What is a possible absolute value inequality to represent −1 ≤ x ≤ 5?

To represent the compound inequality \( -1 \leq x \leq 5 \) using absolute value, you can express it as:

\[
|x - 2| \leq 3
\]

### Explanation:
1. The midpoint between \(-1\) and \(5\) is \(2\). This midpoint helps center the absolute value expression.
2. The distance from \(2\) to either \(-1\) or \(5\) is \(3\), which becomes the boundary for the inequality.

Thus, \( |x - 2| \leq 3 \) captures all the values of \(x\) between \(-1\) and \(5\), inclusive.

Would you like a more detailed explanation or have any questions?

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Here are 5 related questions you might want to explore:
1. How do you solve an absolute value inequality like \( |x - 2| \leq 3 \)?
2. What is the geometric interpretation of absolute value inequalities?
3. How do you convert a compound inequality like \( -4 \leq x \leq 6 \) into an absolute value inequality?
4. What changes if the inequality was \( -1 < x < 5 \) instead of including the endpoints?
5. How do you solve absolute value inequalities with strict inequalities (e.g., \( |x - 2| < 3 \))?

**Tip:** Absolute value inequalities are often used to describe ranges of values that lie within a certain distance from a central point.

Learn how to express the compound inequality −1 ≤ x ≤ 5 using an absolute value inequality. This step-by-step guide covers the midpoint method to convert the inequality into |x - 2| ≤ 3, helping students understand the key algebraic concepts. Suitable for Grades 8-10.

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