using ratinoal approximations, what statement is true?

12−−√ < n
16−−√ < 4
16−−√ > 4
12−−√ > n

1 answer

To analyze the inequalities involving rational approximations, we need to find the approximate values of the square roots involved.

  1. \( \sqrt{12} \) is approximately \( 3.464 \) (since \( 3.464^2 \approx 12 \)).
  2. \( \sqrt{16} = 4 \).

Now, let's evaluate the given statements:

  1. \( \sqrt{12} < n \) - This means \( n \) should be greater than approximately \( 3.464 \).
  2. \( \sqrt{16} < 4 \) - This statement is false because \( \sqrt{16} = 4 \).
  3. \( \sqrt{16} > 4 \) - This statement is false because \( \sqrt{16} = 4 \).
  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 \( n \).