The statement "the distance from \( p \) to 3 is at most 5" can be expressed mathematically using the concept of absolute value. 

The distance between \( p \) and 3 can be represented by \( |p - 3| \). If this distance is at most 5, we can write the inequality:

\[
|p - 3| \leq 5
\]

This inequality means that the distance between \( p \) and 3 is 5 units or less on the number line.

To solve this inequality:

1. Split the absolute value inequality into two separate inequalities:
   \[
   -5 \leq p - 3 \leq 5
   \]
2. Add 3 to all parts of the inequality to solve for \( p \):
   \[
   -5 + 3 \leq p \leq 5 + 3
   \]
   \[
   -2 \leq p \leq 8
   \]

So, \( p \) must be in the range \([-2, 8]\).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve absolute value inequalities with more complex expressions?
2. What is the geometric interpretation of the absolute value on the number line?
3. How would the inequality change if the distance was "at least" 5 instead of "at most" 5?
4. Can you solve for \( p \) if the distance from \( p \) to another number, say 7, is at most 10?
5. How does the concept of distance in absolute value apply to functions in calculus?

**Tip:** When solving absolute value inequalities, always consider both the positive and negative cases to capture the entire solution set.

Explore how to solve the inequality |p - 3| ≤ 5, where p represents a number on the number line. This problem involves understanding absolute value, inequalities, and their applications. Suitable for Grades 7-9.

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