Question

In absolute value
|x-3|-|x-1|<0
Find x

Answer 1

|x-3|-|x-1|<0
|x-3| < |x-1|

I plotted |x-3| < |x-1|

www.wolframalpha.com/input/?i=%7Cx-3%7C+%3D+%7Cx-1%7C

from the graph we can see that x = 2 is the only solution for
|x-3| = |x-1|

so x > 2 is your solution for |x-3| < |x-1|

Answer 2

makes sense, since |x-1| is the distance of x from 1.
same for |x-3|
Since 2 is the same distance (1) from 1 and 3, x>2 will be closer to 3 than to 1.