“Is x Positive”: Why can’t I take the square root of both sides to see if statement 1 and 2 are the same?

Question

Why can’t I take the square root of both sides to see if statement 1 and 2 are the same? Although, this leaves me to choose the wrong answer, I was just wondering why taking the square root of both sides is not allowed.

Answer

The problem is that when we take the square root of something like this, we have to consider both the positive and negative square roots. So if we square both sides of this inequality, we actually get two resulting expressions:

x + y < x - y

and

-(x + y) < -(x - y)

which simplifies to:

x + y > x - y

So overall, we are left with these two expressions, meaning that Statement 1 is not exactly the same as Statement 2.

Link

http://gmat.magoosh.com/questions/969