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Putting this here for the benefit of the next person who faces this issue - since dmesg said
input: AlpsPS/2 ALPS GlidePoint as /devices/platform/i8042/serio2/input/input12
I didn't think I needed a synaptics driver. (And since the basic touchpad functionality worked out of the box, it didn't strike me that I was missing anything in xorg-*; I thought I'd just configured the touchpad wrong.) Took me a lot of googling to find out I needed xf86-input-synaptics, after which everything just worked.
(Also I believe the Inspiron 15 is a rebranded Inspiron 1545, so that probably has an ALPS Glidepoint too)
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