$$ \begin{aligned} \frac{1}{2}(2x-4)+5 &= \frac{2}{3}(x+1)&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{2}(2x-4)+6\cdot5 &= 6 \cdot \frac{2}{3}(x+1)&& \text{cancel out the denominators} \\[1 em]6x-12+30 &= 4x+4&& \text{simplify left side} \\[1 em]6x+18 &= 4x+4&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ 18 }$ to the right} \\[1 em]6x-4x &= 4-18&& \text{simplify left and right hand side} \\[1 em]2x &= -14&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{14}{2}&& \\[1 em]x &= -7&& \\[1 em] \end{aligned} $$

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