1. What Are Absolute Value Inequalities?

Absolute value inequalities are mathematical expressions that involve the absolute value of a linear expression and an inequality relationship. They come in two main forms: |ax + b| < c and |ax + b| > c.

2. How Does the Calculator Work?

The calculator solves inequalities of the form:

Where:

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

3. Solving Absolute Value Inequalities

For |ax + b| < c: The solution is -c < ax + b < c, which translates to a compound inequality.
For |ax + b| > c: The solution is ax + b < -c OR ax + b > c, which translates to two separate inequalities.

4. Using the Calculator

Steps: Enter values for a, b, and c. Select the inequality type. The calculator will provide the solution in interval notation.

5. Frequently Asked Questions (FAQ)

Q1: What if a is zero?
A: The equation becomes |b| < c or |b| > c, which is a simple true/false statement about constants.

Q2: What if c is negative?
A: |ax + b| is always non-negative. |ax + b| < c has no solution when c < 0. |ax + b| > c is always true when c < 0.

Q3: How are the solutions represented?
A: Solutions are shown as compound inequalities (for <) or as two separate inequalities (for >).

Q4: Can this solve other inequality forms?
A: Yes, forms like ≤ and ≥ work similarly - just include equality in the solution.

Q5: What about more complex absolute value inequalities?
A: For more complex forms (like quadratic inside absolute value), different solution methods are needed.