If xy < zy < 0, is y positive? 1. x < z 2. x is negative

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If xy < zy < 0, is y positive?

(1) x < z
(2) x is negative


When you modify the original condition and the question, it becomes (x-z)y<0에서 y>0? --> x-z<0?. x<Z에서 1) is yes, which is sufficient.
For 2), if x<0, y>0 derived from xy<0, which is yes and sufficient. Therefore, the answer is D.


 Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Can you just divide the original equation by Y?
XY < ZY < 0 --> X< Z < 0?

If not, why not?

Dividing an inequality by a variable whose sign is UNKNOWN is the biggest mistake you can make in inequality questions.

The sign of the inequality changes if you divide or multiply by a negative number. This is the reason why you CAN NOT divide by y.

If xy<zy ---> y*(x-z)<0 --> this means that either

Case 1: y>0 and x-z<0 OR
Case 2: y<0 and x-z>0

When you divide by y WITHOUT reversing the signs, you are inherently assuming that y>0. This is the very question asked.

as per the question, it asks y(x-z)<0;
I agree for option 1 is sufficient on the basis that x<z;
for the second option, since x is -ve
x=-5 and z=-4 then y(-5-(-4))<0; hence y is +ve
x=-5 and z=-10 then y(-5-(-10))<0; here y should be -ve

there are conflicting solutions whether y is +ve/-ve

where am I wrong??

From Statement 2, we don't get any information for z & we are not bothered about z, all we need to figure out is whether y > 0

We know from question prompt that xy < 0, hence if x < 0, then y > 0

I hope that helps


Thanks,
GyM

The #'s plugged in are not correct x < z, hence x=-5 and z=-10 is not valid

xy<zy<0
1: x<z, then -x>-z, if we subtract it from above inequality we will get y>0
2: because xy<0 and x<0, therefore y>0

Target question: Is y positive?

Given: xy < zy < 0,
Let's focus on this part of the inequality: xy < zy
Subtract xy from both sides to get: 0 < zy - xy
Factor out the y to get: 0 < y(z - x) (this will come in handy later)

Statement 1: x < z
Subtract x from both sides to get: 0 < z - x
In other words, (z - x) is POSITIVE
This means we can take our given inequality 0 < y(z - x) and divide both sides by (z - x) to get: 0 < y
So, the answer to the target question is YES, y IS positive
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x is negative
The given information tells us that xy < zy < 0
This means xy < 0
In other words, xy is NEGATIVE
So, if x is negative, then it MUST be the case that y is POSITIVE
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

RELATED VIDEO

Official explanation

Solution: D

This difficult inequality problem requires that you properly leverage the information from the question stem. The second statement is relatively easy so you should start there – if x is negative and xy is negative then y must be positive. The second statement is sufficient. Since the second statement is relatively straightforward you should expect the 1st statement to be harder: from the question stem you know that xy < zy. If y were positive then you could divide by y and see that x < z. If y were negative then you could divide both sides to see that x > z. Since statement 1 proves that x < z then it must be the first case and y must be positive. Answer is D. Number picking is awkward and time consuming on first statement so don’t forget to use algebra as your first approach in any inequality problem.

xy<zy<0.

Tip 1: xy<zy, cancel y.. x<z.. dont do that. This is because you dont know what y is (positive or negative)

So, xy<zy,
or, xy-zy<0,
or, y(x-z)<0.

So, Either y<0, therefore in this case, (x-z)>0 .....(i)
OR, (x-z)<0, therefore in this case, y>0 ........(ii)

(1) x<z, which means (x-z)<0, which means equation (ii) is valid. y>0. SUFFICIENT
(2) x is negative, we know xy<0, if x negative, then y must be positive. SUFFICIENT

Answer: D

\(xy - zy < 0\)

\(y(x - z) < 0\)

\(y < 0 or x < z\)

Is \(y > 0?\)

(1) If \(x < z\), we can conclude \(y > 0\). SUFFICIENT.

(2) If \(x < 0\), and we know \(xy < 0\), then \(y > 0\). SUFFICIENT.

Answer is D.

maam in statement 1 how did u know that x and z are both negative it is just given that x is less than z .
z can be positive and still the condition can be satisfied.KarishmaB

We have xy < zy < 0. There are only two cases possible:

Case 1: y is positive. dividing by positive y and keeping the sign gives x < z < 0.
Case 2: y is negative, dividing by negative y and flipping the sign gives x > z > 0.

(1) x < z

We have the first case: y is positive and x < z < 0. Sufficient.

(2) x is negative

We have the first case: y is positive and x < z < 0. Sufficient.

Answer: D.

Hope it's clear.


P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

• Word Problems
• Work Problems
• Distance Problems
• Mixture Problems
• Percent and Interest Problems
• Overlapping Sets Problems
• Statistics Problems
• Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this and similar questions.

Hope it helps.

heyBunuel , need your help . i am preparing for past 4 months had to give the exam in november but was not getting good score in mocks so decided it to push for next 2 months.right now working on weak areas that is inequality ( DS) and work rate. i am facing issues in CR as well. any possible way to remove my fear of CR?


Thanks for sparing some time!!