If 315x is a multiple of 9, where x is a digit, then the value of x will be:

Concept: A number is divisible by 9 if and only if the sum of its digits is a multiple of 9 (the standard digit-sum divisibility rule for 9).

Applying this to 315x:

1. The digit sum of 315x is 3 + 1 + 5 + x, which simplifies to 9 + x.

2. For 315x to be divisible by 9, the sum 9 + x must itself be a multiple of 9.

3. Since x is a single digit (0-9), the only multiples of 9 reachable are 9 (when x = 0) and 18 (when x = 9).

4. Among the given options (1, 3, 6, 9), only x = 9 makes 9 + x a multiple of 9.

Cross-check: substituting x = 9 gives the number 3159, and direct division confirms 3159 divided by 9 equals 351 exactly, with no remainder -- verifying the divisibility independently of the digit-sum rule.

Hence, x = 9.