$$ \begin{aligned} \frac{4}{2}x-1 &= \frac{5}{3}x+1&& \text{multiply ALL terms by } \color{blue}{ 6 }. \\[1 em]6 \cdot \frac{4}{2}x-6\cdot1 &= 6 \cdot \frac{5}{3}x+6\cdot1&& \text{cancel out the denominators} \\[1 em]12x-6 &= 10x+6&& \text{move the $ \color{blue}{ 10x } $ to the left side and $ \color{blue}{ -6 }$ to the right} \\[1 em]12x-10x &= 6+6&& \text{simplify left and right hand side} \\[1 em]2x &= 12&& \text{ divide both sides by $ 2 $ } \\[1 em]x &= \frac{12}{2}&& \\[1 em]x &= 6&& \\[1 em] \end{aligned} $$

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