$$\frac{1}{x}+1+\frac{1}{x}+3 = \frac{1}{x}+2+\frac{1}{x}+4+\frac{1}{x}+5$$

$$ \begin{aligned} \frac{1}{x}+1+\frac{1}{x}+3 &= \frac{1}{x}+2+\frac{1}{x}+4+\frac{1}{x}+5&& \text{multiply ALL terms by } \color{blue}{ x }. \\[1 em]x\cdot\frac{1}{x}+x\cdot1+x\cdot\frac{1}{x}+x\cdot3 &= x\cdot\frac{1}{x}+x\cdot2+x\cdot\frac{1}{x}+x\cdot4+x\cdot\frac{1}{x}+x\cdot5&& \text{cancel out the denominators} \\[1 em]1+x+1+3x &= 1+2x+1+4x+1+5x&& \text{simplify left and right hand side} \\[1 em]4x+2 &= 11x+3&& \text{move the $ \color{blue}{ 11x } $ to the left side and $ \color{blue}{ 2 }$ to the right} \\[1 em]4x-11x &= 3-2&& \text{simplify left and right hand side} \\[1 em]-7x &= 1&& \text{ divide both sides by $ -7 $ } \\[1 em]x &= -\frac{1}{7}&& \\[1 em] \end{aligned} $$

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