It is a fascinating subject but gigantically complicated, especially when it comes to mixing or joining dissimilar plastics. It's why I shy away from solvents, not because they don't work but because it adds yet another layer of complexity. I tend to stick with what I know (no pun intended) and with plastic welding, all you have to get right is the temperature.
That might be "simple" but most people won't just have suitable equipment lying around, soldering irons are not suitable in this context unless they can be temperature controlled. If the material burns, you get effectively a contaminated weld which like metal welding, will be weak and brittle.
Solvent might be the best bet in this case, possibly the easiest to do yourself though you might want to reflect on why it broke in the first place. If it's a weakness in the design, your repair will have to make it stronger...
I'm pretty sure it wasn't misuse or impact damage, but it was out of warranty so I couldn't take it back. I'd thought the same ref the reinforcement, hence buying a large-ish sheet of ABS placcy so I can reinforce both sides, as it would be a massive downer if I fixed one crack and it opened up a new one alongside the repair! Once I get time to have a go I'll keep everyone posted.