$$ \begin{aligned} 3-\frac{2}{3}(x+2) &= 1&& \text{multiply ALL terms by } \color{blue}{ 3 }. \\[1 em]3\cdot3-3 \cdot \frac{2}{3}(x+2) &= 3\cdot1&& \text{cancel out the denominators} \\[1 em]9-(2x+4) &= 3&& \text{simplify left side} \\[1 em]9-2x-4 &= 3&& \\[1 em]-2x+5 &= 3&& \text{ move 5 to the right } \\[1 em]-2x &= 3-5&& \\[1 em]-2x &= -2&& \text{ divide both sides by $ -2 $ } \\[1 em]x &= \frac{-2}{-2}&& \\[1 em]x &= 1&& \\[1 em] \end{aligned} $$

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