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Law of gravitation applies on each planet and it also applies on earth towards the moon but the motion or acceleration of earth towards the moon is very slow and it goes unnoticed. 

The difference in mass makes the pull of earth stronger so it rortates around earth.It does not move closer because its in equilibrium,its roations and revolution counteract the pull of the earth.It remains so because its angular moomentum is conserved.

 

The moon also attract earth but we do not feel. The mass of Earth is too much greater than the moon so we can only see that only Moon revolves around the Earth.

It does move toward the moon and the moon moves toward the earth. They are constantly falling towards their centre of mass. This constant falling appears as revolution around the centre of mass because of the tangential velocity they possess. If you could somehow stop the relative motion of the moon toward the earth, they would 'fall' and collide with each other.

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