Find all real solutions of the equation $\displaystyle \frac{\displaystyle x + \frac{2}{x}}{\displaystyle 3 + \frac{4}{x}} = 5x$
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\begin{equation}
\begin{aligned}
\frac{\displaystyle x + \frac{2}{x}}{\displaystyle 3 + \frac{4}{x}} =& 5x
&& \text{Given}
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\frac{\displaystyle \frac{x^2 + 2x}{x}}{\displaystyle \frac{3x + 4}{x}} =& 5x
&& \text{Simplify the numerator and denominator}
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\frac{x^2 + 2x}{3x + 4} =& 5x
&& \text{Cancel out } x
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x^2 + 2x =& 5x (3x + 4)
&& \text{Multiply both sides by } (3x + 4)
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x^2 + 2x =& 15x^2 + 20x
&& \text{Expand}
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14x^2 + 18x =& 0
&& \text{Combine like terms}
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2x(7x + 9) =& 0
&& \text{Factor out } 2x
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2x =& 0 \text{ and } 7x + 9 = 0
&& \text{Zero Product Property}
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x =& 0 \text{ and } x = \frac{-9}{7}
&& \text{Solve for } x
\end{aligned}
\end{equation}
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