Consider points a(3, 6) and b(8, 4). find point c on the x-axis so ac +bc is a minimum.

Both points are below the x-axis. You can solve this by thinking of the x-axis as a mirror, I suppose? 

Anyway, you have to pick point C:(x,0) on the x-axis. Draw AC and BC. 

Now you have to reflect B:(12,-5) in the x-axis, to B':(12,5), and draw B'C; its the reflection in the x-axis of BC. 

Finding the shortest line in the original problem is connecting A and B', calling its point of intersection with the x-axis, C. 

Then AC+BC will be minimized. 

Find x, and thus, C: 
A: (4,-3)
B: (12,5)
C: (x, 0) 

(0+3)/(x-4) = (5+3)/(12-4) = 1
3 = x - 4
x = 7

C: (7,0)

Well, that took a while. And it sure does look confusing. Hope you'll be able to read it properly.