If the median of Set A above is one less than the mode of Set A, which

The mode of the set is the most common term in that set. Since all of the other numbers in the set only appear once, then in order for the set to have a unique mode, x has to be equal to one of the other numbers in the set. (Otherwise, if x was a different number, all of the numbers in the set would be repeated exactly once, and the set would have more than one mode.)

That lets us eliminate (C) and (E) immediately.

Try the remaining answer choices, and check whether the median is one less than the mode:

(A) if x = 3, the set is 1, 3, 3, 5, 7, 11. The median is (3+5)/2 = 4, and the mode is 3. That's incorrect (we want a median that's 1 less than the mode, not 1 greater.)

(B) if x = 5, the set is 1, 3, 5, 5, 7, 11. The median is (5+5)/2 = 5, and the mode is 5. That's incorrect

(D) if x = 7, the set is 1, 3, 5, 7, 7, 11. The median is (5+7)/2 = 6, and the mode is 7. D is the correct answer.

C and E can be eliminated immediately because if either of them were x, the set would have no mode. From there I went through the values and tested them. 7 is the only one that works, so the answer is (D).