Whole-Number Sides

Often a question adds a condition: the third side is a whole number. Then you only want the whole numbers inside the range. The range 4 < x < 16 holds the whole numbers 5, 6, 7 and so on up to 15. The ends 4 and 16 are not included, because at those values the triangle is flat. So the shortest whole-number side is 5, the longest is 15, and there are 11 possible sides in all.

To count the whole numbers from 5 to 15, subtract and then add 1: 15 - 5 + 1 = 11. It is easy to forget the "+ 1". To see why it belongs, count a short case by hand. The numbers 5, 6 and 7 are three numbers, and 7 - 5 + 1 = 3. With sides 3 and 5, the range is 2 < x < 8. The whole numbers in it are 3, 4, 5, 6 and 7, so there are five choices.