Key Concepts

• The absolute value function is commonly used to measure distances between points.
• Applied problems, such as ranges of possible values, can also be solved using the absolute value function.
• The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction.
• In an absolute value equation, an unknown variable is the input of an absolute value function.
• If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable.
• An absolute value equation may have one solution, two solutions, or no solutions.
• An absolute value inequality is similar to an absolute value equation but takes the form [latex]|A|B,\text{ or }|A|\ge B\\[/latex]. It can be solved by determining the boundaries of the solution set and then testing which segments are in the set.
• Absolute value inequalities can also be solved graphically.

Glossary

absolute value equation

an equation of the form [latex]|A|=B\\[/latex], with [latex]B\ge 0\\[/latex]; it will have solutions when [latex]A=B\\[/latex] or [latex]A=-B\\[/latex]

absolute value inequality

a relationship in the form [latex]|{ A }|<{ B },|{ A }|\le { B },|{ A }|>{ B },\text{or }|{ A }|\ge{ B }\\[/latex]