Correct answer, but also not

[Loki123]

Okay, I am beyond confusion for this one. The first picture is the problem and how I solved it. The second picture it how it's showed to be solved.
Basically the only difference is that I turned sqrt(x-1) on the RHS and the solution turned sqrt(2x-3). Because of this I got a condition x=>5/2 (I outlined it and put a star next to it) and they got a condition x<=3 (i outlined that one too). We get the same answer, however, that difference bugs me. Is there a reason I must turn sqrt(2x-3) to the other side? Like maybe because it is the only negative one? Or is the solution wrong for not including other possibilities? How can we get different conditions with the same problem?
Thank you in advance.

 

[Dr.Peterson]

As I said in the other thread, there is often more than one way to solve a problem!

The fact that you didn't do every step the way they did doesn't mean you are wrong.

I would probably have done what they did, simply because the RHS becomes positive, and I tend to avoid negatives. (You omitted a negative sign on your RHS, though it didn't matter because you then squared it.)

But that doesn't mean you have to do it that way; it's just a convenient habit.