Is the following inequality holds true for all positive integers? If $a , b , c$ and $d$ are positive integers, 
and $ab$ is greater than $cd$,
then,  is   $a+b$  greater than or equal to $c+d$, always true?

Not at all.
Consider $a=b=3$ and $c=1,d=8$. Then $ab=9$ while $cd=8$ however $a+b=6$ while $c+d=9$.
If you want a trivial example, you can take $c=0$, any non zero $a,b$. Then take $d$ very large.
