1. What Are Absolute Value Equations?

Absolute value equations are equations that contain an absolute value expression. The absolute value of a number is its distance from zero on the number line, without considering direction. Absolute value equations can have two solutions.

2. How Does the Calculator Work?

The calculator solves equations of the form:

Where:

• \( a \) — Coefficient of x (cannot be zero)
• \( b \) — Constant term
• \( c \) — Absolute value (must be ≥ 0)

Explanation: The equation \( |ax + b| = c \) has two solutions when c > 0: \( ax + b = c \) and \( ax + b = -c \).

3. Solving Absolute Value Equations

Details: To solve \( |ax + b| = c \):

1. Set up two equations: \( ax + b = c \) and \( ax + b = -c \)
2. Solve both equations separately for x
3. Check both solutions in the original equation

4. Using the Calculator

Tips: Enter the coefficients a, b, and c. The calculator will display both solutions when they exist. Remember that 'a' cannot be zero and 'c' cannot be negative.

5. Frequently Asked Questions (FAQ)

Q1: What if c is negative in the equation?
A: The equation \( |ax + b| = c \) has no solution when c is negative, since absolute value is always non-negative.

Q2: What if a is zero?
A: If a is zero, the equation becomes \( |b| = c \), which is either always true (if \( |b| = c \)) or never true (if \( |b| ≠ c \)). The calculator requires a non-zero value for a.

Q3: Can this calculator handle complex equations?
A: This calculator handles basic absolute value equations of the form \( |ax + b| = c \). More complex equations may need manual solving.

Q4: Why are there sometimes two solutions?
A: Absolute value represents distance from zero, so both the positive and negative cases must be considered, leading to two possible solutions.

Q5: How accurate are the solutions?
A: Solutions are rounded to 4 decimal places for readability, but the calculations use full precision.