Inequalities

by , Feb 6, 2012

Today weve got an inequalities data sufficiency question on tap from GMATPrep. Set your timer for 2 minutes and go!

*Is m + z > 0?

(1) m 3z > 0

(2) 4z m > 0

This is a yes/no data sufficiency question. Im just going to remind myself of the rules: an always yes answer to a statement is sufficient, an always no answer is also sufficient, and a maybe or sometimes yes / sometimes no answer is not sufficient.

Youll note that I havent listed the five answer choices. Do you know what they are? If you havent memorized them yet, add that to your to-do list. (If youre just starting out, stop reading this article and go study data sufficiency in general first. Then come back to this article once you feel comfortable with how DS works. :) )

The question stem is asking whether the sum of m and z is greater than zero. I can also read that as is m + z positive? Thats interesting. What would I need to know in order to answer that question?

Certainly, if I knew what the values of the two variables were, then I could answer the question. Theyre probably not going to give me that, though. What else? Lets see, if I knew that m and z were both positive, then their sum would also have to be positive. What if they are both negative? What if one is positive and one negative? One could even be zero.

If both were negative, then the sum would also have to be negative. If one were positive and one negative, then hmm, I couldnt tell without knowing more (such as the values of the numbers). Finally, one of the variables could be zero (but not both).

Okay, so I have some ideas about whats being tested. I dont really see a way to rephrase the question in a simpler way. Its fairly straightforward as it is.

The first statement doesnt look ridiculously hard, so Ill start with that one. (As a general rule, I start with the first statement unless I dont know what to do with it or it looks much harder than the second one. I know many teachers recommend starting with the easier of the two; I stopped recommending that because I dont want to have to ask myself 15+ times Which of these two statements is easier? I just want to GO! And if that statement is too annoying or hard, then Ill try the other instead.)

(1) m 3z > 0

Hmm. If you subtract something from m, the result is still greater than zero, so maybe that means m is positive? Oh, wait. No, that doesnt have to be true; m could be zero and z could be negative. What Im doing right now is thinking this through theoretically. I could also just try some simple numbers: if m = 5, z = 1, then 5 3(1) > 0. The statement is true using these numbers, som= 5,z= 1are valid numbers to try. Then plug those numbers into the question Ism+z> 0? to get is 5 + 1 > 0? The answer to the question is yes. What ifm= 0,z= -2? Then 0 3(-2) > 0, so these are valid numbers to try. But the answer to the question Ism+z> 0? is no (because 0 + -2 is not greater than zero). Sometimes yes, sometimes no? Statement 1 is insufficient.

I could also rearrange the statement to get m > 3z, especially if I want to think this through theoretically. Its easier to see that I have no idea whether m and z are positive, negative, or zero given this inequality.

Cross off answers A and D.

(2) 4z m > 0

We can use the same approaches for this second statement. Theoretically: the inequality is 4z>m. These variables could be positive, negative or zero using the same reasoning we used above. Trying numbers: ifm= 1,z= 2, then 4(2) 1 > 0, which is true. The answer to the question Ism+z> 0? is yes (because 1 + 2 > 0). Ifm= -1,z= 0, then 4(0) (-1) > 0, which is true, and the answer to the question Ism+z> 0? is no (because 0 + -1 is not greater than zero). Statement 2 is insufficient. Cross off answer B.

Now, we need to look at the two statements together. Whenever we get to this step of a DS problem, our first task is to figure out how to combine the information weve been given. Combining the info might tell us something new.

How can we combine this info? The key is to look at the two rearranged inequalities:

m > 3z

4z > m

What do these inequalities have in common? Both of them have an m on one side. Great! We can combine these into one 3-part inequality:

3z < m < 4z

Wait, how can we do that? Check it out: the first statement tells us that 3z is less than m. Thats also what our new 3-part inequality tells us. And the second statement tells us that m is less than 4z just like our new 3-part inequality! Just be careful to make sure that you're preserving the original relationship (direction of the inequality sign). Now what?

The value of m is somewhere between 3z and 4z. I notice thatzcannot be zero (because thenmdoesnt exist there isnt anything that is both less than zero and greater than zero). Andmcant be zero either. This ones a bit more subtle. 3zand 4zare either both positive or both negative. In either case, you cant have zero in the middle of two positive or two negative numbers.Finally, I also notice that the two variables must have the same sign. We cant have a negative m in the middle with a positive z on either side. Nor can we have a positive m with a negative z on either side.

So, either both are positive or both are negative. Can I tell which? Its certainly possible for them both to be positive. For example, lets say that z = 1 and m = 3.5 which makes 3(1) < 3.5 < 4(1) a true statement. If thats the case, then the answer to the question Is m + z > 0? is yes.

What about two negatives? Lets say that z = -1 and m = -3.5. Then we have the inequality 3(-1) < -3.5 < 4(-1) or -3 < 3.5 < -4. Oops. Thats not a true statement; I cant pick those numbers. Ill have to find some other numbers. Lets see. If I set z = -2 hmm. If z is negative, and I multiply it by both 3 and 4, the 4z number is always going to be farther from zero on the number line; in other words, 4z is always going to be smaller than 3z(when z is negative). But my inequality 3z < m < 4z tells me that 3z is always less than 4z. So z cant be negative; it must be positive! And since m has to have the same sign, m is also positive.

If m and z are both positive, then the answer to the question Is m + z > 0? is always yes. The correct answer is C.

Were not quite done talking about this one. Look at the title of this article again. Now, look down the page a little at what I wrote for the title of the Key Takeaways section. Heres our big lesson on this problem!

In general, most number properties issues are too easy to write in a straightforward way. The test writers instead have to find a way to disguise what it is that theyre really asking. Inequalities are a great way to disguise positive / negative theories. Thats what this problem really ended up being about, right? We didnt have to do a bunch of algebra or weird inequality manipulations. We had to figure out whether these variables represented positive or negative numbers.

I call these Know the Code problems. It takes too long to figure everything out from scratch while the clock is ticking. Weve got to know the code in advance know, for example, that < 0 and > 0 is almost certainly testing us on positive and negative number properties theory, so that we can jump to that immediately. Grab a notebook or open a file on your computer and start keeping a log of Know the Code phrasings. In particular, take note of situations where you finish the problem and think I didnt see that coming or you read an explanation and think, Oh, thats what that question was all about? Then write something down that fills in these blanks: When I see _______ I will think / do _______.

Key Takeaways for Disguised Number Properties Questions:

(1) The GMAT test writers are really good at disguising number properties in general. Inequalities are often used to mask number properties questions, in particular problems that deal with positive and negative. Study from that point of view look for these disguises!

(2) Start keeping a log of all of the ways in which you discover that they distracted you from realizing that a problem was really about number properties (or anything else!).

(3) When you get stuck, try some real numbers to understand whats going on. Dont go over time; if you have to guess, do so and move on. Afterwards (if its not the real test!), you can go back and try numbers until you figure out exactly how the problem works and why.

* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.