Here's the thought process and the scratch they write down:

Here's my starting equation:

x^2 - x = 4/x

I am asked to verify 2 is a solution. Let's plug it in. Are the two sides equal?

2^2 - 2 ?=? 4/2

I don't know. It's not obvious. Let's simplify each side a little. Are the two sides equal?

4 - 2 ?=? 2

I don't know. It's not obvious. Let's simplify each side a little. Are the two sides equal?

2 ?=? 2

YES. It's obvious to me. I am convinced 2 is a solution. BIG CHECKMARK.

If they are doing it right, they are THINKING 2^2-2=4-2=2 and 4/2=2=2, but that is not what they are writing down. They are thinking, okay, I can go down the left hand side and up the right hand side.

For your other example, they are really unsure and start applying functions to both sides that might not be invertible, causing problems.

sqt(9) ?=? 3

Are they equal? I don't know. It's not obvious. Hmmm. I don't know how to take the square root of 9. Let's square both sides, instead. If the square is equal the square root must be equal, right?

9 ?=? 9

YES! So, I can start with 9 = 9 and go up the page to what we were asked to verify. BIG CHECKMARK.

If you can give them an example of where this informal scratch work actually leads "verifying" an incorrect answer, then that might be a good example to share.