$$ \begin{aligned} \frac{3}{2}x+1 &= x+4&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{3}{2}x+2\cdot1 &= 2x+2\cdot4&& \text{cancel out the denominators} \\[1 em]3x+2 &= 2x+8&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ 2 }$ to the right} \\[1 em]3x-2x &= 8-2&& \text{simplify left and right hand side} \\[1 em]x &= 6&& \\[1 em]x &= 6&& \\[1 em] \end{aligned} $$

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