$$ \begin{aligned} \frac{1}{3}+\frac{4}{x-2} &= \frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ 3(x-2)\cdot2 }. \\[1 em]3(x-2)\cdot2\cdot\frac{1}{3}+3(x-2)\cdot2\cdot\frac{4}{x-2} &= 3(x-2)\cdot2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]2x-4+24 &= 3x-6&& \text{simplify left side} \\[1 em]2x+20 &= 3x-6&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 20 }$ to the right} \\[1 em]2x-3x &= -6-20&& \text{simplify left and right hand side} \\[1 em]-x &= -26&& \text{switch signs on both sides} \\[1 em]x &= 26&& \\[1 em] \end{aligned} $$

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