$$x+\frac{5}{8} = \frac{x}{4}+\frac{1}{3}$$

$$ \begin{aligned} x+\frac{5}{8} &= \frac{x}{4}+\frac{1}{3}&& \text{multiply ALL terms by } \color{blue}{ 24 }. \\[1 em]24x+24\cdot\frac{5}{8} &= 24 \cdot \frac{x}{4}+24\cdot\frac{1}{3}&& \text{cancel out the denominators} \\[1 em]24x+15 &= 6x+8&& \text{move the $ \color{blue}{ 6x } $ to the left side and $ \color{blue}{ 15 }$ to the right} \\[1 em]24x-6x &= 8-15&& \text{simplify left and right hand side} \\[1 em]18x &= -7&& \text{ divide both sides by $ 18 $ } \\[1 em]x &= -\frac{7}{18}&& \\[1 em] \end{aligned} $$

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