What is the value of x in the list {36, 47, 51, 65, x}?

What I could understand :
Main Stem:
What is the value of x in the list {36, 47, 51, 65, x}

As per part (1) :
The median of the 5 values is 51.

2 cases are possible :
a) we know as 51 is the median, so in a sorted list we would have something like :
36,47,51,65,x and 36,47,51,x,65

*So can cancel A and D as inconclusive we cannot determine the value of X from (1).

As per part (2) :
The range of the 5 values is 29.
Range of the set = highest term - lowest term

from (1) we can conclude that the lowest term in the list must be 36.
But the greatest can be either X or 65.

But the range is 29, therefore we can say that highest term is 36 + 29 = 65.
So in normal cases we could say that X is 65 and B would be sufficient HOWEVER, here we already have a 65 and maybe the answer is coming from that and not X.
i.e. X may be lesser than 65 or = 65 we cannot say hence B is inconclusive.

Now when we apply (1) and (2) together.
The same thing happens as even when we know that the median is 51 and on the right side we may have x and 65 but, we don't know which one out of them is bigger.
Therefore is x is bigger even than the range is 29 meaning X = 65 and we have 2 identical values in this list.
However, if 65 is bigger and not equal to 65 than it must be between 51<=x<65.
Still we cannot guarantee X's exact value.
We cross out C

Finally, we can conclude that the correct answer is E and X's exact value cannot be evaluated with the given information.

(In my mind )
-- l hope my logics correct, lets see on the answer reveal.
-- If not, lets hope I can get a good reasoning to correct my error.