The length of the sides of a triangle are 2,8, and s, where s is an integer. How many distinct possible values are there for s? a.) 2 b.) 3 c. ) 6 d.) 10 e.) infinitely many Solution b. There are 3 possible integer values for s, 7, 8, and 9. From the triangular inequality, the sum of any 2 sides must be greater than the third. If the two sides are 8 and s, 8+s > 2, or s > -6. From sides 2 and s, we get 2+s > 8 Subtracting 2 from both sides, s>6 Clearly, this interval is a subset of s > -6, so we can ignore the first. Finally, take sides 2 and 8. 2+8>s, or 10 > s Thus, s>6 and 10>s, which we may write 6 < s < 10 This has 3 integer solutions, 7, 8, and 9. .