What is the smallest positive integer $x$ with $3x \equiv 5 \pmod{11}$?

Since $3 \times 4 = 12 \equiv 1 \pmod{11}$, the inverse of $3$ is $4$. Multiplying both sides by $4$: $x \equiv 20 \equiv 9 \pmod{11}$. Check: $3 \times 9 = 27 = 22 + 5$.