For 7x + 96 to be divisible by x, it implies that 96 must be divisible by x, since 7x is always divisible by x. Therefore, x must be a positive divisor of 96. The number of divisors of 96 can be found by first determining its prime factorization (96 = 2^5 * 3^1). The total number of divisors is then (5+1)*(1+1) = 6 * 2 = 12. Hence, there are 12 possible values for x.