1. What Are Absolute Value Inequalities?

Absolute value inequalities are mathematical expressions that involve the absolute value of a variable and an inequality sign. They come in two main forms: less than and greater than inequalities.

2. How Does the Calculator Work?

The calculator solves absolute value inequalities using these rules:

Where:

• \( |x| \) — Absolute value of x
• \( a \) — Positive real number
• \( < \) — Less than inequality
• \( > \) — Greater than inequality

Explanation: The calculator transforms the absolute value inequality into equivalent compound inequalities without absolute value symbols.

3. Importance of Absolute Value Inequalities

Details: These inequalities are fundamental in algebra and appear in various applications including physics (error margins), engineering (tolerance ranges), and economics (acceptable deviations).

4. Using the Calculator

Tips: Select the inequality type (less than or greater than) and enter a positive value for 'a'. The calculator will provide the solution in interval notation.

5. Frequently Asked Questions (FAQ)

Q1: What if 'a' is negative?
A: The calculator requires 'a' to be positive as absolute value is always non-negative.

Q2: How are equalities (≤ or ≥) handled?
A: The same rules apply, just include the endpoints in the solution.

Q3: Can this solve more complex absolute value inequalities?
A: This calculator handles basic forms. More complex forms may require additional steps.

Q4: What about inequalities like |x + b| < a?
A: The same principles apply, but you'd first isolate the absolute value expression.

Q5: Why are there two solutions for > inequalities?
A: Because absolute value represents distance from zero, values can be too far in either direction.