Q:

subtracting inequalities if difference is positive

$$ x \geq y $$
$$ a \geq b $$
$$x+a \geq y+b $$ is valid but $$ x-a \geq y-b $$ is not valid
Can we say the latter is valid if $x-a \geq 0$ ?
Is it a proof or am I wrong? Are there counter examples?
If $x,y,a,b \geq 0 $ ?

A:

If $a \geq b$ then $-a \leq -b$. So you could for sure deduce that $$x-b \geq y-a$$ As for a counter example in the light of the assumption $x-a \geq 0$, try $x = y = a = 1$ with $b = 0$.

