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: |x - h| ≤ k and |x - h| ≥ k, where h is the center point and k is the distance from the center.

2. How Does the Calculator Work?

The calculator solves inequalities of the form:

Where:

• \( h \) — The center point of the inequality
• \( k \) — The distance from the center (must be ≥ 0)
• \( \leq \) — "Less than or equal to" (inclusive range)
• \( \geq \) — "Greater than or equal to" (two separate ranges)

Explanation: The absolute value inequality |x - h| ≤ k represents all numbers x that are within k units of h on the number line, while |x - h| ≥ k represents all numbers x that are k or more units away from h.

3. Importance of Graphing Inequalities

Details: Graphing absolute value inequalities helps visualize the solution set on a number line, making it easier to understand the range of values that satisfy the inequality.

4. Using the Calculator

Tips: Select the inequality type, enter the h (center) value and k (distance) value. The k value must be non-negative. The calculator will display the solution in interval notation.

5. Frequently Asked Questions (FAQ)

Q1: What does |x - h| ≤ k represent?
A: It represents all real numbers x that are within k units of h on the number line, including the endpoints.

Q2: How is |x - h| ≥ k different?
A: This represents all real numbers x that are k or more units away from h, forming two separate intervals.

Q3: What if k is negative?
A: The calculator requires k ≥ 0. If k is negative, |x - h| ≤ k has no solution, while |x - h| ≥ k is true for all real numbers.

Q4: Can this calculator handle other inequality forms?
A: This version handles basic forms. More complex forms like a|x - h| + b ≤ c would need additional inputs.

Q5: How do I graph the solution?
A: For ≤ inequalities, shade between h-k and h+k. For ≥ inequalities, shade outward from these points.