Connect an external monitor and see if there is a display on it with the laptop turned on.
If there is no display on the external monitor either then you have either a GPU or motherboard problem.
If there is a display on the external monitor, shine a torch at an angle close to the laptop's LCD screen and see if you can detect the desktop image. If it is there it will be very faint so it might be easier to try this in a darkened room.
If there is a display detected then you have a backlight problem. This could be caused by a faulty backlight power supply on the motherboard, faulty backlight power cable (maybe combined in the video cable), faulty backlight power cable connection at either end (motherboard or screen), faulty lid sensor or faulty LCD screen.
If there is no display detected then you may have a faulty video cable, video cable connection at either end or a faulty screen.
Here is a link to the ifixit ThinkPad X1 Carbon Repair guide which may be of some help.
Here is a link to the service manual for the laptop which may also help
Here is a link that may be of interest to you.
see the post on the page by "morganae" (2nd post down the page) for a solution
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