Assume that the sides of a triangle are 5, x, and y where x and y are integers such that 1Triangle Inequality Theorem.
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
The illustration of the theorem is given below.
XY+YZ>XZ
YZ+XZ>XY
XZ+XY>YZ
Applying the theorem, we have three inequalities.
Inequality I:x+y>5 ⇒ x+y>5
Inequality II:y+5>x ⇒ 5 > x-y
Inequality III:5+x>y ⇒ 5 > y-x
We can also combine the inequalities as one inequality.
Combined Inequality: |x-y|< 5 < x+y
We can interpret this combined inequality as the difference between x and y is less than 5 and the sum of x and y is greater than 5. As a result we can write (x,y) pairs as follows.
x-values:
2, 3, 4
y-values:
3, 4, 5, 6, 7, 8
List of (x,y) pairs:
(2,4), (2,5), (2,6), (3,3), (3,4), (3,5), (3,6),
(3,7), (4,3), (4,4), (4,5), (4,6), (4,7), (4,8)
