Discussion
Consider the following inequalities
\(p^{2} - 4q \lt 4\)
\(3p + 2q \lt 6\)
where \(p\) and \(q\) are positive integers.
The value of \((p + q)\) is _______.
Answer: A
The value of \(p + q\) is 2.
Both \(p\) and \(q\) are positive integers, so each is at least 1. The second inequality is the tighter one:
\[ \begin{aligned} 3p + 2q &\lt 6 \\ p = 1, q = 1:\ 3 + 2 &= 5 \lt 6 \\ p = 1, q = 2:\ 3 + 4 &= 7 \end{aligned} \]
Any larger \(p\) or \(q\) pushes \(3p + 2q\) to 6 or more, so \(p = q = 1\) is the only candidate. It satisfies the first inequality too: \(1 - 4 = -3 \lt 4\). Hence \(p + q = 2\).
A sum of 1 would need one of the numbers to be 0, which is not positive, and sums of 3 or 4 need \(p = 2\) or \(q = 2\), which break \(3p + 2q \lt 6\).
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