$$ \begin{aligned} \frac{5}{2}x+1-\frac{x}{4} &= 0&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4 \cdot \frac{5}{2}x+4\cdot1-4\frac{x}{4} &= 4\cdot0&& \text{cancel out the denominators} \\[1 em]10x+4-x &= 0&& \text{simplify left side} \\[1 em]9x+4 &= 0&& \text{ move the constants to the right } \\[1 em]9x &= -4&& \text{ divide both sides by $ 9 $ } \\[1 em]x &= -\frac{4}{9}&& \\[1 em] \end{aligned} $$

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