Giá trị $x$ thỏa mãn $\dfrac{{11}}{3}:\left( {\dfrac{1}{2}x - \dfrac{5}{3}} \right) + 1 = \dfrac{1}{9}$ là:

$\begin{array}{l}\dfrac{{11}}{3}:\left( {\dfrac{1}{2}x - \dfrac{5}{3}} \right) + 1 = \dfrac{1}{9}\\\dfrac{{11}}{3}:\left( {\dfrac{1}{2}x - \dfrac{5}{3}} \right) = \dfrac{1}{9} - 1\\\dfrac{{11}}{3}:\left( {\dfrac{1}{2}x - \dfrac{5}{3}} \right) = \dfrac{{ - 8}}{9}\\\dfrac{1}{2}x - \dfrac{5}{3} = \dfrac{{11}}{3}:\dfrac{{ - 8}}{9}\\\dfrac{1}{2}x - \dfrac{5}{3} = \dfrac{{11}}{3}.\dfrac{{ - 9}}{8}\\\dfrac{1}{2}x - \dfrac{5}{3} = \dfrac{{ - 33}}{8}\\\dfrac{1}{2}x = \dfrac{{ - 33}}{8} + \dfrac{5}{3}\\\dfrac{1}{2}x = \dfrac{{ - 99}}{{24}} + \dfrac{{40}}{{24}}\\\dfrac{1}{2}x = \dfrac{{ - 59}}{{24}}\\x = \dfrac{{ - 59}}{{24}}:\dfrac{1}{2}\\x = \dfrac{{ - 59}}{{24}}.2\\x = \dfrac{{ - 59}}{{12}}.\end{array}$.

Vậy $x = \dfrac{{ - 59}}{{12}}$.