Analyzing the Inequalities

We are given two inequalities:

• Inequality (i): $3p - q < 4$
• Inequality (ii): $3q - p < 12$

We need to find an expression for $p+q$ that satisfies both these conditions.

Deriving the Expression for p+q

To find a relationship involving $p+q$, we can add the two given inequalities:

1. Add Inequality (i) and Inequality (ii):

   $(3p - q) + (3q - p) < 4 + 12$

2. Simplify the combined inequality:

   $3p - q + 3q - p < 16$

   Combine like terms:

   $2p + 2q < 16$

3. Factor out 2 from the left side:

   $2(p + q) < 16$

4. Divide both sides by 2 to isolate $p+q$:

   $\frac{2(p + q)}{2} < \frac{16}{2}$

   $p + q < 8$

This result, $p + q < 8$, represents the condition that must be satisfied by $p$ and $q$ to meet the initial inequalities.

Comparing with Options

Now, let's compare our derived condition $p + q < 8$ with the given options:

• Option 1: $p+q<8$ - This exactly matches our derived condition.
• Option 2: $p+q=8$ - This does not satisfy $p+q < 8$.
• Option 3: $8 \le p+q < 16$ - This contradicts our finding that $p+q$ must be less than 8.
• Option 4: $p+q \ge 16$ - This also contradicts our finding that $p+q$ must be less than 8.

Therefore, the expression $p+q < 8$ is the one that satisfies the given inequalities.