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?

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.