In a triangle with integer side lengths, one side is three times as long as a second  side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?
This is called the triangle inequality. So, if I call the second side x and the first side 3x, then I have that x + 3x > 15 and 3x + 15 > x. Simplifying, I get that 4x > 15 and 2x > -15. The second inequality is always true for any positive x, so I only need to focus on the first one. Dividing both sides by 4, I get that x > 3.75. But x has to be an integer, so the smallest possible value for x is 4. That means the smallest possible value for 3x is 12, and the smallest possible perimeter is 4 + 12 + 15 = 31. To get the greatest possible perimeter, I want to maximize x. But I also have to remember the triangle inequality. If x gets too large, then x + 15 might not be greater than 3x. So I need to find the largest x such that x + 15 > 3x. Subtracting x from both sides, I get that 15 > 2x. Dividing both sides by 2, I get that 7.5 > x. So the largest possible value for x is 7, since it has to be an integer. That means the largest possible value for 3x is 21, and the largest possible perimeter is 7 + 21 + 15 = 43. 