Is x < 0? (1) x^2 - x > 0 (2) |x| < 1

Bunuel wrote:
Is x<0? ?

(1) x^2 - x > 0
(2) |x| < 1

This is yes/no question. (Is x a negative number)

Analysis of Statement (1):
x^2 - x > 0
=> x^2 > x
[img]blob:https://www.symbolab.com/e21e63c7-dcff-4a20-91eb-bb3b7a044e54[/img]
As presented in the above number line, what we can get from statement-1 is that either x> 1 or x< 0 & that x can neither be 0 nor be a positive proper fraction. Hence, statement-1 is clearly insufficient.

Analysis of Statement (2):
|x| < 1
=> -1<x<1
[img]blob:https://www.symbolab.com/6695dbd9-9e62-4376-ae99-7a6b382aa35d[/img]
As presented in the above number line, what we can get from this statement-2 is that x can be either 0 or any positive proper fraction or any negative number greater than -1. We do not get the definite answer to the question stem. Hence, statement-2 is clearly insufficient.

Statement-1 & Statement-2 combined rule out the possibility that x can either be positive or be 0. Statement-1 & Statement-2 combined tell us the definite answer that x is negative (x<0).
So, the answer is C.

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Since we have 1 variable (x) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)

\(x^2 - x > 0\)
\(⇔ x(x-1) > 9\)
\(⇔ x < 0\) or \(x > 1\)

Since condition 1) does not yield a unique solution, it is not sufficient.

Condition 2)
\(|x| < 1\)
\(⇔ -1 < x < 1\)

Since condition 2) does not yield a unique solution, it is not sufficient.

Conditions 1) & 2)
The intersection of two solutions from conditions 1) and 2) is \(-1 < x < 0\).
Then x is less than 0 all times and the answer is 'yes'.
Since both conditions together yield a unique solution, they are sufficient.

Therefore, C is the answer.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.­

Is \(x<0?\) ?

(1) \(x^2 - x > 0\) -->insuff: x(x-1)>0 => x <0 & x>1

(2) \(|x| < 1\) --> insuff: -1<x<+1

Combining (1) & (2) => -1<x<0, so sufficient
Answer: C

Hi

thanks for explanation.
will you please explain this equation
x(x-1)>0 => x <0
why x<0 why not x>0?

I know when my aim is to appear for GMAT I should have some basic knowledge. Yet will you plz explain how does first statement imply x<0.

Thanks­

Hello tannumunu,

\(x^2\) – x>0 is a quadratic inequality. When you have a quadratic inequality, factorise the expression to obtain the roots of the expression. These represent the critical points on the number line. Since the expression is quadratic, it will clearly have two roots.

When you mark two points on a number line, you will see that these points will divide the ENTIRE number line into three segments. Consider the right most segment as positive, the segment in the middle as negative and the left most segment as positive. Reason enough to call this the wavy curve method since the graph sweeps down from right to left and then rises up.

If your inequality has a ‘>’ sign, the segments with the positive sign represent the solutions to your inequality. On the other hand, if your inequality has a ‘<’ sign, the segment with the negative sign represents the solutions to your inequality.

Let’s simplify \(x^2\)-x>0 to obtain x(x-1)>0. What are the roots here? They are 0 and 1, right? Let’s mark them on the number line which will look like this:



Since we have a ‘>’ sign in our inequality, the range of x which will satisfy our inequality is x>1 OR x<0. Important keyword here – OR. x can be greater than 1 OR lesser than 0.
As we do not know which range it is, we cannot conclusively say if x>0 or not.

Talking about your other argument – why not x<0?
x(x-1)>0. What does this mean? This means that the product of x and (x-1) has to be positive. This can happen only when both are positive or both are negative.

If x<0, (x-1)<0. If (x-1)<0, x<1. But, if x<0, then x<1 automatically, isn’t it? But, is x<0 the only range? No.

If x>0, (x-1)>0. If (x-1)>0, x>1. But, if x>1, then x>0 automatically, isn’t it?

Hope that clarifies your doubt. Here’s a link to our post on Quadratic inequalities. This will help you understand this concept better.
https://bit.ly/2X35jno

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