Math Problem Statement

Write an absolute value inequality for the graph below. Use x for your variable.

Solution

The number line in the image shows the solution to an absolute value inequality. The endpoints are at x=2x = -2 and x=2x = 2, and the region between them is shaded, indicating the values of xx that satisfy the inequality.

To write the absolute value inequality that represents this graph, you can follow these steps:

  1. The center of the interval is the midpoint between 2-2 and 22, which is 0.
  2. The distance from the center (0) to either endpoint (2 or 2-2) is 2.

Thus, the absolute value inequality that describes the graph is:

x2|x| \leq 2

This inequality means that the distance of xx from 0 is less than or equal to 2, which matches the shaded region between 2-2 and 22 on the graph.

Do you need further details or explanations? Here are five related questions to expand on this concept:

  1. How would you write an absolute value inequality for a range between -5 and 5?
  2. What would change in the inequality if the points were open circles instead of closed?
  3. How can you express the same inequality x2|x| \leq 2 without using absolute value?
  4. What is the general form of an absolute value inequality for a range between aa and bb?
  5. How would the inequality change if the interval was shifted to the right?

Tip: The absolute value inequality xcr|x - c| \leq r describes all values of xx within rr units of cc.

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Math Problem Analysis

Mathematical Concepts

Algebra
Absolute Value Inequalities

Formulas

|x - c| ≤ r

Theorems

Absolute Value Inequality Theorem

Suitable Grade Level

Grades 7-9