Question
$$2(x+1) = \frac{1}{3}(6x-2)$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} 2(x+1) &= \frac{1}{3}(6x-2)&& \text{simplify left and right hand side} \\[1 em]2x+2 &= \frac{6x-2}{3}&& \text{multiply ALL terms by } \color{blue}{ 3 }. \\[1 em]3\cdot2x+3\cdot2 &= 3 \cdot \frac{6x-2}{3}&& \text{cancel out the denominators} \\[1 em]6x+6 &= 6x-2&& \text{move the $ \color{blue}{ 6x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]6x-6x &= -2-6&& \text{simplify left and right hand side} \\[1 em]6x-6x &= -8&& \\[1 em]0 &= -8&& \\[1 em] \end{aligned} $$
Since the statement $ \color{red}{ 0 = -8 } $ is FALSE, we conclude that the equation has no solution.
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