Received a PM asking me to reply. Both examples are from MGMAT Strategy Guides and, as you can see, they are simply math problems, not GMAT-format problems. They're in the text of the chapter, simply describing how to do various kinds of math. (hariharakarthi is asking what would happen IF these were given as DS statements. They weren't actually presented that way in the book.)

#1

The rule cited here is true for math in general, not just the GMAT. When we have quadratics (esp. square roots!), we have to check the solutions in the original equations to ensure that they're valid. If they're not, then they're not actually considered valid solutions to the original equation and they have to be discarded. So, yes, in this case, there's only one valid solution and, if this were a DS statement and the original question were "what is x?" then this statement would be sufficient.

#2
Again, the rule cited here is true for math in general, not just the GMAT. When you solve an equation that contains an absolute value, and you make assumptions when testing the scenarios, those assumptions must fit the final answer. If they don't, then the final answer is invalid.

When trying to solve the specific equation given above, there are two scenarios: |N+9| is positive or |N+9|is negative.

If |N+9| is positive, then when you solve, you get N = 3. Plug that back in and |N+9| = |3+9| = 12, which is positive, so N = 3 is a valid answer.

If |N+9| is negative, then when you solve, you get N = -3. Plug that back in and |-3+9| = 6, which is positive. But we began this solution by assuming that |N+9|is negative, not positive. So this is not a valid solution.

The only valid solution is 3.