Absolute Value Equation Calculator
Solve equations of the form a|bx + c| + d = e with step-by-step solutions
Equation Input
Current Equation:
Cannot be zero
Cannot be zero
Any real number
Any real number
Any real number
Solution
Verification:
Step-by-Step Solution:
Quick Examples
Simple Equation
Complex Equation
Negative Coefficient
No Solution
Solution Types
No Solution
When (e - d)/a < 0
Absolute value cannot be negative
One Solution
When (e - d)/a = 0
Expression inside equals zero
Two Solutions
When (e - d)/a > 0
Both positive and negative cases
Key Properties
|x| = x when x ≥ 0
|x| = -x when x < 0
|x| ≥ 0 always
|x| = 0 only when x = 0
Distance from zero on number line
Understanding Absolute Value Equations
What are Absolute Value Equations?
An absolute value equation contains an expression within absolute value bars. The general form we solve is a|bx + c| + d = e, where a, b, c, d, and e are real coefficients.
Solution Method
- Isolate the absolute value expression
- Check if the right side is negative (no solution)
- If right side is zero, solve bx + c = 0
- If right side is positive, solve two cases:
- bx + c = positive value
- bx + c = negative value
Step-by-Step Process
a|bx + c| + d = e
↓
|bx + c| = (e - d)/a
If (e - d)/a < 0: No solution exists
If (e - d)/a = 0: One solution: x = -c/b
If (e - d)/a > 0: Two solutions from ±cases
Real-World Applications
Distance Problems
Finding points at a specific distance from a reference point on a number line.
Error Analysis
Determining acceptable ranges in measurements and tolerances.
Optimization
Minimizing deviations in engineering and manufacturing processes.