<s>[INST] 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!
 [/INST] I'd start by multiplying by $2/2$, to make the expression easier to read.
$$\frac{x + 1/2}{1/x + 2} = \frac{2x + 1}{2/x + 4} = \frac{2x^2 + x}{4x + 2}$$
Factoring the top and bottom:
$$\frac{2x^2 + x}{4x + 2} = \frac{x(2x + 1)}{2(2x + 1)}$$
And cancel:
$$\frac{x(2x + 1)}{2(2x + 1)} = \frac{x}{2}$$
</s>