Who is right about baby waking up at night?

So my boyfriend and I live with my mom because of financial difficulties, and our baby is going to be 5 months tomorrow. My mom helps out a lot which is awesome.

But the last three nights he has woken up between 12:30-1:20 to be rocked back to sleep. First two nights, I didn’t mind so much, hoping it wouldn’t become a habit. He wasn’t hungry. But it took me like 3 times of rocking him to sleep for 20 minutes and setting him down for him to stay asleep.

So last night I rocked him to sleep twice and set him down, and finally decided to just let him cry it out because I was exhausted and he was going to wake up in like an hour anyways to eat.

But my mom won’t let him cry it out. She can’t stand to hear him cry and after 5-10 minutes she came out and fed him and put him to sleep and told me “If you can’t take care of him you shouldn’t have him.”

This pissed me off. I know we were both just tired but I want to know if I’m wrong. Should I be letting him cry it out when I know he doesn’t need to eat? Or is my mom right and do I need to put him back to sleep if he doesn’t stop crying after like 10 minutes? I read that it’s okay to let them cry it out after 4 months. I don’t mind waking up to feed him but I also don’t want to lose more sleep than I need to. I do stay awake and listen to him to make sure he’s okay when he cries.