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Let's analyze and solve each part of the question:

### Given Information:
- \( p \) and \( q \) are integers.
- The range for \( p \) is \( 2 \leq p \leq 9 \).
- The range for \( q \) is \( -5 \leq q < 3 \).

### Part (a)
**Find the greatest possible value of \( q - p \):**

To maximize \( q - p \), we want \( q \) to be as large as possible and \( p \) to be as small as possible, as this will maximize the difference \( q - p \).

- The maximum value for \( q \) is \( 2 \).
- The minimum value for \( p \) is \( 2 \).

So,
\[
q - p = 2 - 2 = 0
\]

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

### Part (b)
**Find the smallest possible value of \( p^2 + q^2 \):**

To minimize \( p^2 + q^2 \), we need to choose values of \( p \) and \( q \) that make \( p^2 \) and \( q^2 \) as small as possible, as the expression \( p^2 + q^2 \) involves the squares of \( p \) and \( q \).

- The smallest value for \( p \) within its range that gives a minimum square is \( p = 2 \), so \( p^2 = 4 \).
- The smallest value for \( q \) within its range that gives a minimum square is \( q = 0 \), so \( q^2 = 0 \).

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

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

### Summary of Answers:
- **(a)** The greatest possible value of \( q - p \) is \( 0 \).
- **(b)** The smallest possible value of \( p^2 + q^2 \) is \( 4 \).

Would you like further details or have any questions?

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Here are some related questions to deepen understanding:

1. What would happen to \( q - p \) if the range of \( q \) was extended beyond \( 3 \)?
2. How would the minimum value of \( p^2 + q^2 \) change if \( p \) or \( q \) had additional constraints?
3. What are the effects of expanding the range of \( p \) on the smallest possible value of \( p^2 + q^2 \)?
4. Can we have a negative value for \( q - p \) given the current range of \( p \) and \( q \)?
5. How would the calculations change if \( p \) and \( q \) were not restricted to integers?

**Tip:** When maximizing or minimizing an expression involving multiple variables, consider the extremes of each variable's range.

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, and (b) the smallest possible value of p^2 + q^2.

Learn how to find the greatest possible value of q - p and the smallest possible value of p^2 + q^2 for integers p and q, where 2 ≤ p ≤ 9 and -5 ≤ q < 3. This solution involves integer optimization and algebraic calculations, making it ideal for Grades 8-10 students.

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