Discussion
Consider the following inequalities.
(i) \(3p - q \lt 4\)
(ii) \(3q - p \lt 12\)
Which one of the following expressions below satisfies the above two inequalities?
Answer: A
The expression that follows is \(p + q \lt 8\).
Add the two inequalities, which is allowed because both point the same way:
\[ \begin{aligned} (3p - q) + (3q - p) &\lt 4 + 12 \\ 2p + 2q &\lt 16 \\ p + q &\lt 8 \end{aligned} \]
So every pair \((p, q)\) satisfying both inequalities has \(p + q\) strictly less than 8. The other options all allow \(p + q \ge 8\), which the sum rules out: \(p + q = 8\) would need \(2(p + q) = 16\), but the sum must be strictly less than 16.
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