$$ \begin{aligned} \frac{1}{3}-\frac{2}{3}(x+1) &= \frac{1}{6}&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6\cdot\frac{1}{3}-6\frac{2}{3}(x+1) &= 6\cdot\frac{1}{6}&& \text{cancel out the denominators} \\[1 em]2-(4x+4) &= 1&& \text{simplify left side} \\[1 em]2-4x-4 &= 1&& \\[1 em]-4x-2 &= 1&& \text{ move -2 to the right } \\[1 em]-4x &= 1+2&& \\[1 em]-4x &= 3&& \text{ divide both sides by $ -4 $ } \\[1 em]x &= -\frac{3}{4}&& \\[1 em] \end{aligned} $$

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