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Let's go through the problem step-by-step.

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### Problem Analysis

The problem provides us with two variables, \( p \) and \( q \), which are integers that satisfy the following conditions:

1. \( 2 \leq p \leq 9 \)
2. \( -5 \leq q < 3 \)

We are tasked with finding:

1. **The greatest possible value of \( q - p \).**
2. **The smallest possible value of \( p^2 + q^2 \).**

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### Solution

#### Part (a): Greatest Possible Value of \( q - p \)

To maximize \( q - p \), we need to maximize \( q \) and minimize \( p \), as this will make \( q - p \) as large as possible.

- The maximum value of \( q \) within the range \( -5 \leq q < 3 \) is \( q = 2 \).
- The minimum value of \( p \) within the range \( 2 \leq p \leq 9 \) is \( p = 2 \).

Therefore:
\[
q - p = 2 - 2 = 0
\]

Now, let's verify if other combinations give a larger value:
- For any increase in \( p \), \( q - p \) will decrease.
- Similarly, reducing \( q \) will also decrease \( q - p \).

Thus, the **greatest possible value of \( q - p \) is 0**.

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#### Part (b): Smallest Possible Value of \( p^2 + q^2 \)

To minimize \( p^2 + q^2 \), we need to choose values of \( p \) and \( q \) that keep both \( p^2 \) and \( q^2 \) as small as possible.

- Since \( p \) ranges from 2 to 9, the smallest value of \( p^2 \) occurs at \( p = 2 \), giving \( p^2 = 4 \).
- For \( q \), since it ranges from -5 to 2, the smallest value of \( q^2 \) occurs at \( q = 0 \), giving \( q^2 = 0 \).

Thus:
\[
p^2 + q^2 = 4 + 0 = 4
\]

Therefore, the **smallest possible value of \( p^2 + q^2 \) is 4**.

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### Final Answers

1. **The greatest possible value of \( q - p \) is 0.**
2. **The smallest possible value of \( p^2 + q^2 \) is 4.**

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Would you like a further breakdown of any of these steps?

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### Follow-Up Questions

1. What if the range for \( p \) or \( q \) were expanded? How would that affect the values?
2. How would the solution change if the inequalities for \( p \) and \( q \) were strict (i.e., \( < \) instead of \( \leq \))?
3. What would be the smallest possible value of \( q - p \) under the given conditions?
4. How could we generalize this method for similar problems with different ranges?
5. What if we were asked for the maximum value of \( p^2 + q^2 \) instead of the minimum?

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### Tip

When optimizing expressions with inequalities, focus on maximizing or minimizing each term individually within the range limits.

Given that p and q are integers such that 2 ≤ p ≤ 9 and -5 ≤ q < 3, find (a) the greatest possible value of q - p, (b) the smallest possible value of p^2 + q^2.

This solution demonstrates how to find the greatest possible value of q - p and the smallest possible value of p^2 + q^2 given integer constraints on p and q. This problem involves integer optimization within specified inequality ranges and is suitable for students in Grades 10-12.

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