$$ \begin{aligned} \frac{5}{x}+1 &= \frac{4}{x}-1&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{5}{x}+x\cdot1 &= x\cdot\frac{4}{x}-x\cdot1&& \text{cancel out the denominators} \\[1 em]5+x &= 4-x&& \text{move the $ \color{blue}{ -x } $ to the left side and $ \color{blue}{ 5 }$ to the right} \\[1 em]x+x &= 4-5&& \text{simplify left and right hand side} \\[1 em]2x &= -1&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= -\frac{1}{2}&& \\[1 em] \end{aligned} $$

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