The value(s) of x satisfying the equation  $x^9+\frac{9}{8}{x}^6+\frac{27}{64}{x}^3-x+\frac{219}{512}=0$  is

 $$\begin{gathered} x^9+\frac{9}{8}{x}^6+\frac{27}{64}{x}^3-x+\frac{219}{512}=0 \\ {(x^3+\frac{3}{8})}^3=x-\frac{3}{8} \\ \Rightarrow x^3+\frac{3}{8}={(x-\frac{3}{8})}^{1/3} \\ f\left(x\right)=f^{-1}\left(x\right) \end{gathered}$$
Solutions will be along f (x) = x
$$\begin{gathered} x^3+\frac{3}{8}=x \\ x=\frac{-1+\sqrt{13}}{4},\frac{-1-\sqrt{13}}{4},\frac{1}{2} \end{gathered}$$