In the UK, under The Copyright, Designs and Patents Act 1988, are you legally allowed to sell a book that you have previously bought? why/why not?

Yes, you are legally allowed to sell a book that you have previously bought. This is because the Copyright, Designs and Patents Act 1988 grants the owner of a copyright the exclusive right to sell or rent copies of their work. This means that you are allowed to resell a book that you have previously bought, as long as you are not infringing on any other copyright laws.