\(\frac{1}{6}:\left(x-\frac{1}{3}\right)-\frac{5}{6}=\frac{5}{9}\times2\frac{8}{6}\)
\(\Leftrightarrow\frac{1}{6}:\left(\frac{3x-1}{3}\right)=\frac{5}{6}+\frac{5}{9}\times2\frac{4}{3}\)
\(\Leftrightarrow\frac{1}{6}\times\frac{3}{3x-1}=\frac{5}{6}+\frac{5}{9}\times\frac{10}{3}\)
\(\Leftrightarrow\frac{1}{2}\times\frac{1}{3x-1}=\frac{45+100}{54}\)
\(\Leftrightarrow\frac{1}{3x-1}=\frac{2\cdot145}{54}=\frac{145}{27}\)
\(\Leftrightarrow3x-1=\frac{27}{145}\Leftrightarrow3x=\frac{27}{145}+1=\frac{172}{145}\)
\(\Leftrightarrow x=\frac{172}{435}\).