3x + 4|y| = 33. How many integer values of (x, y) are possible?
Since x and y are integers, and since |y| is always positive regardless of the sign of y, this means that when you subtract a multiple of 4 from 33, you will get a multiple of 3.
Since 33 is already a multiple of 3, in order to obtain another multiple of 3, you will have to subtract a multiple of 3 from it. So, y has to be a positive or a negative multiple of 3.
y = 3, -3, 6, -6, 9, -9, 12, -12...etc.