To analyze the inequalities involving rational approximations, we need to find the approximate values of the square roots involved.
- \( \sqrt{12} \) is approximately \( 3.464 \) (since \( 3.464^2 \approx 12 \)).
- \( \sqrt{16} = 4 \).
Now, let's evaluate the given statements:
- \( \sqrt{12} < n \) - This means \( n \) should be greater than approximately \( 3.464 \).
- \( \sqrt{16} < 4 \) - This statement is false because \( \sqrt{16} = 4 \).
- \( \sqrt{16} > 4 \) - This statement is false because \( \sqrt{16} = 4 \).
- \( \sqrt{12} > n \) - This means \( n \) should be less than approximately \( 3.464 \).
Depending on the value of \( n \), the following are possible conclusions:
- If \( n < 3.464 \), then \( \sqrt{12} > n \).
- If \( n > 3.464 \), then \( \sqrt{12} < n \).
Given that \( \sqrt{16} = 4 \), options 2 and 3 can be immediately ruled out.
Therefore, if we are looking for a true statement with respect to \( n \) only, we conclude that if \( n \) is not defined, both \( \sqrt{12} < n \) and \( \sqrt{12} > n \) can be true depending on the value of \( n \). However, the only absolutely false statements are:
- \( \sqrt{16} < 4 \) (false)
- \( \sqrt{16} > 4 \) (false)
So based on rational approximations and the values we calculated:
- \( \sqrt{12} < n \) or \( \sqrt{12} > n \) can be true depending on \( n \).
- The false statements are the ones involving the square root of 16 compared with 4.
Thus, the only definitively true statement when comparing \( \sqrt{12} \) and a variable \( n \), is conditional:
- \( \sqrt{12} > n \) if \( n < 3.464\)
- \( \sqrt{12} < n \) if \( n > 3.464\)
Without specific values for \( n \), we cannot conclusively say one is always true.
To summarize, the only non-conditional true statement is about \( \sqrt{16} \):
- False statements: \( \sqrt{16} < 4 \) and \( \sqrt{16} > 4 \).
- Conditional on \( n \): \( \sqrt{12} \) could be less than or greater than \( n \).