Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 20 Jul 2019, 11:04 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If x/|x|<x which of the following must be true about x?

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Kconfused wrote:
Bunuel wrote:
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

B. $$x>0$$ --> $$|x|=x$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

Bunnel, if I were to multiply the original stem with |x| (since |x| is always positive) it would result in x*(|x|-1) > 0.
This would mean x > 0 and |x| > 1
|x| > 1 would lead to x < -1 and x > 1 . This is completely different from the answer you've reached. I see that your method is accurate and the answer justified, but can you please correct my method here.

It would give the same answer.

$$x*(|x|-1) > 0$$. This implies that both multiples have the same sign.

$$x>0$$ and $$|x|>1$$ (since we consider positive x, then this transforms to x>1) --> $$x>1$$.
$$x<0$$ and $$|x|<1$$ (since we consider negative x, then this transforms to -x<1 --> -1<x) --> $$-1<x<0$$.

The same ranges as in my solution.

Hope it's clear.
_________________
Manager  Joined: 28 May 2014
Posts: 52
Schools: NTU '16
GMAT 1: 620 Q49 V27 Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Bunuel,

What if we square the inequality x/ |x| < x. Then we get (x^2 / x ) < x^2 which implies that x^2 < x^3. Is this correct? Please explain. Thanku
Math Expert V
Joined: 02 Sep 2009
Posts: 56304
If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

1
sri30kanth wrote:
Bunuel,

What if we square the inequality x/ |x| < x. Then we get (x^2 / x ) < x^2 which implies that x^2 < x^3. Is this correct? Please explain. Thanku

We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality), which is not the case here.

Also, the second step in your solution: never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it, we don't know the sign of x, so we cannot multiply x^2/x < x^2 by x here.

For more check here: inequalities-tips-and-hints-175001.html
_________________
Manager  S
Joined: 25 Mar 2013
Posts: 235
Location: United States
Concentration: Entrepreneurship, Marketing
GPA: 3.5
If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

As Bunnel states.
Two cases for \frac{x}{|x|}<x:

A. x<0 --> |x|=-x --> \frac{x}{-x}<x --> -1<x --> -1<x<0;

B. x>0 --> |x|=x --> \frac{x}{x}<x --> 1<x.

Concept is the absolute value of -5 equals 5, or, in mathematical
symbols, I-51 = 5.
From above A.
x<0 mean x is negative Assume x = -1
lxl = l -x l becz x is negative /positive = X is always positive
But not |x|=-x ?????
_________________
I welcome analysis on my posts and kudo +1 if helpful. It helps me to improve my craft.Thank you
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9446
Location: Pune, India
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

kanusha wrote:
x<0 mean x is negative Assume x = -1
lxl = l -x l becz x is negative /positive = X is always positive
But not |x|=-x ?????

You assumed x = -1
You got |x| = 1

Is |x| = x? No. |x| is 1 but x is -1
Then what is |x| in terms of x?

|x| = -x
1 = -(-1) = 1

That is why you say that |x| = -x when x is negative because then -x becomes positive.
_________________
Karishma
Veritas Prep GMAT Instructor

Intern  Joined: 21 Apr 2014
Posts: 39
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

So, since we know that the absolute value is positive, we can multiply both sides by abs(x) without having to change the sign.

x<x * |x| This means that x has to be greater than 1 or in between -1 and 0. You can figure this out from intuition or by testing number. 0 and -1 don't work because that would make both sides equal.

We are looking for something that must be true, so if we can find a scenario for x that works outside the given parameters, we can eliminate it right away.
A) doesn't have to be true, because x could because -1/2 works for x
B) does have to be true there is no value for x that works and is below -1
C) doesn't have to be true, because -1/2 works
D) doesn't have to be true, because x=1 doesn't even work
E) doesn't have to be true because -1/2 works
_________________
Eliza
GMAT Tutor
bestgmatprepcourse.com
Director  Joined: 07 Aug 2011
Posts: 518
GMAT 1: 630 Q49 V27 Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

Attachments gmatclub.jpg [ 77.27 KiB | Viewed 2416 times ]

Intern  Joined: 01 Apr 2014
Posts: 4
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

CAN I SOLVE THE QUESTION IN THIS WAY..
x/|x|<x---->|x|/X>1/X----->|x|>1

Two Cases---> X>1 or X<-1

since,

|ax+b|>s--->ax+b>1 or ax+b<-1
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9446
Location: Pune, India
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

akshay4gmat wrote:
CAN I SOLVE THE QUESTION IN THIS WAY..
x/|x|<x---->|x|/X>1/X----->|x|>1

Two Cases---> X>1 or X<-1

since,

|ax+b|>s--->ax+b>1 or ax+b<-1

You cannot cancel off x's from the denominator without knowing the sign of x.
_________________
Karishma
Veritas Prep GMAT Instructor

Retired Moderator V
Joined: 27 Oct 2017
Posts: 1233
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Hi

The main point in this question is to understand what the question is asking.
which of the following must be true about x?

