Step by step
The question
Question: write the opposite of ∀x (x > 2 → x² > 4).
The upside-down A means: for every.
1 Put not in front
The opposite of a statement: we put not in front of all of it.
Now we move the not inside, one piece at a time.
2 Move the not inside
Not true for every x means: there is some x where it fails.
So ¬∀ becomes ∃¬.
Careful!
Not every does not mean none.
If not every student is here, some student is missing.
It does not mean every student is missing.
3 Turn the inside around
An if-then fails only when the if-part is true and the then-part is false.
So the opposite of A → B is A ∧ ¬B.
And the opposite of x² > 4 is x² ≤ 4.
4 Say it in words
In words: there is some x, with x > 2 ∧ x² ≤ 4.
5 Check
Let's check with x = 3.
The first inside: 3 > 2 → 9 > 4, that is true.
The new inside: 3 > 2 ∧ 9 ≤ 4, that is false.
One true, one false: opposite, as they must be.
Right!
Let's look back.
First: put not in front.
Then: move the not inside.
Then: turn the inside around.
Then: say it in words.
Last: check.