written in algebra as $15 > 9$.
• To show that your age, $a$, must be at least 18 to vote, you would write $a \geq 18$.
• The inequality $x < 5$ means that the value of $x$ can be any number that is strictly less than 5.

Explanation

These symbols are used to show the relationship between two values when they are not perfectly equal. They tell us whether one value is bigger or smaller than another, or if there is a limit. A helpful trick is to remember that the small, pointed end of $<$ or $>$ always faces the smaller number, like an arrow pointing to the lesser value. The line underneath $\leq$ and $\geq$ is simply half of an equal sign!

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

A solution of an inequality is a value of a variable that makes a true statement when substituted into the inequality. To determine whether a number is a solution to an inequality:
Step 1. Substitute the number for the variable in the inequality.
Step 2. Simplify the expressions on both sides of the inequality.
Step 3. Determine whether the resulting inequality is true.

• If it is true, the number is a solution.
• If it is not true, the number is not a solution.

Examples