# why can't I square both sides of the inequality in the case?

here's the question:

sqrt (x-1) + 1 > 0

I subtracted 1 to both sides, giving me:

sqrt (x-1)> -1

I then decided to square both sides, so that I could get rid of the square root.

that gives me:

(x-1)>1

I then added 1 to both sides, giving me x>2

I read somewhere that square rooting here was a mistake, but I'm not sure exactly why?
here's the question:

sqrt (x-1) + 1 > 0

I subtracted 1 to both sides, giving me:

sqrt (x-1)> -1

I then decided to square both sides, so that I could get rid of the square root.

that gives me:

(x-1)>1

I then added 1 to both sides, giving me x>2

I read somewhere that square rooting here was a mistake, but I'm not sure exactly why?

Do you have any ideas? Have you plotted the original expressions involved in the inequaity, subbed the answer in, worked backwards to see if you can get back to the original inequality, ...?

I presume you mean squaring was a mistake in the working as opposed to square rooting? Can you post the original question?
(edited 10 months ago)
here's the question:

sqrt (x-1) + 1 > 0

I subtracted 1 to both sides, giving me:

sqrt (x-1)> -1

I then decided to square both sides, so that I could get rid of the square root.

that gives me:

(x-1)>1

I then added 1 to both sides, giving me x>2

I read somewhere that square rooting here was a mistake, but I'm not sure exactly why?

The squaring step is what causes the trouble. More caution is needed.

Take x=1 for instance it satisfies sqrt(x-1) > -1 but does not satisfy x-1 > 1.
Original post by RDKGames
The squaring step is what causes the trouble. More caution is needed.

Take x=1 for instance it satisfies sqrt(x-1) > -1 but does not satisfy x-1 > 1.

For (hopefully obvious reasons concerning the range of the sqrt function), it seems extremely likely something has gone wrong *before* arriving at the inequality sqrt(x-1)+1 > 0.
Original post by DFranklin
For (hopefully obvious reasons concerning the range of the sqrt function), it seems extremely likely something has gone wrong *before* arriving at the inequality sqrt(x-1)+1 > 0.

I think the OP is at uni and the question reads like a tutorial question to think about. But it seems like you could argue either about the sqrt or the squaring being increasing and it would pretty much amount to the same thing. The former is probably more obvious though the latter is more interesting in terms of what operations you can do to an inequality.
(edited 10 months ago)
When is the LHS 0 or less? Well, never... (why?). So really, the question is, when is the LHS a valid expression?
Original post by mqb2766
I think the OP is at uni and the question reads like a tutorial question to think about.
It doesn't seem it to me - it reads like either a poorly written question, or (more likely) a place they've reached in their working (probably incorrectly).

I feel a bit hamstrung in avoiding spoilers, but if that was the original question, it seems ill-posed on multiple levels, so I'd certainly hope it wasn't a university/tutorial question.

But it seems like you could argue either about the sqrt or the squaring being increasing and it would pretty much amount to the same thing. The former is probably more obvious though the latter is more interesting in terms of what operations you can do to an inequality.
To my mind, once you square, you bring in a whole lot of unnecessary considerations. By definition, sqrt is {spoiler-about-range-of-sqrt}. What more needs to be said?
Original post by DFranklin
For (hopefully obvious reasons concerning the range of the sqrt function), it seems extremely likely something has gone wrong *before* arriving at the inequality sqrt(x-1)+1 > 0.

Hard to say if the first equality is the start of a problem or a consequence in a wider problem. Certainly if it is a consequence then I would check for errors.

If it’s just a problem on its own then yeah no need to square just argue about the range, but nevertheless it’s perhaps an interesting problem to pose at A-Level to get students thinking about the exact logic trap OP has fallen into.
(edited 10 months ago)