How does $\frac{x + \frac{1}{2}}{\frac{1}{x} + 2} = \frac{x}2$? How does simplifying $\dfrac{x+\frac12}{\frac1x+2}=\dfrac{x}2$
Plugging and chugging seems to prove this, but I don’t understand the algebra behind it.
How would you simplify $\dfrac{x+\frac12}{\frac1x+2}$ to get $\dfrac{x}2$?
I tried multiplying the expression by $x/x$, but that left me with $\frac{x^2+\frac{x}2}{1+2x}$
Is the fact that $\frac1x$ and $2$ are the reciprocals of $x$ and $\frac12$ respectively of any significance?
Thanks!

If $x \neq \frac{-1}{2}$, then $\frac{(x+1/2)}{(1/x+2)} = \frac{x^2+x/2}{1+2x} = \frac{2x^2+x}{2+4x}$. This is multiplying by 2 on top and on the bottom. Note that we can write $x = \frac{2x+4x^2}{2+4x}$. Therefore $\frac{x}{2} = \frac{x+2x^2}{2+4x}$, and the expressions are equivalent.
