https://file.aiquickdraw.com/mathbot/file/1728322655464wk1huq1a.jpg

Let's break down the exercise from the image and solve each part.

### Information from the image:
- **Given sets:**
  - \( x = [2, \infty) \) (the set of numbers greater than or equal to 2)
  - \( y = [-2, 3] \) (the set of numbers between -2 and 3, inclusive)

- **Instructions:**
  You are asked to find the following on the number line:
  1. \( x \cap y \) (the intersection of \( x \) and \( y \))
  2. \( x \cup y \) (the union of \( x \) and \( y \))
  3. \( x - y \) (the difference between \( x \) and \( y \), meaning elements in \( x \) but not in \( y \))
  4. \( y - x \) (the difference between \( y \) and \( x \), meaning elements in \( y \) but not in \( x \))

### Solutions:

1. **\( x \cap y \) (intersection of \( x \) and \( y \)):**
   - \( x = [2, \infty) \) and \( y = [-2, 3] \).
   - The intersection is the overlap between the two intervals.
   - Since \( x \) starts at 2 and \( y \) ends at 3, the intersection is \( [2, 3] \).

   \[
   x \cap y = [2, 3]
   \]

2. **\( x \cup y \) (union of \( x \) and \( y \)):**
   - The union includes all elements in both sets.
   - \( x = [2, \infty) \) and \( y = [-2, 3] \), so the union covers from \( -2 \) to infinity.
   
   \[
   x \cup y = [-2, \infty)
   \]

3. **\( x - y \) (elements in \( x \) but not in \( y \)):**
   - \( x = [2, \infty) \) and \( y = [-2, 3] \).
   - The part of \( x \) that is not in \( y \) is from 3 to infinity, as \( y \) covers \( [-2, 3] \).
   
   \[
   x - y = (3, \infty)
   \]

4. **\( y - x \) (elements in \( y \) but not in \( x \)):**
   - \( x = [2, \infty) \) and \( y = [-2, 3] \).
   - The part of \( y \) that is not in \( x \) is from \( -2 \) to 2, since \( x \) starts at 2.
   
   \[
   y - x = [-2, 2)
   \]

Would you like any additional details or explanations on these results?

### Follow-up Questions:
1. What does the union of two sets represent on a number line?
2. How do you interpret set differences visually on a number line?
3. Can two sets have an empty intersection? What does it signify?
4. How would you calculate \( x \cap y \) if both intervals were open (e.g., \( (2, \infty) \) and \( (-2, 3) \))?
5. How does set notation change when dealing with finite versus infinite sets?

### Tip:
When calculating intersections, always focus on the overlap between intervals to ensure you capture the common region.

If x = [2, ∞) and y = [-2, 3], find the following using the number line: x ∩ y, x ∪ y, x - y, y - x.

Learn how to find the intersection, union, and differences between the sets x = [2, ∞) and y = [-2, 3] using the number line. This problem covers essential set theory concepts like interval notations and set operations, ideal for Grades 10-12.

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