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: From inequality (2).
\[ 3p + 2q < 6 \] Since \(p, q\) are positive integers, smallest values are at least 1. If \(p=1\): \[ 3(1) + 2q < 6 \Rightarrow 3 + 2q < 6 \Rightarrow 2q < 3 \Rightarrow q < 1.5 \] Thus, \(q = 1\).

Step 2: Check inequality (1).
\[ p^2 - 4q < 4 \] Substitute \(p=1, q=1\): \[ 1^2 - 4(1) = 1 - 4 = -3 < 4 \text{(satisfied).} \]

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