Simple verification: is this equivalence always true?

Question

I have a constrained optimization problem and I am trying to reduce the number of contraints and am afraid to be losing information by doing so. If we have two constraints as the following $$A \geq B \mbox{ and } C \geq D $$ Is it equivalent to write: $$ A + C \geq B + D $$ It is clear that $ A \geq B \Rightarrow A + C \geq B + C \geq B + D \Rightarrow A + C \geq B + D$. Does the implication work the other way round (it seems to me that the answer is yes)? If not, can you think of a counter example?

Thank you

Answer 1

No. An example: $$5+2\geq 1+4$$ where $A=5, B=1, C=2, D=4$. $C$ is not greater than or equal to $D$.

Answer 2

It is not equivalent. Obviously the first two inequalities prove the second, but this doesn't work in the opposite direction.