Answer
True
Work Step by Step
**Proof by Contradiction:**
Assume for the sake of contradiction that
\[
3\sqrt{2} - 7
\]
is rational. Then there exists a rational number \(q\) such that
\[
3\sqrt{2} - 7 = q.
\]
Adding 7 to both sides, we have
\[
3\sqrt{2} = q + 7.
\]
Since the sum of rational numbers is rational, \(q + 7\) is rational. Dividing both sides by 3 (a nonzero rational) yields
\[
\sqrt{2} = \frac{q + 7}{3}.
\]
This shows that \(\sqrt{2}\) is rational, which is a contradiction because it is a well-known fact that \(\sqrt{2}\) is irrational.
Thus, our assumption is false, and \(3\sqrt{2} - 7\) is irrational.
