1. What is an Absolute Value Equation?

An absolute value equation is an equation that contains an absolute value expression. The general form is |ax + b| = c, where a, b, and c are real numbers. The absolute value of a number is its distance from zero on the number line, without considering direction.

2. How Does the Calculator Work?

The calculator solves equations of the form:

Where:

• \( a \) — Coefficient of x (must not be zero)
• \( b \) — Constant term
• \( c \) — Absolute value (must be non-negative)

Explanation: The equation is solved by considering both the positive and negative cases of the expression inside the absolute value.

3. Solving Absolute Value Equations

Method: To solve |ax + b| = c:

1. Check that c ≥ 0 (if not, no solution exists)
2. Create two equations: ax + b = c and ax + b = -c
3. Solve both equations separately
4. Check solutions in the original equation

4. Using the Calculator

Tips: Enter valid coefficients (a ≠ 0) and a non-negative value for c. The calculator will display both solutions when they exist.

5. Frequently Asked Questions (FAQ)

Q1: What if c is negative?
A: The equation |ax + b| = c has no solution when c < 0 because absolute value is always non-negative.

Q2: What if a is zero?
A: If a = 0, the equation becomes |b| = c, which has no solution if |b| ≠ c, or infinitely many solutions if |b| = c.

Q3: Can absolute value equations have one solution?
A: Yes, when c = 0, there is exactly one solution: x = -b/a.

Q4: How are extraneous solutions handled?
A: The calculator automatically checks for valid solutions and will indicate when no real solutions exist.

Q5: Can this calculator handle complex numbers?
A: No, this calculator only provides real number solutions.