The question asks for the set of numbers that are either less than or equal to 6, or greater than or equal to 12.

This can be expressed in mathematical notation as: $\{ x \,|\, x \leq 6 \, \text{or} \, x \geq 12 \}$

Looking at the choices:

1. $\{ x \,|\, x \leq 6 \, \text{or} \, x \geq 12 \}$ – This is the correct set, matching the condition "less than or equal to 6, or greater than or equal to 12."
2. $\{ x \,|\, x < 6 \, \text{and} \, x > 12 \}$ – This is incorrect because there are no numbers that can be simultaneously less than 6 and greater than 12.
3. $\{ x \,|\, x < 6 \, \text{or} \, x > 12 \}$ – This set is incorrect because it excludes the numbers 6 and 12.
4. $\{ x \,|\, x \leq 6 \, \text{and} \, x \geq 12 \}$ – This is impossible because no number can satisfy both conditions at the same time.

So, the correct answer is the first option: $\{ x \,|\, x \leq 6 \, \text{or} \, x \geq 12 \}$

Would you like more details or have any questions about this?

Here are 5 related questions:

1. How would the set notation change if we wanted numbers strictly less than 6 or strictly greater than 12?
2. What is the difference between "or" and "and" in set notation?
3. Can a number satisfy both $x \leq 6$ and $x \geq 12$ at the same time?
4. How would you represent the set of numbers greater than or equal to 6 but less than or equal to 12?
5. How does the number line representation for this set look?

Tip: When dealing with inequalities in sets, always check whether the conditions are inclusive (≤, ≥) or exclusive (<, >) to avoid errors.