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

Question

GMAT CLUB'S FRESH QUESTION

If  x  is an integer such that  x ≠ 1  and  x ≠ 0 , what is the value of  \frac{|x| - 1}{x - 1} ?

(1)  x^x < 0 
 
(2)  \frac{|x|}{x}=-1

M36-100

Bunuel · Accepted Answer

Official Solution:

If  x  is an integer such that  x ≠ 1  and  x ≠ 0 , what is the value of  \frac{|x| - 1}{x - 1} ? 
 
(1)  x^x < 0 
 
Notice that  x  cannot be positive because  positive^{positive}=positive . 
 
Also, notice that  x  cannot be negative even integer because  negative^{even}=positive  
 
So, above works for negative odd integers only: -1, -3, -5, ... Each of which give different value for  \frac{|x| - 1}{x - 1} . Not sufficient.
 
(2)  \frac{|x|}{x}=-1 
 
 |x|=x . This implies that  x  is negative. Different negative values give different values for  \frac{|x| - 1}{x - 1} . Not sufficient.
 
(1)+(2) The second statement adds no additional info to what we already knew from the first one, so even taken together the statements are not sufficient.

Answer: E

WizakoBaskar · Answer

The data is sufficient if we get a Unique value for the expression.

Statement 1:  x^x < 0 
When x is an odd negative integer,  x^x < 0 
Example: x = -3 will satisfy the condition in statement 1. i.e.,  (-3)^{(-3)} < 0  The value of the expression will be 2/(-4) = -1/2
Counter example: x = -5 will also satisfy the condition in staement 1. The value of the expression will be 4/(-6) = -2/3

We are not able to get a unique value for the expression using information about x in statement 1. Statement 1 ALONE is not sufficient.

Statement 2:  \frac{|x|}{x}=-1 
We can infer that x is a negative number. We can use the same example and counter example of statement 1 for statement 2 as well to establish that statement 2 is not sufficient.

Combining the two statements: 
Example x = -3 satisfies information in the two statements. Value of expression -1/2
Counter example: x = -5 also satisfies information in the two statements. Value of expression -2/3.

Together the statements are not sufficient.
Choice E.

GyMrAT · Answer

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

Bunuel · Answer

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

GMATin · Answer

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

ksrutisagar · Answer

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

Basshead · Answer

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.

natheodore · Answer

(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

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

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GMAT Data Sufficiency (DS) Questions

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

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