When can we manipulate the Qstem in inequalities?

Sometimes you can interpret a question that uses an inequality even if you don't know the signs. Consider the following question stem:

Is x/y > 1? (There would probably be a disclaimer that y is not 0, but the GMAT isn't going to ask us about division by 0.)

We can't multiply both sides by y without knowing if it's positive or negative, but we can look at the two cases.

If y is positive, the question is now "Is x > y?"
If y is negative, the question is now "Is x < y?"

So now we have a two-part question. We need to know whether y is positive or negative. Now let's look at two statements:

1) x < y

This answers the second part, but since we don't know whether y is + or -, it doesn't give us a full answer. If y is positive, then this gives us a NO: x is not greater than y. If y is positive, we get YES: x is less than y.
INSUFFICIENT

2) y is negative.

By itself, this of course tells us nothing about x.
INSUFFICIENT

1&2) Putting both statements, together, we see we're in the second case: y is negative, so we want to answer the question "Is x < y?" Statement 1 gives us a YES to that question.
SUFFICIENT.

Wow that's a tricky one.. These things make me hate inequalities, when ones thinks it's all under control there is always something like this that shows up...

Thank you so much for this example! Hope there aren't many like this on the actual GMAT

I've seen a real GMAT q with a similar trick, but it's not the norm. However, you could probably nail this one by testing numbers without getting too theoretical. Just ask yourself "If x<y, does x/y have to be greater than 1?"

Then you could test:

If you plug in two positive integers that fit the statement, say x=2 and y=3, then you'll get a NO. x/y is not greater than 1.
If you plug in two negative integers, such as x=-3 and y=-2, then you get a YES.
Insufficient.

2) Again, you can test + and -. I can get a YES by making both variables negative. If x is negative, I'll get a negative fraction, so that's a NO.

1&2) Now you're more limited. y is negative, and x has to be less than y. No matter what you plug in, x/y will always be > 1.

For instance, if y=-2 and x=-3, then x/y=3/2.
If y=-4 and x=-100, then x/y=25.
SUFFICIENT

What did you understand when the question asked you "is x/y > 1"? In what cases will you get a yes and in what cases will you get a no?

From the first glance, x/y >1 (if its true) should tell you:

1. x and y MUST be of same sign as only then a fraction is >0. Think : negative/negative or positive / positive)
2. If x/y > 1 ---> the absolute value of x > absolute value of y ---> |x| > |y| ---> this statement when taken in isolation should mean that when y<0, x<y and when y>0 , x>y. If you dont get this straightaway, just test a few cases out.

Build into such understanding and you will get better at quant.

Dmitry and Engr,

Thank you very much for your input on this.

By default I don't like to test values, I use to do it a lot when I started, but if the stem doesn't have restrictions then you have tricky combinations to test like -1<y<0 and x>1, then reversed, then same sign, then both integers, then both fractions etc etc and for me 2 minutes would be up. So I always try to get the logic and then if I'm stuck go to testing values as a last resort.

This case here is very interesting from the abstract point of view, if you don't test values.

Ones again, thank you both!

Check here: inequalities-tips-and-hints-175001.html

Hope it helps.

Bunuel, thank you very much for the link. Just checked it, it has all the answers I needed!

:D

I think the replies by DmitryFarber are sufficient.. I personally rely on Manhattan resources when prepared for my GMAT exam.

Imagine that we have a>0 b>0 c>0

\(\frac{a}{c}\)=\(\frac{d}{b}\)

So we know that a,b,c are positive but we know nothing about d

Are we allowed to cross-multiply b by \(\frac{a}{c}\) to get \(\frac{ba}{c}\)=d

(Assume that the expression in this form becomes useful to evaluate something) So basically can we multiply\divide the terms that we are sure to be positive even if there is another term we know nothing about? Or a situation like this is unlikely on GMAT? I am asking this to avoid not manipulating terms when I cold have.

Thank you!

Multiplying/dividing by positive/negative is relevant only for inequalities, when we are concerned about the sign of the inequality.

iliavko, if you meant that to be an inequality, then yes, that's a valid move. (Of course you could do that with an equation, too.)

We only have to worry about flipping the sign when we actually multiply/divide by a negative. Let's take a look at an example:

x/y < 1

If y is positive, then we can just move it over and get x<y. For x/y to be less than 1, x would have to be either a smaller positive number (yielding a conventional fraction) or a negative number (yielding a negative, which of course will always be <1). For instance, x could be 2 and y could be 3, or x could be -2 and y could be any positive number.

If y is negative, we have to flip the sign when we move it over. x >y. Why is this so? Think about the kind of values you'd need to make it true. If we copy from our last example and just make the values negative, we could say x= -2 and y =-3. This of course will give us 2/3 and satisfy the inequality. But notice that while 2<3, -2 is actually GREATER THAN -3.

So basically, the whole reason we flip the sign is that negative numbers are counted backwards. The "smallest" ones are actually the biggest, making -1 much bigger than -100. Since the whole system runs opposite from the way we count positive numbers, whatever holds true on the positive side is reversed on the negative side.

For that reason, when we don't know if a variable is positive or negative, we can't multiply/divide it to the other side, because we don't know if we need to switch scenarios. No other action is going to cause the same problem (unless we're unsquaring or removing and absolute value sign), so the other moves we know are safe.

That was supposed to be an inequality, sorry! Thank you very much for your explanations!

Btw, assume that my example is an equation, so the signs wouldn't be important and we know that denominators are never 0. But would we have to know that the numerators are not 0 to cross-multiply? Could it be a trap?

And look, sorry for these questions, but I am constantly discovering these "holes" in my math basics.. And it's tough to find the exact answer in a book or something

No, we wouldn't have to worry about that. One easy way to think about it is that if a variable isn't "moving," then you don't need to know anything about its sign for the manipulation to work. You only need to worry about the elements you are actively manipulating.

By the way, if one of the numerators in that equation did equal zero, then they would both have to. Since the denominators can't be 0, it would have to be both numerators to make the sides equal. So we could get something like ba/c = d or ba = cd, but it wouldn't really be giving us any information about b and c. Zero times anything is equal to zero.

Thank you very much for the clarifications, Dmitry!

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