Absolute value equation

Definition

An absolute value equation is an equation where the unknown variable appears inside absolute value bars, e.g., $|x| = a$. Solutions to these equations require considering both the positive and negative scenarios.

5 Must Know Facts For Your Next Test

1. The general form of an absolute value equation is $|ax + b| = c$.
2. To solve $|ax + b| = c$, set up two separate equations: $ax + b = c$ and $ax + b = -c$.
3. Absolute value equations may have two solutions, one solution, or no solution depending on the value of c.
4. If $c < 0$, the equation $|ax + b| = c$ has no real solution since absolute values cannot be negative.
5. Graphically, solving an absolute value equation involves finding the points where the V-shaped graph intersects with a horizontal line.

Review Questions

• What is the first step in solving the absolute value equation $|2x - 3| = 7$?
• How many solutions can an absolute value equation have if it is set equal to a positive number?
• Explain why the equation $|x + 4| = -2$ has no solution.

Related terms

Absolute Value: The distance of a number from zero on the number line, always non-negative.

Piecewise Function: A function composed of multiple sub-functions, each applying to a certain interval of the main function's domain.

Inequality: A mathematical statement indicating that two expressions are not equal and involving symbols such as >, <, ≥, or ≤.