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 by first observing the midpoint of the interval \([-1, 5]\). The midpoint is:

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

This means that the interval is symmetric around \(x = 2\), with a total length of \(6\). The absolute value inequality that describes all values of \(x\) within 3 units (half of the total length) of \(2\) is:

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

This inequality describes all values of \(x\) that are at most 3 units away from 2, which matches the range \([-1, 5]\).

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

Here are some related questions you might find useful:
1. How can you solve absolute value inequalities?
2. What is the geometric interpretation of an absolute value inequality?
3. How would you solve \( |x - a| \leq b \) for specific values of \(a\) and \(b\)?
4. How does an absolute value inequality differ from a regular inequality?
5. Can absolute value inequalities have no solution?

**Tip:** When solving absolute value inequalities, consider splitting the inequality into two cases based on the definition of absolute value.

Discover how to represent the inequality −1 ≤ x ≤ 5 using an absolute value inequality. Learn step-by-step how to find the midpoint of the interval and construct the absolute value inequality |x - 2| ≤ 3. This explanation is ideal for students in Grades 9-10 and covers essential algebraic concepts.

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