Negate ∃x (2x + 1 = 7)

Step by step

The question

Question: write the opposite of ∃x (2x + 1 = 7).

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

The opposite of 2x + 1 = 7 is 2x + 1 ≠ 7.

4 Say it in words

In words: for every x, with 2x + 1 ≠ 7.

5 Check

Let's check with x = 3.

The first inside: 7 = 7, that is true.

The new inside: 7 ≠ 7, 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.