If you take 7 minus the result of a dice throw, then that's another valid dice throw. So, for two dice throws, the probability that the sum is greater than 8, is the same as the probability of 14 minus the sum being larger than 8, which is the same as the probability of the sum being less than 6.
If we then add the probability for the sum being larger than 8 and smaller than 6, we get twice the probability. If you subtract that from 1, you get the result:
1-2 p = probablity that the sum is 6 7 or 8
where p is the desired probability.
Now the probability of the sum being 6, P(6)is the same as the sum being 8,
P(8), because 14-8 = 6, so we have:
1-2p = 2 P(6) + P(7) ------->
p = 1/2 - P(6) - 1/2 P(7)