Replacing Motobecane 500 HT derailleur hanger Can I just replace my rear derailer hanger in order to fix my derailer, is that an option?  It looks like it broke and it definitely is detachable.
Yes, but you will need to buy a replacement part that is specifically designated for your bicycle vendor, model and even model year combination as there is close to none compatibility between derailleurs.

You can search the net for the words "[bicycle name] + derailleur". Some companies in some countires offer wide catalogs of compatible parts. Some of these folks can even cut a new part with a CNC machine for you. In this case, contact them and provide the information they will request. Most likely they will need photoes and measurements of the derailleur and/or dropout where it sits.

Note that replacements derailleurs are relatively cheap, and even making a custom part is much cheaper than buying a new bike. And while you are at it, why not order an extra part, just for the future?