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The problem presented involves finding the solution set for the following compound inequalities:

\[
(x \geq 2 \text{ and } x < 5) \text{ and } (x > 3 \text{ and } x \leq 8)
\]

### Step-by-Step Solution:

1. **Inequality 1: \( x \geq 2 \text{ and } x < 5 \)**
   - This represents the interval \( [2, 5) \), meaning \( x \) is greater than or equal to 2 but less than 5.

2. **Inequality 2: \( x > 3 \text{ and } x \leq 8 \)**
   - This represents the interval \( (3, 8] \), meaning \( x \) is greater than 3 and less than or equal to 8.

3. **Combining the Inequalities:**
   - The solution requires that both inequalities be satisfied simultaneously.
   - The first inequality limits \( x \) to the interval \( [2, 5) \), and the second limits \( x \) to \( (3, 8] \).
   - To satisfy both conditions, we need the overlap (intersection) of these intervals.

4. **Intersection of \( [2, 5) \) and \( (3, 8] \):**
   - \( [2, 5) \) and \( (3, 8] \) overlap in the interval \( (3, 5) \).
   
   Therefore, the solution is:

   \[
   \boxed{(3, 5)}
   \]

This is the set of values for \( x \) that satisfy both inequalities.

Would you like further details on any part of the solution or clarification on how the intervals were determined?

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### Related Questions:
1. How do you find the intersection of two sets of inequalities?
2. What is the graphical representation of this solution on a number line?
3. Can you explain how compound inequalities are solved in general?
4. What happens if there is no overlap between two inequalities?
5. How do inequalities change when you multiply or divide by a negative number?

### Tip:
When solving compound inequalities, always visualize the intervals on a number line to find their intersection easily.

(x ≥ 2 and x < 5) and (x > 3 and x ≤ 8)

This problem involves solving the compound inequality (x ≥ 2 and x < 5) and (x > 3 and x ≤ 8). Learn how to find the intersection of intervals and express the solution in interval notation. Suitable for Grades 6-8 students.

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