Question

Consider the following inequalities.
(i) \( 3p - q<4 \)
(ii) \( 3q - p<12 \)
Which one of the following expressions below satisfies the above two inequalities?

• A. \( p + q<8 \)
• B. \( p + q = 8 \)
• C. \( 8 \leq p + q \leq 16 \)
• D. \( p + q \geq 16 \)

Hint:

When working with inequalities involving two variables, try to express one variable in terms of the other and then combine the results to check which conditions hold true.

Answer 1

The Correct Option is A

We are given two inequalities: \[ (i) \quad 3p - q<4, \quad (ii) \quad 3q - p<12. \] Let's manipulate these inequalities step by step.
Step 1: Solve inequality (i).
From inequality (i), we can express \( q \) in terms of \( p \): \[ 3p - q<4 \quad \Rightarrow \quad q>3p - 4. \]
Step 2: Solve inequality (ii).
From inequality (ii), we can express \( q \) in another form: \[ 3q - p<12 \quad \Rightarrow \quad 3q<p + 12 \quad \Rightarrow \quad q<\frac{p + 12}{3}. \]
Step 3: Combine the two inequalities.
We now have two expressions for \( q \): \[ q>3p - 4 \quad \text{and} \quad q<\frac{p + 12}{3}. \] For both inequalities to hold, the following must be true: \[ 3p - 4<q<\frac{p + 12}{3}. \]
Step 4: Check the expression \( p + q \).
From the above inequality, we can try combining the bounds for \( q \) and check which expression satisfies the condition \( p + q \). After solving and substituting various values, we find that the expression \( p + q<8 \) satisfies the given inequalities. Therefore, the correct answer is \( p + q<8 \), which corresponds to option (A).