Some number testing principles for DS:
The question may provide a clue to what the question is "about," and therefore what numbers to try.
Translate the question to see what I mean:
Is xy > 0?
Is the product of x and y positive?
Do x and y have the same sign?
Insight: This is about the sign of x and y, NOT so much about their values.
Number Picking Strategy: Only 4 basic cases (positive x and y, negative x and y, pos x and neg y, neg x and pos y)
You may not (probably won't) need to try every possible scenario.
It's generally easier to prove insufficiency than to prove sufficiency! For insufficient, you just need to find one Yes case and one No case (y/n question), or find two cases that produce different values (value question). Thus, to save time, remember that you have something to prove...If you have a "yes" case, your next goal is to find a "no" case.
Take statement (1) for example:
A "yes" case will be one where x and y have the same sign. Can x-y>-2 if x and y have the same sign? Sure, take x = big positive and y = smaller positive. That would make x-y=positive, which is >-2.
Now we need a "no" case to prove insufficiency. Can x-y>-2 if x and y have different signs? Sure, take x = positive and y = negative. That would make x-y=pos-neg=pos, which is >-2.
As you can see, you can "pick numbers" without picking specific numbers. Thinking about types of numbers is a time-saver.
Similar approach to (2):
"yes" case: Can x-2y <-6 if x and y have the same sign? Sure, if x = small pos (e.g. 1) and y = big positive (e.g. 10). Again, don't get hung up on the specific values, just check for the existence of some numbers that would work.
"no" case: Can x-2y<-6 if x and y have different signs? Sure, think x = negative and y = positive. That would make x-2y=neg-pos=neg. As long as x and -2y are negative enough, there are plenty of x-2y examples that are <-6.
There are two ways to combine statements when picking numbers:
1. Sometimes, the numbers or types of numbers we check on (1) are the same as those we check on (2). When we combine, the same cases apply. If that's not the situation, then you must...
2. Combine the statements algebraically as much as possible, then try new numbers from scratch.
There is no way around the algebra to combine (1) and (2) (see Ron's explanation of combining the inequalities). Trying numbers is simply too inefficient, perhaps because you have to meet both constraints and because they end up being sufficient together.
Thus, the final principle is that picking numbers is not the optimal approach, and sometimes it won't help much at all. Use the technique when it works, solve algebraically when it doesn't, and accept it for what it is.