Absolute Value Inequality Calculator

Absolute Value Inequality:

\[ |expr| < k \quad \Rightarrow \quad -k < expr < k \]

Solution:

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1. What is Absolute Value Inequality?

An absolute value inequality describes a range of values that satisfy the inequality condition. The form |expr| < k means the expression is between -k and k.

2. How Does the Calculator Work?

The calculator uses the absolute value inequality rule:

\[ |expr| < k \quad \Rightarrow \quad -k < expr < k \]

Where:

\( expr \) — Any mathematical expression
\( k \) — Positive real number

Explanation: The absolute value inequality is converted to a compound inequality showing the range of possible values.

3. Importance of Absolute Value Inequalities

Details: Absolute value inequalities are fundamental in mathematics, used in solving real-world problems involving tolerances, error margins, and range constraints.

4. Using the Calculator

Tips: Enter any valid mathematical expression and a positive k value. The calculator will show the equivalent compound inequality.

5. Frequently Asked Questions (FAQ)

Q1: What if k is negative?
A: The inequality |expr| < k has no solution when k is negative, as absolute value is always non-negative.

Q2: How does this work for greater than inequalities?
A: For |expr| > k, the solution would be expr < -k OR expr > k. This calculator focuses on the less than case.

Q3: Can I use variables in the expression?
A: Yes, the calculator accepts variables, though the solution will be in terms of those variables.

Q4: What's the difference between |expr| < k and |expr| ≤ k?
A: The latter includes the endpoints -k and k in the solution.

Q5: Can this handle complex expressions?
A: Yes, as long as the expression is mathematically valid, the calculator will process it.