I'm not using RH's editor. I'd learnt to avoid it.
All my CSS is worked on in Dreamweaver CC. Also tried just using Notepad.
You'll need to make sure you link to the image in your project. If you have your stylesheet in a subfolder and you added the baggage file by clicking on Project Files, it should be something like the following, but check where each of the files are in the project:
background-image: url('../livechat_question.gif');
Also, background-image doesn't need the file to be a baggage file, RH knows enough to pick up the file correctly if the file is in your project and you point to the directory it's in.
However, I've done some research on the initial / in your path, and I suspect RH doesn't understand that syntax. So assuming this is a resource you want to pick up from your webserver, then you could change the URL to an absolute path (including the domain). I'm not sure if that would display correctly in preview or when run locally, but it should definitely be fine when published.
Basing my guess on this page: Relative URLs | WebReference
(EDIT: SNAP! 
)
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