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The sets \( B \) and \( C \) are defined as:

\[
B = \{ x \mid x \geq 4 \}
\]
\[
C = \{ x \mid x > 9 \}
\]

We are tasked with finding \( B \cup C \) (union of \( B \) and \( C \)) and \( B \cap C \) (intersection of \( B \) and \( C \)) using interval notation.

### 1. \( B \cup C \) (Union):
The union of two sets includes all elements that are in either set. Since \( B = \{ x \mid x \geq 4 \} \) and \( C = \{ x \mid x > 9 \} \), the union will cover all values starting from \( x = 4 \) (the lower bound of \( B \)) and going to infinity. Hence, the union is:

\[
B \cup C = [4, \infty)
\]

### 2. \( B \cap C \) (Intersection):
The intersection of two sets includes only the elements that are common to both sets. Since \( B \) includes all values greater than or equal to 4, and \( C \) includes all values greater than 9, their intersection will be all values greater than 9. Hence, the intersection is:

\[
B \cap C = (9, \infty)
\]

### Final Answer:
- \( B \cup C = [4, \infty) \)
- \( B \cap C = (9, \infty) \)

Would you like more details or further clarifications?

Here are five follow-up questions:
1. What is the difference between open and closed intervals?
2. How do we determine the union and intersection when sets are defined with inequalities?
3. What happens if two sets have no common elements?
4. How do unions and intersections apply to more than two sets?
5. How do we handle cases where the sets are defined on finite intervals?

**Tip:** In set theory, always pay attention to whether the boundaries are included (closed intervals) or not (open intervals), as this changes the result significantly.

Write B ∪ C and B ∩ C using interval notation. The sets are defined as follows: B = {x | x ≥ 4} and C = {x | x > 9}.

Learn how to express the union and intersection of two sets using interval notation. The sets B and C are defined as B = {x | x ≥ 4} and C = {x | x > 9}. This problem is suitable for high school students in grades 9-12.

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