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?

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.

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.

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.