Difficult maths problem - requires some thought

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To determine when the hour and minute hands of a clock are pointing in opposite directions after 8 o'clock, one must calculate the angles of both hands relative to 12 o'clock. The angle of the minute hand, denoted as m, changes with time x minutes past the hour, while the hour hand's angle, h, also shifts as time progresses. The key equation to solve is h - m = 180 degrees. It is noted that the hands are opposite at 6 o'clock and occur 11 times in a 12-hour period. Solving the problem can be approached through angle equations or by understanding the rates at which the hands move.
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After the hour hand has passed 8 on your watch, when will the minute hand and the hour hand be pointing in exactly opposite directions?

I did initially try formulating equations based on the change in the angle of the hour hand after x minutes, but it doesn't seem to be taking me anywhere.

I don't even know where to start on this question, any pointers in what direction I should head would be appreciated.
 
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When the time is x minutes past the hour, what is the angle (in degrees) between the minute hand and 12 o'clock (measured clockwise)? Call this angle m. (Note that x need not be a whole number.)

When the time is 8 o'clock and x minutes, what is the angle (in degrees) between the hour hand and 12 o'clock (measured clockwise)? Call this angle h.

For what value of x do we have h-m=180?
 
The easy way is to notice the hour and minute hand are pointing in opposite directions at 6 o'clock, and every 12 hours there are exactly 11 times they are pointing in opposite directions. Now you can figure out the times. But you can also do it with rates. Post your attempt and someone will probably show you where you went wrong.
 

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