Question

Consider the following inequalities: \[ p^2 - 4q < 4, 3p + 2q < 6 \] where \(p\) and \(q\) are positive integers. The value of \((p+q)\) is \underline{\hspace{1cm}}.

• A. 2
• B. 1
• C. 3
• D. 4

Hint:

In inequality problems with integer constraints, test small integer values systematically. Often only one pair satisfies both conditions.

Answer 1

The Correct Option is A

Step 1: Analyze inequality (2).
\n\[\n3p + 2q < 6\n\] \nGiven that \(p\) and \(q\) are positive integers, their minimum value is 1. \n\nFor \(p=1\): \n\[\n3(1) + 2q < 6 \Rightarrow 3 + 2q < 6 \Rightarrow 2q < 3 \Rightarrow q < 1.5\n\] \nTherefore, the only possible integer value for \(q\) is 1. \n\n \n

Step 2: Verify inequality (1).
\n\[\np^2 - 4q < 4\n\] \nSubstituting \(p=1\) and \(q=1\): \n\[\n1^2 - 4(1) = 1 - 4 = -3 < 4 \text{(condition met).}\n\] \n\n \n

Step 3: Calculate \(p+q\).
\n\[\np+q = 1+1 = 2\n\] \n\n \n\[\n\boxed{2}\n\]