Inequalities With Parity-Specific Domains

Chelonian

When I was working on a rather difficult real-life math problem, I nearly found the solution. What I came up with was two inequalities: ##X≥\frac{2b-2}{2a+1}-1## and ##Y≥\frac{2b}{2a+1}-2## and the fact that ##X>Y##. However, ##X## must be an even integer and ##Y## must be an odd integer. Is there any way of combining all of these statements into one nice, neat equation, or do I have to leave it in the unpleasant form in currently exhibits? I would truly hate to have all of the work I went through thus far on this problem end with such an ugly solution. I have struggled for about two or three hours, so any help you can give on this topic is greatly appreciated. Thank you for your time, and for any assistance you can provide.
P.S. Last time I posted a topic, it was recommended that I post my mathematical ability, so I am a mathematically advanced student in tenth grade.