Since \(x\) is directly proportional to \(y\) and inversely proportional to \(z\), we can write an equation of the form:

\(x = \frac{ky}{z}\), where \(k\) is a constant of proportionality.

We are told that \(x = 9\) when \(y = 24\) and \(z = 8\), so we can use this information to find the value of \(k\): \(9 = \frac{k \times 24}{8}\). Solving for \(k\), we get \(k = \frac{9 \times 8}{24} = 3\).

Now that we know the value of \(k\), we can use it to find the value of \(x\) when \(y = 5\) and \(z = 6\):

\(x = \frac{3 \times 5}{6} = \frac{15}{6} = 2\frac{1}{2}\).

Therefore, when \(y = 5\) and \(z = 6\), the value of \(x\) is \(2\frac{1}{2}\).