If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x

Hello Bunuel, I tried to solve like this:
Basically the question asks whether -1<x<1?
Condition 1: x/|x|<x
case 1: if x>0; x/x<x ==> 1<x ==> x>1
case 2: if x<0; x/-x<x; since x<0, -x >0 therefore, x<-x2; x+x2 <0 ==> x(1+x)<0, therefore, -1<x<0 {Please confirm if I have derived this right}
Given that we do not know the sign we cannot determine, hence condition 1 is insufficient

Condition 2: |x|>x ==> which means x cannot be > then 0. thus, x<0, independently condition 2 is not sufficient;

Combining C1 and C2: we arrive at case 2; but that proves that x does not lie between -1 and 1 but between -1 and 0.

Hence C. Please let me know in case I have done anything conceptually wrong!

You've done everything right.

Though for case 2 you could do as follows: \(x<0\) --> \(\frac{x}{-x}<x\) --> \(-\frac{x}{x}<x\) --> x is simply reduced: \(-1<x\). Since we consider the range \(x<0\) then \(-1<x<0\).

Also for (1)+(2) we have that \(-1<x<0\). So we can answer yes to the question whether \(-1<x<1\).

Hope it helps.
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Here is a video explanation that takes a slightly different approach using graphs of basic functions:

https://www.gmatquantum.com/shared-posts ... -xx-x.html

Cheers,
Dabral

if x<0, doesn't x/|x|<x become -x/x<-x which simplifies to -1<-x or x<1 ?

When \(x<0\), then \(|x|=-x\), thus \(\frac{x}{|x|}<x\) becomes \(\frac{x}{-x}<x\) --> \(-1<x\) but since \(x<0\), then \(-1<x<0\).

Hope it's clear.
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Hi,

There are a handful of Number Property rules in this DS question that you can find by either TESTing VALUES or taking lots of notes.

We're told that X cannot = 0. We're asked if |X| < 1. This is a YES/NO question.

Fact 1: X/|X| < X

This inequality requires a thorough examination of the possibilities.

X CANNOT = 1 since 1/|1| is NOT < 1

X can be ANY number > 1 though

If X = 2....
2/|2| < 2
And the answer to the question is NO.

X CANNOT be a positive fraction...

If X = 1/2
(1/2)/|1/2| = 1 which is NOT < 1/2

X could be ANY negative fraction though

If X = -1/2
(-1/2)/|-1/2| = -1 which IS < -1/2
And the answer to the question is YES.

X CANNOT be -1 since -1/|-1| is NOT < -1

X CANNOT be < -1...

X = -2
(-2)/|-2| = -1 which is NOT < -2

According to all of this data, the possibilities are X > 1 or -1 < X < 0
Fact 1 is INSUFFICIENT

Fact 2: |X|> X

Here, we know that X can be ANY negative, but CANNOT be positive.

If X = -1/2 then the answer to the question is YES (we did the work already in Fact 1)
If X = -2 then the answer to the question is NO
Fact 2 is INSUFFICIENT

Combining Facts, we have...
X > 1 or -1 < X < 0
AND
X MUST be negative

The ONLY possibilities that fit BOTH Facts ARE negative fractions. Thus the answer to the question will be ALWAYS YES.
Combined, SUFFICIENT

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Thanks for your nice explanations. I am not clear on one issue though. As we can multiply an inequality by a variable only if we know its sign, how can we multiply both side of an inequality by an absolute value? Would you please explain..

An absolute value of a number cannot be negative (it's 0 or positive), and since we are given that x is not 0, then |x| is positive only.
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Hi All,

This DS question is built around some interesting Number Properties and patterns. If you can spot those patterns, then solving this problem should take considerably less time. This also looks like a question that can be beaten by TESTing VALUES.

We're told that X CANNOT = 0. We're asked if |X| < 1. This is a YES/NO question.

Fact 1: X/|X| < X

Before TESTing VALUES, I want to note a pattern in this inequality:

X/|X| will either equal 1 (if X is positive) OR -1 (if X is negative). This will save us some time when it comes to TESTing VALUES, since there are many values of X that will NOT fit this information.

If X = 2, then the answer to the question is NO.

X cannot be 1, any positive fraction, 0, or any negative integer…..

So what's left to TEST….?

If X = -1/2, then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: |X| > X

This tells us that X CANNOT be positive or 0.

