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The question asks for the set of numbers that are either less than or equal to 4 or greater than 7.

Let's analyze each option:

1. **Option 1: \(\{x \mid x \leq 4 \text{ or } x > 7\}\)**  
   This describes numbers that are either less than or equal to 4 or greater than 7. This matches the given condition of "less than or equal to 4 or greater than 7."

2. **Option 2: \(\{x \mid x \leq 4 \text{ or } x < 7\}\)**  
   This includes numbers less than or equal to 4 or numbers less than 7. It would include numbers like 6, which doesn't satisfy "greater than 7."

3. **Option 3: \(\{x \mid x < 4 \text{ or } x \geq 7\}\)**  
   This describes numbers less than 4 or greater than or equal to 7. It doesn't include numbers like 4, which should be included since we need "less than or equal to 4."

4. **Option 4: \(\{x \mid x < 4 \text{ and } x \geq 7\}\)**  
   This is a conjunction, meaning it requires numbers to satisfy both conditions at the same time (less than 4 and greater than or equal to 7), which is impossible, so this option is incorrect.

### Conclusion:
The correct answer is **Option 1**: \(\{x \mid x \leq 4 \text{ or } x > 7\}\).

Would you like further details or have any questions?

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Here are five related questions:
1. What is the difference between "and" and "or" in set notation?
2. How do we represent the solution to an inequality on a number line?
3. How do interval and set notation relate when describing ranges of numbers?
4. What does a union of two sets mean in the context of inequalities?
5. Can a number be part of two different intervals at the same time?

**Tip:** In inequalities, "or" expands the solution set, while "and" narrows it down, requiring both conditions to be true simultaneously.

Which is the set of numbers less than or equal to 4 or greater than 7?

Learn how to identify the correct set of numbers that are less than or equal to 4 or greater than 7, with an explanation of set notation and inequalities. Suitable for students in Grades 7-9.

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