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The problem involves solving inequalities to determine the number of sides in a polygonal region formed by them. Here's a step-by-step analysis:

### Inequalities:
1. \( 0 \leq x \leq 2 \) — represents vertical boundaries: \( x = 0 \) and \( x = 2 \).
2. \( 0 \leq y \leq 1 \) — represents horizontal boundaries: \( y = 0 \) and \( y = 1 \).
3. \( y \leq -\frac{1}{2}x + \frac{3}{2} \) — represents a slanted line.

### Steps:
1. **Interpretation**: The region will be constrained by the lines \( x = 0 \), \( x = 2 \), \( y = 0 \), \( y = 1 \), and \( y = -\frac{1}{2}x + \frac{3}{2} \).
2. **Find Intersections**:
   - \( x = 0 \): Substitute into \( y = -\frac{1}{2}x + \frac{3}{2} \):
     \[
     y = -\frac{1}{2}(0) + \frac{3}{2} = \frac{3}{2}.
     \]
     However, \( y = 1 \) is the upper bound, so use \( y = 1 \) instead.
   - \( y = 0 \): Substitute into \( y = -\frac{1}{2}x + \frac{3}{2} \):
     \[
     0 = -\frac{1}{2}x + \frac{3}{2} \implies x = 3.
     \]
     But \( x = 2 \) is the boundary, so use \( x = 2 \).
   - Intersection of \( y = -\frac{1}{2}x + \frac{3}{2} \) and \( y = 1 \):
     \[
     1 = -\frac{1}{2}x + \frac{3}{2} \implies x = 1.
     \]
   - Intersection of \( y = -\frac{1}{2}x + \frac{3}{2} \) and \( y = 0 \):
     \[
     0 = -\frac{1}{2}x + \frac{3}{2} \implies x = 3.
     \]
3. **Vertices**: The vertices of the polygon are:
   - \( (0, 0) \) from \( x = 0, y = 0 \),
   - \( (2, 0) \) from \( x = 2, y = 0 \),
   - \( (2, 1) \) from \( x = 2, y = 1 \),
   - \( (1, 1) \) from \( y = -\frac{1}{2}x + \frac{3}{2}, y = 1 \),
   - \( (0, 1) \) from \( x = 0, y = 1 \).

### Conclusion:
The polygon has **5 sides**. The correct answer is **D (5)**.

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Would you like a diagram for this solution or further clarification?  
Here are 5 related questions to expand your understanding:  
1. What is the general method to solve systems of inequalities graphically?  
2. How do you find the intersection of two lines algebraically?  
3. What determines the shape of a polygon formed by inequalities?  
4. Can you identify if the polygon is convex or concave?  
5. How would the polygon change if \( y \geq -\frac{1}{2}x + \frac{3}{2} \) instead?  

**Tip**: Always start by graphing each inequality to visualize the solution region.

The solutions to the inequalities 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, and y ≤ -1/2 x + 3/2 form a polygonal region in the plane. How many sides does this polygon have?

Explore the solution to a polygonal region formed by solving 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, and y ≤ -1/2 x + 3/2. Learn how to determine the number of sides, calculate vertices, and understand graphing of inequalities, suitable for Grades 9-12.

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