Algebra

First off, we need to put a in terms of b. This helps us identify a range. 

We get

$a+b=1\\ a=1-b$

Now, plugging that into the expression, we get

$\frac{1}{a}+\frac{1}{b}= \frac{1}{1-b}+\frac{1}{b}=\frac{b+1-b}{b-b^2}=\frac{1}{b-b^2}$

Now, we can easily find the set of all real numbers. We have

$\{ R>1\} |\{a,b\} \subset R$ where $1> a,b > 0$

Thanks! :)