[Heye] bought an LCD projector for $60 off AliExpress, and it turned out to be rather noisy: the air fan that sucked in air to cool the LED light source made a whooshing noise.

Per screen inch, projectors are much cheaper than a big LCD, even when you include the screen. Hideable. With little effort, you can make the projector and screen disappear when you're not using them.

A projector is not like an LCD TV in that you can just stick it in any room and plug it into the wall. Projectors require mounting, and a corresponding screen across the room.