$$ \begin{aligned} \frac{1}{2}(4x+6)-4x &= \frac{3}{2}(2x-2)&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{1}{2}(4x+6)-2\cdot4x &= 2 \cdot \frac{3}{2}(2x-2)&& \text{cancel out the denominators} \\[1 em]4x+6-8x &= 6x-6&& \text{simplify left side} \\[1 em]-4x+6 &= 6x-6&& \text{move the $ \color{blue}{ 6x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]-4x-6x &= -6-6&& \text{simplify left and right hand side} \\[1 em]-10x &= -12&& \text{ divide both sides by $ -10 $ } \\[1 em]x &= \frac{-12}{-10}&& \\[1 em]x &= \frac{6}{5}&& \\[1 em] \end{aligned} $$

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