## Inequality

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### Inequality

by yuliawati » Wed Nov 10, 2010 2:16 am
The official explanation is very long. When I met this question, I can't get the answer within 2 min, then I just guessed and moved. I can eliminate A, B, and D. But, out of time to combine both statements together. Expert, please help!

If a and b are integers, and |a| > |b|, is a Â· |b| < a - b?

(1) a < 0

(2) ab >= 0

Source: MGMAT

OA: E

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by selango » Wed Nov 10, 2010 3:13 am
Combining 1 and 2,

a<0 and ab>=0

-->b<=0

a=-1,b=0; a Â· |b| > a - b

a=-1,b=-1; a Â· |b| < a - b

Not sufficient

Pick E
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by [email protected] » Wed Nov 10, 2010 11:01 am
Hey yuliawati,

Good question - and if you didn't want to grind through the algebra on this one, you can also get a pretty good feel for it by paying particular attention to the specifics of the statements. To me, statement 2 offers some tremendous insight into this one:

ab > OR = 0

Well, if we know that a is negative, then that means that b is either negative OR it is 0. And 0 is the great equalizer when it comes to multiplication - as soon as I see >= or <= or "nonnegative" I immediately start thinking about how I can use 0 to prove that a statement is not sufficient. So if we're taking these statements together, let's try an example with b as negative and an example with b as 0 and see if that does, indeed, provide the difference we need:

Both Negative:

We need a to be less than b to satisfy the absolute-value inequality above, so let's try:

a = -2, b = -1

Then a * absval (b) = -2*1 = -2

and

a - b = -2 - (-1) = -1

So we get the answer "YES" the first statement is less than the second.

Now we want to get that answer "NO", and my hunch is that 0 is going to play a major factor...it almost always does when the authors of a question have to make specific allowances for it. So let's try:

a = -2, b = 0

The first term multiplies -2 * 0, so that's 0.

The second term is a - b = -2 - 0, so that's -2

And in this case the first term is NOT less than the second, so we get the answer NO, and can prove that both together are still insufficient.

To me, the biggest key is that recognition that if a question makes a specific allowance for 0 as a potential variable, you have to try 0 because it's probably a difference-maker. This question is littered with a few 0-clues:

-"Absolute value" means "distance from 0", so 0 may well come into play
-ab >=0 is very different from ab > 0 . That = makes a specific allowance for 0
-You have an inequality with multiplication by a variable on the left as part of the question...that should at least prompt you to ask "what if a variable is 0 and that whole term is just 0 as a result?"

Learn to love zero in Data Sufficiency - this blog post may be helpful: https://www.veritasprep.com/blog/2010/09 ... ts-denard/
Brian Galvin
GMAT Instructor