Step by step
The question
Question: write the opposite of ∃x (x² = 9 ∧ x < 0).
The backwards E means: there is some.
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
There is no x means: for every x, it fails.
So ¬∃ becomes ∀¬.
Careful!
There is none means every one fails, not just one.
3 Turn the inside around
De Morgan: the not goes onto each part, and and turns into or.
Now each part: x² = 9 becomes x² ≠ 9, and x < 0 becomes x ≥ 0.
4 Say it in words
In words: for every x, with x² ≠ 9 ∨ x ≥ 0.
5 Check
Let's check with x = 3.
The first inside: 9 = 9 ∧ 3 < 0, that is false.
The new inside: 9 ≠ 9 ∨ 3 ≥ 0, that is true.
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.