About the solution
This is a fairly straightforward question and the hints do help students to see the truth / falsity of the statements easily.
The thing that I object is that if I were a student, I wish to first see what I should answer in the exam, which should not be too wordy.
Then I would perhaps also like to see further elaborations and other cases in a subsequent paragraph, but I would prefer it to be short and succinct. Discussing a possible wrong route in the last paragraph do not seem helpful at all. I suggest removing the paragraph with the first sentence
"If you wanted to have a statement of that kind where a unique real number x works for any choice of w then you have to first state the existence of that number x."
Count the number of prepositions and punctuations! This type of English sentence only comes from hard-core mathematicians!