To solve this problem, we can use the principle of inclusion-exclusion.

First, we find the total number of ways to invite X's friends to dinner. Since X can invite one or more friends, each friend has two options (either invited or not invited), so the total number of ways is 2^4 = 16.

Next, we consider the number of arrangements in which C is seated next to X. To do this, we can treat X and C as a single entity, and then arrange the other three friends around the circle. This gives us 3! = 6 arrangements. However, we must also consider the possibility that X and C are seated in the opposite order (i.e. C is to X's right instead of left). This gives us another 6 arrangements. So the total number of arrangements in which C is seated next to X is 12.

Similarly, we can find the number of arrangements in which D is seated next to X. Again, we can treat X and D as a single entity, and arrange the other three friends around the circle. This gives us another 12 arrangements.

However, we have now counted twice the arrangements in which both C and D are seated next to X. To correct for this, we must subtract the number of arrangements in which both C and D are seated next to X. To do this, we can treat X, C, and D as a single entity, and arrange the remaining friend around the circle. This gives us 2! = 2 arrangements. So the total number of arrangements in which both C and D are seated next to X is 2.

Finally, we can use the principle of inclusion-exclusion to find the total number of arrangements in which neither C nor D is seated next to X. This is given by:

16 - 12 - 12 + 2 = 6

So there are 6 arrangements in which neither C nor D is seated next to X.

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