Inequality sign doubt

If they were both ≥ inequalities, then you'd need to preserve the "≥", but if one of them is "≥" and the other is ">", then when you add the two inequalities, you can safely make the result ">" always. You might be able to see conceptually why this should be true; if you know

• Amy has more money than Bimal
• Carlos has at least as much money as Daaria

then Amy and Carlos combined will surely have strictly more money than Bimal and Daaria combined, so if we had these inequalities

A > B
C ≥ D

then we can be sure A+C > B+D is true. Of course, A+C ≥ B+D is also true, but that's slightly less informative. Or you can see why algebraically: if A > B and C ≥ D, then either A > B and C > D, in which case A+C > B+D just by adding the two inequalities, or A > B and C = D. But A + C > B + C is clearly true, because we're just adding the same thing on both sides of an inequality, and if C = D, we just proved A + C > B + D.

As for your other questions, I'm not quite sure what you mean. We cannot, in general, multiply inequalities at all. For example these two inequalities are true:

1 > -3
2 > -5

but if you multiply them, you won't get something true. So the 'rules' about how to multiply inequalities when one is "≥" and the other is ">" would depend on the restrictions on your numbers. If negatives could be involved, you couldn't multiply safely at all. If zero could be involved, you'd need to preserve the "≥" and if everything was positive, then it would work like the addition case above. Perhaps I've misunderstood your question though.