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

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 $p^2 + q^2$.

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.

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.

Final Answers

1. The greatest possible value of $q - p$ is 0.
2. The smallest possible value of $p^2 + q^2$ is 4.

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

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?

Tip

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