How does one find the largest point of intersection of two curves?

[Phoenix]

I have two functions of the form, p(x) = \sum_{i=1}^{n} \frac{1}{x-a_i}  and q(x) = \sum_{i=1}^{n} \frac{1}{x-b_i}

Given two such functions I fix a = max{a_i,b_i}

I want to ask if the equation p(x) - q(x) has a root or not above x = a

- Can something in Sage help?