Number Picking for DS

I have seen that it could be a valuable skill on some tough DS questions esp involving inequality/absolute value etc. However, number picking doesnt come naturally to me. For instance in problems such as

If u(u+v) different from 0 and u >0, is 1/(u+v) < 1/u + v?
1) u+v >0
2) v>0

for st1 since the algebraic approach isnt very easy, when I try to use number plugging-
I have to hold the following three statements in mind:
u(u+v)# 0
u >0
u+v > 0
And then think of numbers that will give two different answers to 1/(u+v) < 1/u + v?
This entire process flips me out. So picking numbers, while relatively easy to put in practice, is uncomfortable, error prone and time consuming method for me.

Where as algebra is far more natural. For instance in St2- algebra is quite straightforward. But algebra is not always obvious and could be really time consuming especially on options where the statement is insufficient. That is probably also why it takes me a lot of time to solve DS problems where answer is E.

Any suggestions to improve number picking skills? Any drills would you guys suggest? Im not looking for advice such as pick all numbers +ve and -ve intigers , +ve and -ve fractions , roots , and zero etc etc. I am looking for how to practically improve this skill

Thanks