$$\frac{1}{2}+\frac{3}{x}-\frac{1}{x^2} = \frac{1}{4x}+\frac{1}{2x^2}$$

$$ \begin{aligned} \frac{1}{2}+\frac{3}{x}-\frac{1}{x^2} &= \frac{1}{4x}+\frac{1}{2x^2}&& \text{multiply ALL terms by } \color{blue}{ 2xx^2\cdot4 }. \\[1 em]2xx^2\cdot4\cdot\frac{1}{2}+2xx^2\cdot4\cdot\frac{3}{x}-2xx^2\cdot4\cdot\frac{1}{x^2} &= 2xx^2\cdot4\cdot\frac{1}{4x}+2xx^2\cdot4\cdot\frac{1}{2x^2}&& \text{cancel out the denominators} \\[1 em]4x+24-\frac{8}{x^1} &= 2+4x&& \text{multiply ALL terms by } \color{blue}{ x^1 }. \\[1 em]x^1\cdot4x+x^1\cdot24-x^1\cdot\frac{8}{x^1} &= x^1\cdot2+x^1\cdot4x&& \text{cancel out the denominators} \\[1 em]4x^2+24x-8 &= 2x+4x^2&& \text{simplify right side} \\[1 em]4x^2+24x-8 &= 4x^2+2x&& \text{move all terms to the left hand side } \\[1 em]4x^2+24x-8-4x^2-2x &= 0&& \text{simplify left side} \\[1 em]4x^2+24x-8-4x^2-2x &= 0&& \\[1 em]22x-8 &= 0&& \text{ move the constants to the right } \\[1 em]22x &= 8&& \text{ divide both sides by $ 22 $ } \\[1 em]x &= \frac{8}{22}&& \\[1 em]x &= \frac{4}{11}&& \\[1 em] \end{aligned} $$

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