Question
$$\frac{1}{4}x+\frac{2}{3} = \frac{1}{4}x-2$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1}{4}x+\frac{2}{3} &= \frac{1}{4}x-2&& \text{multiply ALL terms by } \color{blue}{ 12 }. \\[1 em]12 \cdot \frac{1}{4}x+12\cdot\frac{2}{3} &= 12 \cdot \frac{1}{4}x-12\cdot2&& \text{cancel out the denominators} \\[1 em]3x+8 &= 3x-24&& \text{move the $ \color{blue}{ 3x } $ to the left side and $ \color{blue}{ 8 }$ to the right} \\[1 em]3x-3x &= -24-8&& \text{simplify left and right hand side} \\[1 em]3x-3x &= -32&& \\[1 em]0 &= -32&& \\[1 em] \end{aligned} $$
Since the statement $ \color{red}{ 0 = -32 } $ is FALSE, we conclude that the equation has no solution.
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