$$ \begin{aligned} \frac{1}{3}(x+2) &= \frac{2}{5}(x+1)&& \text{simplify left and right hand side} \\[1 em]\frac{x+2}{3} &= \frac{2x+2}{5}&& \text{multiply ALL terms by } \color{blue}{ 15 }. \\[1 em]15 \cdot \frac{x+2}{3} &= 15 \cdot \frac{2x+2}{5}&& \text{cancel out the denominators} \\[1 em]5x+10 &= 6x+6&& \text{move the $ \color{blue}{ 6x } $ to the left side and $ \color{blue}{ 10 }$ to the right} \\[1 em]5x-6x &= 6-10&& \text{simplify left and right hand side} \\[1 em]-x &= -4&& \text{switch signs on both sides} \\[1 em]x &= 4&& \\[1 em] \end{aligned} $$

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