Lesson 3: Inequalities

Property

An equation is a mathematical statement that two expressions are equal ($=$), and it typically has one specific solution.

An inequality is a mathematical statement that two expressions are not equal ($<, >, \leq, \geq$), and it often has a range of many solutions.

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