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Is \(zx > yx\)? --> is \(zx-yx>0\)? --> is \(x(z-y) >0\)?

(1) z > 0. Clearly insufficient.

(2) y < 0. Clearly insufficient.

(1)+(2) Now, remember that we can subtract inequalities with the signs in opposite direction --> subtract (2) from (1): \(z-y>0\). So, we know that the second multiply in \(x(z-y)\) is positive, but we still don't know the sign of \(x\): from the stem (\(x>y\)) and the second statement (\(y<0\)) the only thing we can deduce is that \(x\) is greater than negative number \(y\), hence \(x\) could be negative, positive or zero. Not sufficient.

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Does anyone have advice on how to tackle these problems faster? I always end up resorting to plugging in numbers and various conditions and even though that helps me ensure more accuracy, it usually takes me about 3 min. per such problem (inequalities with only variables).

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St.1 z>0. INSUFF (we do not know whether x,y>0) St.2 y<0. INSUFF (we do not need other variables because if one is negative we cannot answer definitely) St.1+St.2. Still y<0, so INSUFF

just to make sure i understood it: zx > yx --> we cannot simplify to z > y because we don't know if x is positive or negative, right?

Yes exactly. In inequalities be very careful with cancelling variables without knowing for sure what signs are they.

In the current question, zx>xy --> x(z-y)>0

After combining the 2 statements, you do know that -y>0 and z>0 ---> z-y>0 but without knowing the sign of 'x', you will not be able to answer yes or no for the question asked.

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