Question
$$4x-\frac{7}{4}x+14 = \frac{7}{4}(x+8)+\frac{1}{2}x$$
Answer
The equation has an infinite number of solutions.
Explanation
$$ \begin{aligned} 4x-\frac{7}{4}x+14 &= \frac{7}{4}(x+8)+\frac{1}{2}x&& \text{multiply ALL terms by } \color{blue}{ 4 }. \\[1 em]4\cdot4x-4 \cdot \frac{7}{4}x+4\cdot14 &= 4 \cdot \frac{7}{4}(x+8)+4\frac{1}{2}x&& \text{cancel out the denominators} \\[1 em]16x-7x+56 &= 7x+56+2x&& \text{simplify left and right hand side} \\[1 em]9x+56 &= 9x+56&& \text{move the $ \color{blue}{ 9x } $ to the left side and $ \color{blue}{ 56 }$ to the right} \\[1 em]9x-9x &= 56-56&& \text{simplify left and right hand side} \\[1 em]9x-9x &= 0&& \\[1 em]0 &= 0&& \\[1 em] \end{aligned} $$
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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