Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + ... + 1/(x+a_n) = 1/x$ are all real Prove that the roots of the equation:
$$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$
are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.

Your question is solved in
Number of real roots of $\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+...+\frac{a_n}{a_n-x}=2016$ for $0<a_1<...<a_n$? 
because 
$1=\sum\limits_{k=1}^n \frac{x}{x+a_k}=\sum\limits_{k=1}^n \frac{1/|a_k|}{1/|a_k|-1/x}$ .
