Question

Let x and y be positive real numbers such that x + y = 1.

Prove that (1 + (1/x)) * (1 + (1/y)) ≥ 9

Answer 1

Since both x and y are positive and real, the largest possible value of y is 1 and occurs when x = 0. Similarly, the largest possible value of x is 1 and occurs when y = 0. Both x and y are limited to the domain from 0 to 1.
By letting x or y go to 1 (and the other variable go to 0), the expression
(1 + (1/x)) * (1 + (1/y)) goes to infinity.
For symmetry reasons, the minimum value of
(1 + (1/x)) * (1 + (1/y))
will occur when x = y = 1/2.
This corresponds to
(1 + (1/x)) * (1 + (1/y))
=(1 + 2)(1 + 2) = 9