Is x negative? - DS Question

[nte.wang]

Is x negative?

(1) At least one of x and x2 is greater than x3.

(2) At least one of x2 and x3 is greater than x.
___________
The answer to this problem states, "All non-negative values for x have been eliminated from contention, so it is impossible to find a NO answer. Therefore, the two statements together guarantee that the answer is YES and are therefore sufficient."

How can you assume that the two statements together guarantee the answer is YES? Don't you have to go through and test all the possible negative options to ensure they work?

Thank you!

[tim]

I think you may just not have read the solution thoroughly. All possible negative options have been dealt with.

[RonPurewal]

How can you assume that the two statements together guarantee the answer is YES? Don't you have to go through and test all the possible negative options to ensure they work?

Thank you!

no. if you know that a number is neither zero nor positive, then ... it's negative.

maybe what you're asking is, "how do we know that at least one of the negative numbers will work?", or, equivalently, "what if there aren't any numbers that satisfy both statements?"
if that's what you're asking, then you can rest assured that it's a non-issue: there will ALWAYS be at least one solution to the two statements together. (even in problems where you never have to combine them in the first place!) you don't have to test things to establish that; there will never be a "no solution" type of situation on DS.
so, if you know that there aren't any non-negative solutions, then all existing solutions must be negative -- and you know that such solution(s) must actually exist.

if that's not what you meant, then, unfortunately, that means you'll have to go back and review the basics of DS.