$$\frac{x}{x+2}+\frac{1}{x-1} = 1$$

$$ \begin{aligned} \frac{x}{x+2}+\frac{1}{x-1} &= 1&& \text{multiply ALL terms by } \color{blue}{ (x+2)(x-1) }. \\[1 em](x+2)(x-1)\frac{x}{x+2}+(x+2)(x-1)\cdot\frac{1}{x-1} &= (x+2)(x-1)\cdot1&& \text{cancel out the denominators} \\[1 em]x^2-x+x+2 &= x^2+x-2&& \text{simplify left side} \\[1 em]x^2+2 &= x^2+x-2&& \text{move all terms to the left hand side } \\[1 em]x^2+2-x^2-x+2 &= 0&& \text{simplify left side} \\[1 em]x^2+2-x^2-x+2 &= 0&& \\[1 em]-x+4 &= 0&& \text{ move the constants to the right } \\[1 em]-x &= -4&& \text{switch signs on both sides} \\[1 em]x &= 4&& \\[1 em] \end{aligned} $$

This page was created using