$$ \begin{aligned} \frac{9}{2}(x+1)-2 &= 4(x+3)&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{9}{2}(x+1)-2\cdot2 &= 2\cdot4(x+3)&& \text{cancel out the denominators} \\[1 em]9x+9-4 &= 8x+24&& \text{simplify left side} \\[1 em]9x+5 &= 8x+24&& \text{move the $ \color{blue}{ 8x } $ to the left side and $ \color{blue}{ 5 }$ to the right} \\[1 em]9x-8x &= 24-5&& \text{simplify left and right hand side} \\[1 em]x &= 19&& \\[1 em]x &= 19&& \\[1 em] \end{aligned} $$

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