$$2(x+1)-3 = x+\frac{1}{2}$$

$$ \begin{aligned} 2(x+1)-3 &= x+\frac{1}{2}&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2\cdot2(x+1)-2\cdot3 &= 2x+2\cdot\frac{1}{2}&& \text{cancel out the denominators} \\[1 em]4x+4-6 &= 2x+1&& \text{simplify left side} \\[1 em]4x-2 &= 2x+1&& \text{move the $ \color{blue}{ 2x } $ to the left side and $ \color{blue}{ -2 }$ to the right} \\[1 em]4x-2x &= 1+2&& \text{simplify left and right hand side} \\[1 em]2x &= 3&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{3}{2}&& \\[1 em] \end{aligned} $$

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