The Earth rotates towards the east. This is why the Sun ‘rises’ in the East. The Moon is a little different than the Sun in relation to its motion in our sky. The Moon rotates the Earth in an eastern direction. Now if the Moon’s orbital speed could match that of the orbital speed of the Earth, in relation to the Earth’s surface (what is called a synchronous orbit radius), the Moon would stay in a fixed point in our sky. If the Moons orbital speed, in relation to the Earth surface, was faster than the Earth’s rotation, the Moon would rise in the east and set quickly in the west. This is what Phobos does on Mars. Here on Earth, the Moon takes about 29 days to orbit the Earth, while the Earth rotates in 24 house. So the Moon appears to rise in the east and set in the west.