📐 Absolute Value Equation Calculator
Solve absolute value equations step by step
|ax + b| = c
How to Use This Calculator
Enter the Coefficient
Type the coefficient (a) of the variable inside the absolute value. For |3x| = 12, enter 3.
Enter the Value
Type the value (c) that the absolute value equals. For |3x| = 12, enter 12.
Click Solve
Press "Solve Equation" to find all solutions. The calculator will show both possible answers.
Review Solutions
See the solutions and step-by-step working. Absolute value equations typically have two solutions.
Formula
|ax + b| = c
If c ≥ 0, then ax + b = c or ax + b = -c
If c < 0, there are no solutions
Example 1: Solve |3x| = 12
Step 1: Recognize |3x| = 12 means 3x = 12 or 3x = -12
Step 2: Solve both equations
• 3x = 12 → x = 4
• 3x = -12 → x = -4
Solutions: x = 4 or x = -4
Example 2: Solve |2x + 1| = 5
Step 1: |2x + 1| = 5 means (2x + 1) = 5 or (2x + 1) = -5
Step 2: Solve both equations
• 2x + 1 = 5 → 2x = 4 → x = 2
• 2x + 1 = -5 → 2x = -6 → x = -3
Solutions: x = 2 or x = -3
Example 3: Solve |x - 3| = -2
Step 1: Absolute value is always non-negative
Step 2: Cannot equal a negative number
No solutions
About Absolute Value Equation Calculator
The Absolute Value Equation Calculator solves equations containing absolute value expressions. Absolute value equations typically have two solutions because the absolute value of a number represents its distance from zero on a number line, and both positive and negative numbers can have the same absolute value.
When to Use This Calculator
- Algebra Homework: Solve absolute value equations quickly
- Test Preparation: Check your answers and verify solutions
- Distance Problems: Solve real-world problems involving distance from zero
- Inequality Preparation: Understand absolute value before solving inequalities
- Graphing: Find x-intercepts of absolute value functions
Why Use Our Calculator?
- ✅ Shows Both Solutions: Displays all possible answers
- ✅ Step-by-Step Working: See how solutions are found
- ✅ Handles Edge Cases: Identifies equations with no solutions
- ✅ Instant Results: Fast and accurate calculations
- ✅ Educational: Perfect for learning absolute value concepts
- ✅ Free to Use: No registration required
Understanding Absolute Value Equations
Key Concept: The absolute value |x| is always non-negative. If |x| = c, then either x = c or x = -c (as long as c ≥ 0). This is why most absolute value equations have exactly two solutions.
- If c > 0, the equation has two solutions: x = c and x = -c
- If c = 0, the equation has one solution: x = 0
- If c < 0, the equation has no solutions (impossible)
- Graphically, absolute value functions form a V-shape
- Solutions represent x-intercepts of the absolute value function
Real-World Applications
Distance Problems: If you're 5 miles away from home, your position could be +5 or -5 miles from the origin point.
Temperature: Finding temperatures that differ from a target by a certain amount.
Error Analysis: Calculating measurement errors or tolerances in manufacturing.
Tips for Solving Absolute Value Equations
- Always check if the value on the right side is non-negative
- Set up two separate equations: one with + and one with -
- Solve each equation independently
- Verify solutions by substituting back into the original equation
- Remember that |x| ≥ 0 for all real numbers
- Watch for special cases like |x| = 0 (one solution) or |x| = negative (no solution)
Frequently Asked Questions
How many solutions does an absolute value equation have?
Most absolute value equations have exactly two solutions. Exceptions: |x| = 0 has one solution (x = 0), and |x| = negative has no solutions.
What if the right side is negative?
If |x| = negative, there are no solutions because absolute value is always non-negative. The calculator will show "No Solutions."
Can I solve more complex absolute value equations?
This calculator handles basic forms like |ax| = c and |ax + b| = c. For more complex equations with nested absolute values, you'll need to split into multiple cases manually.
Why does the equation have two solutions?
Absolute value represents distance from zero. Both positive and negative numbers can be the same distance from zero. For example, both 3 and -3 are 3 units from zero, so |x| = 3 has both x = 3 and x = -3.
What's the difference between absolute value equations and inequalities?
Equations use = (exact value), inequalities use <, >, ≤, or ≥. Inequalities often have a range of solutions rather than just two discrete solutions.
How do I verify my solutions?
Substitute each solution back into the original equation and check if it makes the equation true. Both solutions should satisfy |ax| = c.