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I replaced my macbook air screen because it had a cracked LCD. I replaced the entire display using an ifixit guide, but now when I try to turn on my laptop, I hear it waking up when I have it plugged in and I try pressing the power button, but the screen is black and stays black. How do I know if I made a mistake with the logic board, or with assembly in general, or if my replacement display is faulty? The fan does opperate when I hold down the power button.

If you are able to see apple logo after switching it on follow steps if not then there must be a loose conection to display or the display is not installed correctly Try to use command+option+r button after switching on your Mac possibly it will take you to recovery mode and ask to install OS X in your Mac.