1. **State the problem:** We want to find the positive integer value of $x$ such that the expression $3x + 9$ is a multiple of 7. 2. **Formula and rule:** An expression is a multiple of 7 if it is divisible by 7, i.e., $3x + 9 \equiv 0 \pmod{7}$. 3. **Set up the congruence:** $$3x + 9 \equiv 0 \pmod{7}$$ 4. **Simplify the constant term modulo 7:** $$9 \equiv 2 \pmod{7}$$ So the congruence becomes: $$3x + 2 \equiv 0 \pmod{7}$$ 5. **Rewrite the congruence:** $$3x \equiv -2 \pmod{7}$$ Since $-2 \equiv 5 \pmod{7}$, we have: $$3x \equiv 5 \pmod{7}$$ 6. **Find the multiplicative inverse of 3 modulo 7:** We want a number $k$ such that: $$3k \equiv 1 \pmod{7}$$ Testing values, $3 \times 5 = 15 \equiv 1 \pmod{7}$, so the inverse of 3 modulo 7 is 5. 7. **Multiply both sides by 5:** $$x \equiv 5 \times 5 \equiv 25 \equiv 4 \pmod{7}$$ 8. **Conclusion:** The smallest positive integer $x$ that satisfies the condition is: $$\boxed{4}$$ This means when $x=4$, $3x + 9 = 3(4) + 9 = 12 + 9 = 21$, which is divisible by 7.