If p and q are non-zero integers such that |p| + |q| = 18. What is the

Using Statement 1, we have a 2 equations / 2 unknowns problem, with the equations |p| + |q| = 18, and |p| - |q| = 6. If we add those equations, we learn 2|p| = 24, so |p| = 12, and then by substitution, we learn |q| = 6. But we have no way to determine the signs of p and q, so pq can equal 12*6, or it can equal -(12)(6), and we can get two answers to the question.

From Statement 2, notice we can be certain p and q have opposite signs, which is why Statement 2 will turn out to be sufficient -- we can be certain pq is negative. If p = -2q, we can substitute for p:

|p| + |q| = 18
|-2q| + |q| = 18
|-2|*|q| + |q| = 18
2|q| + |q| = 18
3|q| = 18
|q| = 6

and if q = 6, p = -12, while if q = -6, p = 12, and in either case, pq = -(12)(6).