$$ \begin{aligned} 2x-\frac{x}{2}+\frac{x+1}{4} &= 9x&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4\cdot2x-4 \cdot \frac{x}{2}+4\frac{x+1}{4} &= 4\cdot9x&& \text{cancel out the denominators} \\[1 em]8x-2x+x+1 &= 36x&& \text{simplify left side} \\[1 em]6x+x+1 &= 36x&& \\[1 em]7x+1 &= 36x&& \text{move the $ \color{blue}{ 36x } $ to the left side and $ \color{blue}{ 1 }$ to the right} \\[1 em]7x-36x &= -1&& \text{simplify left side} \\[1 em]-29x &= -1&& \text{ divide both sides by $ -29 $ } \\[1 em]x &= \frac{-1}{-29}&& \\[1 em]x &= \frac{1}{29}&& \\[1 em] \end{aligned} $$

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