$$ \begin{aligned} \frac{x}{2}-\frac{2}{3} &= x+\frac{1}{5}&& \text{multiply ALL terms by } \color{blue}{ 30 }. \\[1 em]30 \cdot \frac{x}{2}-30\cdot\frac{2}{3} &= 30x+30\cdot\frac{1}{5}&& \text{cancel out the denominators} \\[1 em]15x-20 &= 30x+6&& \text{move the $ \color{blue}{ 30x } $ to the left side and $ \color{blue}{ -20 }$ to the right} \\[1 em]15x-30x &= 6+20&& \text{simplify left and right hand side} \\[1 em]-15x &= 26&& \text{ divide both sides by $ -15 $ } \\[1 em]x &= -\frac{26}{15}&& \\[1 em] \end{aligned} $$

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