What is the minimum value of |2z-1| + | 3z -2| ?I have been trying it for days but am not able to get the answer. Please help.

Ok this is a slightly difficult to explain but I will try

Part 1: This is straight forward

now considering these two terms are positive we can apply AM GM and the equality exist when terms are equal


Part 2: Now we need to solve this, thinking intutively the smallest such z would lie between the points (1/2,0) and (2/3,0) cutting it in ratio 3:2
that point can be found easily to be
(3/5,0)
Hence
Thus the minimum value is 2/5