If X = -1, then the answer to the question is NO.
If X = -1/2, then the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we have deal with the "overlapping restrictions" that we noted in the two Facts:
X cannot be….anything positive, 0, or any negative integer.
X can ONLY BE negative fractions between 0 and -1.
ALL of those answers (e.g. -1/2, -.4, etc.) lead to a YES answer.
Combined SUFFICIENT.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

I did mine a little differently.

If x ≠ 0, is |x| < 1?


Means: Is -1 < x < 1?

(1) \(\frac{x}{|x|}< x\)

x < |x|*x
If x is positive, then: \(\frac{x}{x} < |x|\) which is the same as \(1 < |x|\)
This means that \(-1 > x > 1\)

If x is negative, then: \(\frac{x}{x} > |x|\) which is the same as \(1 > |x|\)
This means that \(-1 < x < 1\) (we switch the direction of < to > because we divided by -1 to put |x| by itself).

These two answers are inconsistent: x is both less than 1 and greater than 1. So, insufficient.

(2) \(|x| > x\)
This means that x is negative since the negative value of something is always less than the absolute value of something; whereas, a positive or 0 value of something is equal to its absolute value.

If x is equal to a negative value less than -1, then it's within the range -1 < x < 1. But, if x is equal to a large negative value, |x| can be out of range, for example, x = -1,000. Therefore, (2) is insufficient.

(1) + (2)
|x| > x and |x| > 1
x is negative, and we know that x < -1. So, pick an arbitrary value less than -1, say, -5.
Plug it into equation (1): \(\frac{-5}{|-5|} < -5\). Is \(-1< -5\)? False. x does not refer to values < -1.

|x| > x and |x| < 1
x is negative, and we know that x > -1. Therefore, \(-1 < x < 0\).
Pick an arbitrary value between -1 and 0, non-inclusive, say, -0.5.
Plug it into equation (1): \(\frac{-0.5}{|-0.5|} < -0.5\). Is \(-1 < -0.5\)? Yes, this works. Sufficient.

Answer C.

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 x is not equal to 0, is |x| less than 1?

(1) x/|x| < x
(2) |x| > x

If the range of the condition falls into that of the condition in terms of inequalities, the condition is sufficient.

There is 1 variable in the original condition, and there are 2 equations provided by the 2 conditions, so there is high chance (D) will be our answer.
For condition 1, if x>0, x/|x|<x --> x/x<x --> 1<x, then 1<x
if x<0, x/|x|<x --> x/-x<x --> -1<x, then -1<x<0. Therefore this condition is insufficient because this range does not fall into that of the question.
For condition 2, |x|>x --> x<0. This is insufficient for the same reason.
Looking at them together, however, -1<x<0 falls into the range of the question, so this is sufficient, and the answer becomes (C).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

Hi Bunuel,

For the A, how did you assume that remember that x<0

Thanks

There we consider two cases: (A) when x<0 and (B) when x>0. So, A we discard the range which is not less than 0 and for B discard the range which is not more than 0.

Hope it's clear.
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Given : x is NOT equal to zero

Question : Is |x| < 1?

Statement 1: x/|x|< x

Case 1: If x <0, x/|x| = -1 i.e. x/|x|< x can be rewritten as -1< x i.e. -1< x < 0 Giving answer to the question as YES
Case 2: If x >0, x/|x| = +1 i.e. x/|x|< x can be rewritten as 1< x i.e. 1< x Giving answer to the question as NO
NOT SUFFICIENT

Statement 2: |x| > x
|x| can be greater than x only if x is Negative because in all other cases both will be equal
i.e. x < 0 but x may be -0.5 Giving answer to the question as YES and x may be -1.5 Giving answer to the question as NO. Hence,
NOT SUFFICIENT

Combining the two statements
Combining -1< x < 0 and 1< x and x < 0

we get, only -1< x < 0 Giving answer to the question as YES. Hence,
SUFFICIENT

Answer: option C

For (1), wouldn't x<0 give you:

-x/x < -x
-1 < -x
1 > x?

When I plug in numbers for that, it doesn't work, but I don't see how we get x/-x < x when plugging in x<0?

If x < 0, then |x| = -x.

Substitute |x| by -x in x/|x|< x to get x/(-x) < x and then to get -1 < x. Since we consider the range when x < 0, then -1 < x < 0.
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the red zone indicates that the range is -1<x<0 , how do we arrive at -1<x<1?

The question asks whether \(-1<x<1\) is true. We got that \(-1<x<0\). Any, x from \(-1<x<0\) IS in the range from -1 to 1, so we have an YES answer to the question.
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Similar topic: https://gmatclub.com/forum/if-x-is-diff ... 26715.html
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