$$ \begin{aligned} \frac{1}{2}x+\frac{8}{3}x &= x+52&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{1}{2}x+6\frac{8}{3}x &= 6x+6\cdot52&& \text{cancel out the denominators} \\[1 em]3x+16x &= 6x+312&& \text{simplify left side} \\[1 em]19x &= 6x+312&& \text{move the $ \color{blue}{ 6x } $ to the left side} \\[1 em]19x-6x &= 312&& \text{simplify left side} \\[1 em]13x &= 312&& \text{ divide both sides by $ 13 $ } \\[1 em]x &= \frac{312}{13}&& \\[1 em]x &= 24&& \\[1 em] \end{aligned} $$

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