Lesson 8: Understanding and Solving 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 solve an inequality using addition or subtraction:

1. We can add the same quantity to both sides of an inequality.
2. We can subtract the same quantity from both sides of an inequality.
3. The direction of the inequality sign remains unchanged when adding or subtracting.

Examples

Property

To solve an inequality using multiplication or division, multiply or divide both sides by the same positive or negative number. If you multiply or divide by a negative number, you must reverse the inequality sign. If you multiply or divide by a positive number, the inequality sign stays the same.

Examples