Some problems in algebra lead to inequalities instead of equations. An inequality looks just like an equation, except that in the place of the equal sign is one of the symbols, <, >, <=, or >=. Here is an example of an inequality:

4x + 7 <= 19

To solve an inequality that contains a variable means to find all values of the variable that make the inequality true. Unlike an equation, an inequality generally has infinitely many solutions, which form an interval or a union of intervals on the real line. The following illustration shows how an inequality differs from its corresponding equation: 

To solve inequalities, we use the following rules to isolate the variable on one side of the inequality sign. These rules tell us when two inequalities are equivalent (the symbol => means “is equivalent to”). In these rules the symbols A, B, and C stand for real numbers or algebraic expressions. Here we state the rules for inequalities involving the symbol <= but they apply to all four inequality symbols.

Solving Linear Inequalities

Solving Nonlinear Inequalities

Solving a Quadratic Inequality

Absolute Value Inequalities