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


The inequality \( -1 \leq x \leq 5 \) can be represented as an absolute value inequality.

First, recognize that this inequality can be rewritten in a centered form around some midpoint. The midpoint between \(-1\) and \(5\) is:

\[
\text{Midpoint} = \frac{-1 + 5}{2} = 2
\]

Now, the distance from the midpoint \(2\) to either endpoint (such as \(-1\) or \(5\)) is:

\[
\text{Distance} = 5 - 2 = 3
\]

This leads to the absolute value inequality:

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

This inequality says that the distance between \(x\) and \(2\) is at most \(3\), which is equivalent to the original inequality \( -1 \leq x \leq 5 \).

### Would you like further details or have any questions?

Here are 5 related questions:

1. How do you solve absolute value inequalities?
2. What is the difference between strict and non-strict inequalities?
3. Can an absolute value inequality ever represent two separate intervals?
4. How would the inequality change if it were \( -1 < x < 5 \)?
5. How does graphing absolute value inequalities work?

**Tip**: Always check if you can express a compound inequality as an absolute value inequality—it can simplify solving the problem.

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

Discover how to represent the inequality −1 ≤ 𝑥 ≤ 5 as an absolute value inequality. This guide walks through the concept of midpoints and distances in absolute value inequalities. Suitable for students in Grades 9-11.

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