Hi Renee,
Yes, I have to agree, your hinges are indeed broken. As indicated by the arrow I added to your picture, there's supposed to be a metal bar that extends all the way to the top of the display connected right there. Years of flexing the display as it's opened and closed eventually causes the metal to fatigue and finally break off.
[image|2916783]
The other side is in the same condition.
We'll need to know the model number of your laptop in order to be able to give you instructions on exactly how to fix this, but in general you'll have to open up the bottom of the laptop to get to the screws holding the hinges in place so you can remove them. You'll need to unplug the display from the motherboard at this point.
After that, the front bezel will have to come off so you can remove the LCD from the display housing in order to access the screws holding the hinge arms. Note that I suspect you may have to replace the rear cover on the lid, but can't be quite sure yet. The picture shows PEM nuts that are typically embedded in metal or plastic, so if you can see those nuts, it means they've been ripped out of wherever they were supposed to be mounted. In those cases you usually have to replace the part the PEM nuts were supposed to be embedded in; on your computer I'm guessing it's the rear of the display case.
[image|2916793]
Those are my best guesses based on what we know about your laptop; with the correct model number we can get more specific about replacement parts and possibly even guides to show you how to do the job.