$$ \begin{aligned} \frac{1}{4}(3x-2)-\frac{1}{3}(2x+3) &= \frac{2}{3}-x&& \text{multiply ALL terms by } \color{blue}{ 12 }. \\[1 em]12 \cdot \frac{1}{4}(3x-2)-12\frac{1}{3}(2x+3) &= 12\cdot\frac{2}{3}-12x&& \text{cancel out the denominators} \\[1 em]9x-6-(8x+12) &= 8-12x&& \text{simplify left and right hand side} \\[1 em]9x-6-8x-12 &= -12x+8&& \\[1 em]x-18 &= -12x+8&& \text{move the $ \color{blue}{ -12x } $ to the left side and $ \color{blue}{ -18 }$ to the right} \\[1 em]x+12x &= 8+18&& \text{simplify left and right hand side} \\[1 em]13x &= 26&& \text{ divide both sides by $ 13 $ } \\[1 em]x &= \frac{26}{13}&& \\[1 em]x &= 2&& \\[1 em] \end{aligned} $$

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