$$ \begin{aligned} 4x-\frac{1}{5} &= 2x+1&& \text{multiply ALL terms by } \color{blue}{ 5 }. \\[1 em]5\cdot4x-5\cdot\frac{1}{5} &= 5\cdot2x+5\cdot1&& \text{cancel out the denominators} \\[1 em]20x-1 &= 10x+5&& \text{move the $ \color{blue}{ 10x } $ to the left side and $ \color{blue}{ -1 }$ to the right} \\[1 em]20x-10x &= 5+1&& \text{simplify left and right hand side} \\[1 em]10x &= 6&& \text{ divide both sides by $ 10 $ } \\[1 em]x &= \frac{6}{10}&& \\[1 em]x &= \frac{3}{5}&& \\[1 em] \end{aligned} $$

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