$|x-\frac{1}{2}| - \sqrt[]{\frac{1}{9}} = \sqrt[]{\frac{1}{4}} $
$⇔ |x-\frac{1}{2}| = \frac{5}{6} $
$⇔ \left[ \begin{array}{l}x-\frac{1}{2}=\frac{5}{6}\\x-\frac{1}{2}=\frac{-5}{6}\end{array} \right.$
$⇔ \left[ \begin{array}{l}x=\frac{4}{3}\\x=\frac{-1}{3}\end{array} \right.$