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, 12, and 16, what is the value of r?
Choose the correct option
A) 18
B) 20
C) 22
D) 23
E) 24
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 + 12 &gt; 16   2. 12 + 16 &gt; q   3. q + 16 &gt; 12   Equation 3 is true for any positive value of q.   From equation 1, q &gt; 4   From equation 2, 28 &gt; q   Then 28 &gt; q &gt; 4 so q can be any value from 5 to 27 inclusive.   The number of possible values is 23.   The answer is 23.
<result>D</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, 12, and 16, what is the value of r?