This is actually pretty easy. Just be careful, and have a good backup, as we are messing with a mounted(!) root file system.
First, we need to find some free space in an available physical volume. One way to do this is to decrease the size of one of your other partitions, per my "Reduce an LVM Volume" post in this same Admin/LVM section.
Then add the free space to the root logical volume, for example:
lvextend /dev/mapper/x201i--vg-root /dev/dm-0
where the first parameter is the device associated with the root logical volume, and the second parameter is the name of the physical volume with free space. Without further qualification, lvextend simply adds all the available free space to the specified logical volume.
Now adjust the size of the file system inside the root logical volume, ie.
Now df should show that the root file system is bigger. LVM = OUTSTANDING!!