If x 1, what is the value of (|x| - 1)/(x - 1) ?

Statement 1:

\(x^x < 0\)

Gives us that \(x\) is odd & \(x < 0\). Hence \(x\)can be any negative odd value.

Statement 1 alone is Not Sufficient.

Statement 2:

\(\frac{|x|}{x}=-1\)

Gives us that \(x < 0\). Hence \(x\) can take any negative value.

Statement 2 alone is Not Sufficient.

Combining the 2 statements, doesn't give us any new information.

Combining the 2 statements is Not Sufficient.

Answer E.

Thanks,
GyM

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

Lets get down to statements directly.

1) \(x^x < 0\)
For \(x^x\) to be negative, x needs to satisfy two conditions: a) x needs to be negative and b) x needs to odd so that negative sign of x remains the same.
With these constraints, lets take some values to check sufficiency.

If x=-1. (Note here, x is given not equal to 1, but it can most certainly be -1)
Thus, \(\frac{|x| - 1}{x - 1}\)=0

If x=-3
Thus, \(\frac{|x| - 1}{x - 1}\)=-1/2

Clearly, we cannot arrive at a unique value of the expression. Hence, insufficient

2) \(\frac{|x|}{x}=-1\)

Here, x can take any negative value because the numerator will always be positive (due to mod) and denominator will continue to be negative value. Note here that x doesn't have to be -1 here because the common number in numerator and denominator is essentially cancelling each other and giving us 1 any way.

Now, we can use the same example from statement 1 here (x=-1 and x=-3) and declare that this condition is also insufficient

Combining 1) & 2), we can clearly see that statement 2 is not adding any new information to statement 1 and therefore not changing our result of insufficiency.

Hence, answer is E

Analyzing the given expression \(\frac{|x| - 1}{x - 1}\) it can be seen that if x >= 0, Value =1 ( Since |x| = x)

if x < 0 Value of expression cannot be determined unless value of x is known as |x| = - x

St 1 : \(x^x < 0\)

x could be a negative odd integer ex : - 3

or x could be a negative fraction ex : - (1/3)

In either case x is negative and exact value of x is not known

The value of the expression cannot be determined

Not Sufficient

St 2 : \(\frac{|x|}{x}=-1\)

From this it can be concluded that |x| = -x

or x is negative

But value of x is not known, So value of expression cannot be determined

Not Sufficient

Even if we combine both Statements , no additional information is obtained

Choice E

If x ≠ 1, what is the value of \(\frac{|x| - 1}{x - 1}\)?


(1) \(x^x < 0\)

if \(x = -3\)

\(\frac{|-3| - 1}{-3 - 1}\)?

= \(\frac{2}{-4}\)

if \(x = -1\)

\(\frac{|-1| - 1}{-3 - 1}\)

\(= 0\)

Insufficient.

(2) \(\frac{|x|}{x}=-1\)

This tells us that x = negative. Insufficient.

(1&2) We can still use our two options from statement 1 and arrive at two different answers. Insufficient.

Answer is E.

(1) x^x < 0 x is odd and negative.

However, each value results in a unique number for x

(2) |x| = -x

There is still a a wide variety of -ve number for x.


(1) and (2),

Still able to pick a variety of negative and odd values for x, resulting in differing values. Answer is E