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

$$ \begin{aligned} \frac{2}{3}-3 &= \frac{4}{x+1}&& \text{multiply ALL terms by } \color{blue}{ 3(x+1) }. \\[1 em]3(x+1)\cdot\frac{2}{3}-3(x+1)\cdot3 &= 3(x+1)\cdot\frac{4}{x+1}&& \text{cancel out the denominators} \\[1 em]2x+2-(9x+9) &= 12&& \text{simplify left side} \\[1 em]2x+2-9x-9 &= 12&& \\[1 em]-7x-7 &= 12&& \text{ move -7 to the right } \\[1 em]-7x &= 12+7&& \\[1 em]-7x &= 19&& \text{ divide both sides by $ -7 $ } \\[1 em]x &= -\frac{19}{7}&& \\[1 em] \end{aligned} $$

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