$$ \begin{aligned} \frac{x}{2}+\frac{x}{5} &= \frac{x}{6}-\frac{1}{3}&& \text{multiply ALL terms by } \color{blue}{ 30 }. \\[1 em]30 \cdot \frac{x}{2}+30\frac{x}{5} &= 30 \cdot \frac{x}{6}-30\cdot\frac{1}{3}&& \text{cancel out the denominators} \\[1 em]15x+6x &= 5x-10&& \text{simplify left side} \\[1 em]21x &= 5x-10&& \text{move the $ \color{blue}{ 5x } $ to the left side} \\[1 em]21x-5x &= -10&& \text{simplify left side} \\[1 em]16x &= -10&& \text{ divide both sides by $ 16 $ } \\[1 em]x &= -\frac{10}{16}&& \\[1 em]x &= -\frac{5}{8}&& \\[1 em] \end{aligned} $$

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