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The problem involves two sets of real numbers, \( F \) and \( H \), defined as follows:

- \( F = \{x \mid x \geq 2\} \)
- \( H = \{y \mid y \leq 9\} \)

You are asked to:

1. Write \( F \cup H \) (union of \( F \) and \( H \)) and \( F \cap H \) (intersection of \( F \) and \( H \)) using interval notation.
2. Find the set \( F \setminus H \) (set difference of \( F \) and \( H \)).

### Step 1: Interval notation for \( F \) and \( H \)

- \( F = [2, \infty) \), because \( x \geq 2 \).
- \( H = (-\infty, 9] \), because \( y \leq 9 \).

### Step 2: Union of \( F \) and \( H \)

The union of two sets contains all elements that belong to either set. In interval notation, this is:

- \( F \cup H = (-\infty, \infty) \), because \( F \) includes all numbers greater than or equal to 2, and \( H \) includes all numbers less than or equal to 9. Together, these cover the entire real number line.

### Step 3: Intersection of \( F \) and \( H \)

The intersection of two sets contains all elements that belong to both sets. In this case, the intersection is the overlap between \( F = [2, \infty) \) and \( H = (-\infty, 9] \), which is the set of numbers between 2 and 9, inclusive:

- \( F \cap H = [2, 9] \).

### Step 4: Set difference \( F \setminus H \)

The set difference \( F \setminus H \) contains all elements in \( F \) that are not in \( H \). Since \( H = (-\infty, 9] \), removing the elements of \( H \) from \( F = [2, \infty) \) leaves:

- \( F \setminus H = (9, \infty) \), because only elements greater than 9 remain from \( F \).

### Final Answer

1. \( F \cup H = (-\infty, \infty) \)
2. \( F \cap H = [2, 9] \)
3. \( F \setminus H = (9, \infty) \)

Would you like more details or further explanations?

Here are five related questions:

1. What is the union of \( F \) and \( H \) if \( H = (0, 5] \)?
2. How would the answer change if \( F = (3, \infty) \) and \( H = (-\infty, 7) \)?
3. Can the intersection of two disjoint sets ever be non-empty? Explain.
4. What is the complement of set \( F \) in the real numbers?
5. How would you graph the union and intersection of \( F \) and \( H \)?

**Tip**: When working with intervals, visualize them on a number line to better understand the union and intersection operations.

F and H are sets of real numbers defined as follows:

F = {x | x >= 2}
H = {y | y <= 9}

Write F ∪ H and F ∩ H using interval notation.
Find the set F \ H.

Learn how to find the union (F ∪ H), intersection (F ∩ H), and set difference (F \ H) of the real number sets F and H using interval notation. Suitable for students in Grades 10-12.

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