$$ \begin{aligned} \frac{2}{3}(2x+1)-\frac{4}{9}x &= \frac{2}{9}\cdot(6-x)-\frac{1}{9}&& \text{multiply ALL terms by } \color{blue}{ 9 }. \\[1 em]9 \cdot \frac{2}{3}(2x+1)-9\frac{4}{9}x &= 9 \cdot \frac{2}{9}\cdot(6-x)-9\cdot\frac{1}{9}&& \text{cancel out the denominators} \\[1 em]12x+6-4x &= -2x+12-1&& \text{simplify left and right hand side} \\[1 em]8x+6 &= -2x+11&& \text{move the $ \color{blue}{ -2x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]8x+2x &= 11-6&& \text{simplify left and right hand side} \\[1 em]10x &= 5&& \text{ divide both sides by $ 10 $ } \\[1 em]x &= \frac{5}{10}&& \\[1 em]x &= \frac{1}{2}&& \\[1 em] \end{aligned} $$

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