I've seen examples where Wolfram|Alpha seems like it tends to treat names w,x,y,z as real variables and names m,n, maybe even i,j.k as integer variables. What you are seeing looks like it might be another example of this.

I am not trying to defend what it did or to be critical in any way of what you did..

Your query didn't specify whether the solution should be real or integer. It didn't specify whether it should give the smallest value which satisfied the problem and include that all larger values also satisfied the problem. It didn't specify that you wanted all possible solutions or some solutions or even just one solution.

For problems over the real numbers you might know a variety of different ways of finding solutions. For problems over the integers there are perhaps fewer methods that can be used. Perhaps that might explain the result that it gave, but there is likely no way way of knowing exactly why it did what it did.

If you modified your question slightly then

(.5)^(2(x-1))/(2(x-1))!<10^-7 and 3<x<7

gives a variety of different solutions.