Question
$$\frac{1}{2}x+\frac{1}{2}x = x+1$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{1}{2}x+\frac{1}{2}x &= x+1&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2 \cdot \frac{1}{2}x+2\frac{1}{2}x &= 2x+2\cdot1&& \text{cancel out the denominators} \\[1 em]x+x &= 2x+2&& \text{simplify left side} \\[1 em]2x &= 2x+2&& \text{move the $ \color{blue}{ 2x } $ to the left side} \\[1 em]2x-2x &= 2&& \text{simplify left side} \\[1 em]2x-2x &= 2&& \\[1 em]0 &= 2&& \\[1 em] \end{aligned} $$
Since the statement $ \color{red}{ 0 = 2 } $ is FALSE, we conclude that the equation has no solution.
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