The answer to the statement that "air which enters a low-pressure region will fall and cool" is;
When air enters a region of low pressure, it is forced to rise up and form clouds in the atmosphere.
There is a cooling of this air and after holding up in the atmosphere for some time, it is forced to condense and fall as rain.
Because the earth is trying to balance up the differences in temperature at every point in time, there are changes in pressure thus causing different regions to experience high and low pressure at several points.
Regions of the earth with low-pressure experience more frequent rains.
Conclusively, when air enters a region of low pressure, it rises up and forms clouds that condense and fall as rain.
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