Cutting a closed object, there by making it open

After you do the Difference, there will be a node on the former rectangular path, at the 2 points where the circle originally intersected it. Or at least there should be. Sometimes the path operation put new nodes in strange places. But from your screenshot, it looks like there probably are a cusp/corner node on each side (or top and bottom), where the circle intersected it.

1 - Select both of those nodes (with Node tool).
2 - Click the "Break path at selected nodes button" on Node tool control bar (or the appropriate key shortcut, whatever it is (see Help menu > Key and Mouse Ref)).
3 - There are a couple of different ways to go, from here, depending on your goals. If you want to keep the path segment between those 2 nodes where the path was broken (for some reason), you can do Path menu > Break apart. That will make them 2 separate paths.

Or if you don't need that segment, you could just highlight and delete the nodes.

However, I notice on your screenshot that you have the ends of the former rectangular path, sort of shaped, so that they have tiny arcs on the end. If you do need that arcing on the end of the path, there's a way to do it. But then the path will no longer behave as a single path. It's too hard to explain in words, and there's no way to attach files here, to show you. But you can try this, and you'll see what I mean.

Select the rectangular path, and do Path menu > Stroke to Path. Then overlap the circle and do Path menu > Difference. That's it. But now, the former rectangular path is not a single, open stroked path, with no fill. Now it is a closed path, with a fill and no stroke.

Hopefully 1 of those processes will work for you :-)

Thanks Brynn. I am going to try your suggestion -- should learn something new.

In the meantime, I had cross-posted the question (somewhat in a desperate urgency) on graphic-design SE site. I got 2 answers there, and both work. Posting a link-back for a logical conclusion.

http://graphicdesign.stackexchange.com/questions/32020/how-to-remove-a-portion-of-a-closed-object-to-turn-it-into-an-open-object

You're welcome.

FYI -- Clipping is indeed an option, but would not be my 1st choice, because it can cause issues in a complex drawing, if there are other objects nearby, or grouped, or etc.

You could also probably do it with the eraser tool, although precision would be difficult with that....still sort of buggy tool. Also could get there with Cut Path and maybe Division. This is why I don't like to participate in the stack exchange sites. It makes getting the right answer into a competition, when sometimes there is no right or wrong answer. Sometimes it's just the personal preference of the user.

The 1st response in the SE is incorrect. Difference can be applied to shapes, even if not converted to path. (A little known fact :-).)