The given sides are 3 and 2 and the third side is x.
The triangular equality says essentially, if you were walking along the sides
of a triangle ABC,
1. If you walk from A to B and then to C, you will have walked farther than if
you had just walked straight from A to C.
So AB + BC > AC
2. If you walk from A to C and then to B, you will have walked farther than if
you had just walked straight from A to B.
So AC + CB > AB
3. If you walk from B to C and then to A, you will have walked farther than if
you had just walked straight from B to A.
So BC + CA > BA
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The sides are 3, 2 and x. So by the very obvious triangular inequality:
1. 3 + 2 > x
2. 3 + x > 2
3. x + 2 > 3
Simplifying:
1. 5 > x
2. x > -1
3. x > 1
Ignore the second one because the lengths of every side of every triangle is
greater than a negative number. So we only need consider the other two:
5 > x and x > 1
You can write that together as 5 > x > 1, or if you prefer, 1 < x < 5.
Since they are whole numbers, the set of integers that satisfy 1 < x < 5,
are {2, 3, 4}. The smallest whole number length for the third side in 2.
Edwin
