Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 10, and 12, what is the value of r?  Pick one.
A) 17
B) 18
C) 19
D) 20
E) 21
From the triangle inequality rule, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So, the three inequalities are:   1. q + 10 &gt; 12   2. 10 + 12 &gt; q   3. q + 12 &gt; 10   Equation 3 is true for any positive value of q.   From equation 1, q &gt; 2   From equation 2, 22 &gt; q   Then 22 &gt; q &gt; 2 so q can be any value from 3 to 21 inclusive.   The number of possible values is 19.   The answer is 19.
<result>C</result>
Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 10, and 12, what is the value of r?