First, find the derivative \(f'(x)\):

\(f(x) = \frac{8}{x-2} + 2\)

\(f'(x) = \frac{-8}{(x-2)^2}\)

Next, find the inverse function \(f^{-1}(x)\):

Let \(y = \frac{8}{x-2} + 2\)

\(y - 2 = \frac{8}{x-2}\)

\(x - 2 = \frac{8}{y-2}\)

\(f^{-1}(x) = \frac{8}{x-2} + 2\)

Substitute into the inequality:

\(6f'(x) + 2f^{-1}(x) - 5 < 0\)

\(6 \left( \frac{-8}{(x-2)^2} \right) + 2 \left( \frac{8}{x-2} + 2 \right) - 5 < 0\)

\(\frac{-48}{(x-2)^2} + \frac{16}{x-2} + 4 - 5 < 0\)

\(\frac{-48}{(x-2)^2} + \frac{16}{x-2} - 1 < 0\)

Multiply through by \((x-2)^2\):

\(-48 + 16(x-2) - (x-2)^2 < 0\)

\(-48 + 16x - 32 - (x^2 - 4x + 4) < 0\)

\(-48 + 16x - 32 - x^2 + 4x - 4 < 0\)

\(x^2 - 20x + 84 < 0\)

Factor the quadratic:

\((x-6)(x-14) < 0\)

Find the intervals where the inequality holds:

\(2 < x < 6, \; x > 14\)