October 4th, 2012 my brother and I observed the Moon at 10am PST in the Western sky (our location Southern Nevada, USA). Later the same evening at 10pm PST the Moon was in the Eastern sky. How can this be possible?

Orbital Properties of the Earth

Your'e basically asking how can the moon be above the horizon at both 10AM and 10PM, right? The answer is the same for how the sun can be above the horizon for more than 12 hours - during the summer you might observe the sun in the eastern sky at 7AM and again in the western sky at 7PM. In both cases this is possible because the earth is tilted - it's tilted relative to the its orbt around the sun and the also tilted relative to the moon's orbit around the earth. The consequence is that at certain times of the year the sun (or moon) is up for more than 12 hours each day, and at other times they are set for more than 12 hours.

According to the US Naval Observatory for an observer in Las Vegas the moon set at about 9:52 AM on October 4th and rose again at about 10:06 PM. See: http://aa.usno.navy.mil/data/docs/RS_OneYear.php

You miss the point... the sun and moon always appear to rise in the east and set in the west. East to west. On this day the moon's position was above the western horizon at 10am. Later in the evening the moon's position was above the eastern horizon at 10pm. The positions in both instances was well above the horizon. How is this possible?

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Originally Posted by LVTLM

On this day the moon's position was above the western horizon at 10am.

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Yes - just before the moon set in the west.

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Originally Posted by LVTLM

Later in the evening the moon's position was above the eastern horizon at 10pm.

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Just after it rose again.

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Originally Posted by LVTLM

How is this possible?

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Are you thinking that the moon somehow traveled from west to east during the day of October 4? That's not what happened at all. The moon had actually risen on the evening of October 3 at about 10:30 PM, so next morning (Oct 4) you observed it at 10AM in the west just before it set. Then about 12 hours later in the evening (still October 4) it rose agan in the east, and you saw it at 10PM in the east. If you had stayed up to watch it throughout the night of October 4 and early hours of October 5 you would have seen it cross from east to west and set around noon on October 5.