$$\frac{x}{x+1}+1 = \frac{1}{0+1}$$

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

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