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

To verify if a value is a solution to an inequality, substitute the value for the variable and check if the resulting statement is true.
Test the boundary value and values on both sides to confirm the complete solution set.

Examples

Property

To show the solution set of an inequality on a number line:

• For inequalities with $\leq$ or $\geq$, use a filled-in dot to show the endpoint is included in the solution.
• For inequalities with $<$ or $>$, use an open circle to show the endpoint is not included.
• Shade the region of the number line that represents all possible solutions, often with an arrow to show it continues infinitely.

Examples

• To graph $x < 2$, place an open circle at 2 and shade the number line to the left.
• The graph for $x \geq -1$ has a filled-in dot at -1 and shading to the right.
• The solution set for $x > 5$ is shown with an open circle at 5 and an arrow shading everything to the right.

Explanation

Think of the dot as a gate. A filled-in (closed) dot means the gate is part of your property (the solution). An open circle means it's just a post marking the boundary, but it isn't included.