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Inequalities

by tonebeeze » Thu Jan 13, 2011 8:22 am
If xy>0 and yz<0, which of the following must be negative?

a. xyz
b. (x)(y)(z^2)
c. (x)(y^2)(z)
d. (x)(y^2)(z^2)
e. (x^2)(y^2)(z^2)

When testing positives or negatives in "variable in the answer choice" inequalities questions, what is the best approach to solve these problems? I got the problem correctly, but the testing took over 3 mins.
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by anshumishra » Thu Jan 13, 2011 8:38 am
tonebeeze wrote:If xy>0 and yz<0, which of the following must be negative?

a. xyz
b. (x)(y)(z^2)
c. (x)(y^2)(z)
d. (x)(y^2)(z^2)
e. (x^2)(y^2)(z^2)

When testing positives or negatives in "variable in the answer choice" inequalities questions, what is the best approach to solve these problems? I got the problem correctly, but the testing took over 3 mins.
xy>0 => x,y -> same sign
yz < 0 => y,z -> diff sign

So, xyz = (+ve)*z = can be anything
xy(z^2) = xy*(+ve) = xy = +ve
x(y^2)z = xz*(+ve) = xz = -ve (As x and y have same sign and y and z different, so x and z have different sign).
C
Thanks
Anshu

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by MAAJ » Thu Jan 13, 2011 1:18 pm
The best approach? discart incorrect choices and test answer choices that may be correct. This would be easier if you understand the information given in the question stem, and other fundaments.

In this specific case, I bet they are testing:

Negative * negative = positive AND positive * negative = negative
Negative number powered to a even number is always positive

and... ehmmm... that's about it....

If you know this, then you can easly discart choices b) and e) and d) as they all yield positive numbers!!!

So test answer choices a) and c)!!!! and pick the right one which is C)

Note: It's important to rephrase things, like in the comment above xy> 0 means that x and y have the same sign and yz<0 means that they have different sign!
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by Adam@Knewton » Thu Jan 13, 2011 2:20 pm
The reason all GMAT test-prep programs stress Testing Cases is that it always works, even when you're totally stumped on what Number Properties they're testing. However, here, as MAAJ pointed out, it shouldn't be that hard to deduce the Number Properties themselves and speed through the choices:
xy>0
This means that x and y are the same sign.
yz<0
This means that y and z are different signs.

Learning to read these kinds of "<0" or ">0" statements in this way will really speed you up on test day. It's not a very difficult deduction to make, now, that if x and y are the same and y is different from z, x and z are different signs.

Now, the answers:

A) xyz --> xy>0, we know that, so it all depends on z. We don't know the sign of z, so, we don't know. Eliminate.
B) xy>0, we know that, and z^2>0 because anything^2 is always positive. Eliminate.
C) y^2>0, by rule, and xz will be negative since they are different signs. This must be negative, and is the answer.
D) y^2>0 and z^2>0 by rule, but we don't know the sign of x, so it depends. Eliminate.
E) All three terms are positive by rule, so this must always be positive. Eliminate.

On test day, we wouldn't go through all 5 choices; I just did it here to show you that a small amount of number theory can really make this type of problem very doable, and very efficient, regardless of what they ask us for or what they give us.
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