$$ \begin{aligned} 4x-\frac{5}{3}+2x &= 7+\frac{2}{9}x+2&& \text{multiply ALL terms by } \color{blue}{ 9 }. \\[1 em]9\cdot4x-9\cdot\frac{5}{3}+9\cdot2x &= 9\cdot7+9 \cdot \frac{2}{9}x+9\cdot2&& \text{cancel out the denominators} \\[1 em]36x-15+18x &= 63+2x+18&& \text{simplify left and right hand side} \\[1 em]54x-15 &= 2x+81&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ -15 }$ to the right} \\[1 em]54x-2x &= 81+15&& \text{simplify left and right hand side} \\[1 em]52x &= 96&& \text{ divide both sides by $ 52 $ } \\[1 em]x &= \frac{96}{52}&& \\[1 em]x &= \frac{24}{13}&& \\[1 em] \end{aligned} $$

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