Lesson 5: Writing and Graphing Inequalities

Property

An inequality is a statement that compares two expressions, asking for which values of the unknowns the comparison is true.
The set of all such values is the solution set.
The main types of inequalities are:

• $A < B$ ($A$ is less than $B$)
• $A > B$ ($A$ is greater than $B$)
• $A \leq B$ ($A$ is at most $B$, or less than or equal to $B$)
• $A \geq B$ ($A$ is at least $B$, or greater than or equal to $B$)

Examples

• Does $x=10$ make the inequality $x > 8$ true? Yes, because 10 is greater than 8.
• Does $x=7$ make the inequality $x \leq 7$ true? Yes, because 7 is equal to 7, which fits the 'less than or equal to' condition.
• Does $x=3$ make the inequality $x < 3$ true? No, because 3 is not strictly less than 3.

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

Property

When graphing inequalities on a number line, we use different symbols to show whether the boundary point is included in the solution set.
For strict inequalities like $x > 3$ or $x < 3$, we use an open circle at the boundary point to show it is not included.
For non-strict inequalities like $x \geq 3$ or $x \leq 3$, we use a closed circle (or filled dot) at the boundary point to show it is included.
We then shade the number line in the direction that contains all the solutions.

Examples