Question
$$-2x+3+4x = \frac{1}{2}(4x+6)$$
Answer
The equation has an infinite number of solutions.
Explanation
$$ \begin{aligned} -2x+3+4x &= \frac{1}{2}(4x+6)&& \text{simplify left and right hand side} \\[1 em]2x+3 &= \frac{4x+6}{2}&& \text{multiply ALL terms by } \color{blue}{ 2 }. \\[1 em]2\cdot2x+2\cdot3 &= 2 \cdot \frac{4x+6}{2}&& \text{cancel out the denominators} \\[1 em]4x+6 &= 4x+6&& \text{move the $ \color{blue}{ 4x } $ to the left side and $ \color{blue}{ 6 }$ to the right} \\[1 em]4x-4x &= 6-6&& \text{simplify left and right hand side} \\[1 em]4x-4x &= 0&& \\[1 em]0 &= 0&& \\[1 em] \end{aligned} $$
Since the statement $ \color{blue}{ 0 = 0 } $ is TRUE for any value of $ x $, we conclude that the equation has infinitely many solutions.
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