I have a comment and question to make about some recent posts solving inequalities with a radical in them.

Consider the inequality:

I think we would all agree that

as a reasonable condition on the solutions to this inequality.

But from there I have seen several instances of the following procedure:

Thus the solution set for x would be .

But if you look at the graph of you will see that the only solutions are for .

So why does this happen? The only thing I can think of is that when then by squaring both sides of the inequality we are multiplying both sides by a negative number, switching the to . So the interval is discarded from our previous solution since this interval makes .

The new inequality gives

The solution set to this is . But this solution does not contribute at all to the solution set. What has gone wrong?

-Dan