Absolute Value Equation Calculator
Solve equations in the form |ax + b| = c
An absolute value equation is a mathematical equation that contains an absolute value expression. The absolute value of a number is its distance from zero on a number line, regardless of whether the number is positive or negative.
In the form |ax + b| = c, the equation asks for what values of x will the absolute value of the linear expression (ax + b) equal the constant c. This type of equation can have zero, one, or two solutions depending on the values of a, b, and c.
To solve an absolute value equation |ax + b| = c:
- Check if c is negative (no solution if it is)
- If c = 0, solve ax + b = 0 directly
- If c > 0, create two equations:
- ax + b = c
- ax + b = -c
- Solve each equation for x
- Check solutions in the original equation
Absolute value equations can have three types of solutions:
- No Solution: When c is negative (impossible since absolute values are never negative)
- One Solution: When c = 0 (the expression inside the absolute value equals zero)
- Two Solutions: When c > 0 (the expression inside the absolute value can equal either c or -c)
Absolute value equations are used in various real-world contexts:
- Error Analysis: Calculating acceptable margins of error in measurements
- Quality Control: Setting tolerance limits for manufacturing processes
- Distance Problems: Finding locations that are a specific distance from a point
- Temperature Variation: Analyzing temperature fluctuations around a mean
- Financial Analysis: Setting price ranges and analyzing deviations
Why can't an absolute value equation equal a negative number?
The absolute value represents the distance from zero, which is always positive or zero. It's impossible for a distance to be negative, so absolute value equations have no solutions when the right side is negative.
When does an absolute value equation have exactly one solution?
An absolute value equation has exactly one solution when the right side equals zero. This occurs because there's only one way for an expression to have an absolute value of zero: the expression itself must equal zero.
How do you check if your solutions are correct?
Substitute each solution back into the original equation. The absolute value of the left side should equal the right side. If it doesn't, either the solution is incorrect or you may have included an extraneous solution.