How to prove a solution exist between an interval.

Question

For example f(x)= (5/x-1) +(7/x-2) has a solution between 1 and 2. I can see this graphically but do not know how to proof it mathematically

Answer 1

The easiest way is to solve the equation.

Cross multiply to get 7x - 7 + 5x - 10 = 0 (as long as x is not equal to 1 or 2, you can ignore the denominator)

Then x =17/12 = 1.41666666.

Another way is to calculate f(1.1) which is greater than zero and f(1.6) which is less than zero and since f is continuous on the interval there is a root between.