To solve the inequality \( |x - 2.5| \leq 0.5 \), we can proceed by breaking it down into two cases based on the absolute value definition.

The inequality \( |x - 2.5| \leq 0.5 \) means that the distance between \( x \) and 2.5 is at most 0.5. Therefore, we can rewrite this as a compound inequality:

\[
-0.5 \leq x - 2.5 \leq 0.5
\]

Now, let's solve this inequality step-by-step:

1. Add 2.5 to each part of the inequality:

   \[
   -0.5 + 2.5 \leq x \leq 0.5 + 2.5
   \]

2. Simplify:

   \[
   2 \leq x \leq 3
   \]

So, the solution to the inequality \( |x - 2.5| \leq 0.5 \) is:

\[
2 \leq x \leq 3
\]

In interval notation, this can be written as:

\[
x \in [2, 3]
\]

Would you like more details on any step or have further questions?

Here are some related questions:
1. How would you solve a similar inequality if it was \( |x - 4| < 2 \)?
2. Can you solve \( |x + 1| \geq 3 \) and interpret its solution?
3. How would the solution change if the inequality was \( |x - 2.5| < 0.5 \) instead of \( \leq \)?
4. How does the interval notation help in interpreting the solution set?
5. How do absolute values affect inequalities and solutions in general?

**Tip:** When solving absolute value inequalities, remember that \( |A| \leq B \) translates to \( -B \leq A \leq B \).

Discover how to solve the absolute value inequality |x - 2.5| ≤ 0.5 with a step-by-step breakdown. This solution explains the approach to rewriting the inequality as a compound inequality and solving for x, suitable for students in Grades 9-10.

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