Solution:
We are given that
1/x + 1/y = 1/12.
Multiplying both sides by 12xy, we get 12y + 12x = xy, which we can write as
xy - 12x - 12y = 0.
We can then apply Simon's Favorite Factoring Trick, by adding 12*12 = 144 to both sides:
xy - 12x - 12y + 144 = 144.
The left-hand side factors:
(x - 12)(y - 12) = 144.
We want to minimize x + y, which is equivalent to minimizing the sum of the two factors x - 12 and y - 12.
Since the product of x - 12 and y - 12 is a constant, we can minimize their sum by making them as close to each other as possible. Normally, we could set x - 12 = y - 12 = 12, but the problem states that x does not equal y. The next two factors of 144 (whose product is 144) that are close to 12 as possible are 9 and 16. Hence, we can set x - 12 = 9 and y - 12 = 16, to get x = 21 and y = 28, so the minimum value of x + y is 21 + 28 = 49.