1. State the problem: Find the smallest prime number $x$ such that $3 - 4x < -41$. 2. Solve the inequality: $$3 - 4x < -41$$ Subtract 3 from both sides: $$-4x < -44$$ Divide both sides by $-4$ (remember to reverse the inequality because dividing by a negative number): $$x > 11$$ 3. Find the smallest prime number greater than 11. The prime numbers greater than 11 are 13, 17, 19, ... 4. Conclusion: The smallest prime number $x$ satisfying the inequality is $\boxed{13}$.