We need to determine which one of the given numbers is not rational. Let's analyze each number step by step.
1. [tex]\(-5 \frac{4}{11}\)[/tex]:
This number can be written as a mixed number: [tex]\(-5 + \frac{4}{11}\)[/tex]. Any mixed number can be expressed as a fraction of two integers. Specifically, [tex]\(-5 + \frac{4}{11}\)[/tex] equals to [tex]\(\frac{-55 + 4}{11} = \frac{-51}{11}\)[/tex], which is a ratio of two integers. Therefore, [tex]\(-5 \frac{4}{11}\)[/tex] is a rational number.
2. [tex]\(\sqrt{31}\)[/tex]:
The square root of 31 is an irrational number because 31 is not a perfect square. An irrational number is a number that cannot be exactly expressed as a ratio of two integers. Therefore, [tex]\(\sqrt{31}\)[/tex] is not a rational number.
3. 7.608:
This number can be written as a decimal. Any finite decimal can be expressed as a fraction of two integers. Specifically, 7.608 equals [tex]\(\frac{7608}{1000}\)[/tex], which can be simplified to a ratio of two integers. Therefore, 7.608 is a rational number.
4. [tex]\(18.4 \overline{6}\)[/tex]:
This is a repeating decimal where the digit 6 repeats indefinitely. Repeating decimals can always be expressed as fractions of two integers. Hence, [tex]\(18.4 \overline{6}\)[/tex] is a rational number.
In conclusion, the number that is not rational is:
[tex]\[
\boxed{\sqrt{31}}
\][/tex]
