Absolute value equation

Definition

An absolute value equation is an equation that contains an absolute value expression, which represents the distance of a number from zero on the number line. Solutions are obtained by considering both the positive and negative scenarios of the expression inside the absolute value.

5 Must Know Facts For Your Next Test

1. Absolute value equations have two cases: one for the positive value and one for the negative value of the expression within the absolute value.
2. The basic form of an absolute value equation is $|ax + b| = c$, where $c$ is a non-negative constant.
3. To solve $|ax + b| = c$, you must split it into two separate linear equations: $ax + b = c$ and $ax + b = -c$.
4. If $c < 0$, then $|ax + b| = c$ has no solution, because absolute values cannot be negative.
5. Always check your solutions in the original equation, as extraneous solutions can arise during solving.

Review Questions

• What are the two cases you need to consider when solving an absolute value equation?
• How do you solve $|3x - 4| = 5$?
• What does it mean if you encounter a negative constant on the right side of an absolute value equation?

Related terms

Linear Equation: An equation between two variables that gives a straight line when plotted on a graph.

Extraneous Solution: A solution derived from an algebraic process that is not a valid solution to the original equation.

Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$.