Two sides of a triangle measure 8 and 12. How many different integer values are possible for the length of the third side?

By the triangle inequality, 12 − 8 < $x$ < 12 + 8, that is 4 < $x$ < 20. The integers strictly between 4 and 20 are 5, 6, …, 19, which is 19 − 5 + 1 = 15 values.