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, 9, and 12, what is the value of r? Choose the correct option:
A) 5 B) 17 C) 18 D) 22 E) 29
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 + 9 &gt; 12   2. 9 + 12 &gt; q   3. q + 12 &gt; 9   Equation 3 is true for any positive value of q.   From equation 1, q &gt; 3   From equation 2, 21 &gt; q   Then 21 &gt; q &gt; 3 so q can be any value from 4 to 20 inclusive.   The number of possible values is 17.   The answer is 17.
<result>B</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, 9, and 12, what is the value of r?