jmc

First, we know that $x$ is even. Every prime number larger than $3$ is odd, so the sum of three members has to be even.

Now, $12$ does not work because $(2+10+12)+1=25$, which is not prime.

Similarly, $14$ does not work because $(2+4+14)+1=21$, which is not prime.

But $16$ works because all the possible sums involving $x=16$ generate prime results: $(2+4+16)+1=23$, $(2+10+16)+1=29$, and $(4+10+16)+1=31$.

So the smallest possible value of $x$ is $\boxed{16}$.