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 ................

• A. 2
• B. 1
• C. 3
• D. 4

Hint:

When solving a system of inequalities with integer constraints, always start with the most restrictive inequality. A linear inequality like \(ax + by<c\) is often more restrictive for positive integers than a quadratic one, as it limits the variables more severely.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are given two inequalities and the condition that \(p\) and \(q\) must be positive integers. We need to find the specific integer values of \(p\) and \(q\) that satisfy both inequalities and then calculate their sum. Positive integers are {1, 2, 3, ...}.
Step 2: Key Formula or Approach:
The best approach is to use the simpler inequality to narrow down the possible values for \(p\) and \(q\), and then test those values in the second inequality. The second inequality, \(3p + 2q<6\), is linear and has stronger constraints for positive integers.
Step 3: Detailed Calculation:
1. Analyze the second inequality: \(3p + 2q<6\)
Since \(p\) and \(q\) must be positive integers, the smallest possible values are \(p=1\) and \(q=1\).
Let's test possible values for \(p\):
- If \(p=1\), the inequality becomes \(3(1) + 2q<6\), which simplifies to \(3 + 2q<6\).
Subtracting 3 from both sides gives \(2q<3\).
Dividing by 2 gives \(q<1.5\).
Since \(q\) must be a positive integer, the only possible value for \(q\) is 1.
- If \(p=2\), the inequality becomes \(3(2) + 2q<6\), which simplifies to \(6 + 2q<6\).
This implies \(2q<0\), or \(q<0\). This is not possible, as \(q\) must be a positive integer.
- If \(p>2\), the value of \(3p\) will be even larger, making it impossible for the inequality to hold for any positive \(q\).
Therefore, the only possible integer solution from the second inequality is \(\mathbf{p=1}\) and \(\mathbf{q=1}\).
2. Verify this solution with the first inequality: \(p^2 - 4q<4\)
Substitute \(p=1\) and \(q=1\) into the inequality:
\[ (1)^2 - 4(1)<4 \] \[ 1 - 4<4 \] \[ -3<4 \] This statement is true. So the values \(p=1\) and \(q=1\) satisfy both conditions.
3. Calculate the required value of \((p+q)\):
\[ p+q = 1+1 = 2 \] Step 4: Final Answer:
The value of \((p+q)\) is 2.
Step 5: Why This is Correct:
By systematically testing the constraints for positive integers on the simpler linear inequality, we found a unique solution pair (\(p=1, q=1\)). This pair was then verified to satisfy the first inequality as well. Thus, it is the only valid solution.