$$\frac{2}{3}x-1 = 4+\frac{4}{9}x$$

$$ \begin{aligned} \frac{2}{3}x-1 &= 4+\frac{4}{9}x&& \text{multiply ALL terms by } \color{blue}{ 9 }. \\[1 em]9 \cdot \frac{2}{3}x-9\cdot1 &= 9\cdot4+9 \cdot \frac{4}{9}x&& \text{cancel out the denominators} \\[1 em]6x-9 &= 36+4x&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ -9 }$ to the right} \\[1 em]6x-4x &= 36+9&& \text{simplify left and right hand side} \\[1 em]2x &= 45&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{45}{2}&& \\[1 em] \end{aligned} $$

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