It means the question is asking about a Set of Values, which contains all the value of x which satisfy the given inequality.
But the Big Idea is that the set may contain other values also which does not satisfy the inequality.

It simply means the set must contains all the value of x satisfying the inequality but vice versa is not required.

only after solving the inequality as explained above,
-1<x<0 , when x is negative , or
x>1 when x is positive.

So option B x>-1 is the only the set of values which contains all the above required values of x.

_________________
Manager  B
Joined: 02 Jan 2016
Posts: 124
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Its always a good idea to simply or manipulate the Question Stem

X divided by |X| will either give a "1" or "-1", depending on "sign of X"

Incase "1" then X > 1 and Incase "-1" X > "-1",

If you think on this X >1 might be true but not always true, but X > -1 will always be true.

Even if "X" is a fraction, X>-1 is true and this also matches our answer.
Intern  B
Joined: 26 Jul 2018
Posts: 13
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

If x=0, then

0/0 < 0

undefined < 0

makes the choice x>-1 absurd.
Intern  B
Joined: 06 Aug 2018
Posts: 15
Concentration: Strategy, Technology
GMAT 1: 680 Q49 V34 GMAT 2: 740 Q51 V39 GPA: 2.24
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Quote:
Think again.
Every value greater than -1 need not satisfy the inequality but every value satisfying the inequality must be greater than -1.

Earlier I marked A as the answer, but this line made it crystal clear that the correct choice should be B
Intern  B
Joined: 30 Apr 2017
Posts: 6
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Bunuel wrote:
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

B. $$x>0$$ --> $$|x|=x$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

in the 1 case when x<0, why arent changing the sign of the inequality. i thought when we open a mod with a negative sign we change the sign of the inequality.
p.s- this question has jolted all my concepts of mod!
Retired Moderator V
Joined: 27 Oct 2017
Posts: 1233
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Hi

We change the sign of inequality only when we divide or multiply by a negative number.

Jasveensingh wrote:

in the 1 case when x<0, why arent changing the sign of the inequality. i thought when we open a mod with a negative sign we change the sign of the inequality.
p.s- this question has jolted all my concepts of mod![/quote]
_________________
Intern  B
Status: when you say,"I can or I can't", Both times you are right!
Joined: 26 Nov 2018
Posts: 31
Location: India
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

x/lxl<x
=x/x<lxl
=1<lXl
or
lxl>1

which gives -1>x>1

how could it be "B"?
Retired Moderator V
Joined: 27 Oct 2017
Posts: 1233
Location: India
GPA: 3.64
WE: Business Development (Energy and Utilities)
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Did you read the above posts?

Rupesh1Nonly wrote:
x/lxl<x
=x/x<lxl
=1<lXl
or
lxl>1

which gives -1>x>1

how could it be "B"?

Posted from my mobile device
_________________
Manager  S
Joined: 11 Aug 2018
Posts: 111
Location: Pakistan
GPA: 2.73
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

A tricky Mod multiplication.
_________________
If you like this post, be kind and help me with Kudos!

Cheers!
Manager  B
Joined: 29 Nov 2016
Posts: 140
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

chetan2u

I have a totally different doubt and usually I face it whenever I solve modulus questions.

we say |X|=-X, when X<0
|X|=X , when X>0

Why don't we replace these values in actual X i.e. why do we open only |X| if X<0. Like in this question why don't we substitute
X/|X| , if X<0 would lead to -X/|X| which will give -1. In this question the result is same but I have seen questions where these substitution leads to different result

Thanks

Posted from my mobile device
Math Expert V
Joined: 02 Aug 2009
Posts: 7764
Re: If x/|x|<x which of the following must be true about x?  [#permalink]

### Show Tags

Mudit27021988 wrote:
chetan2u

I have a totally different doubt and usually I face it whenever I solve modulus questions.

we say |X|=-X, when X<0
|X|=X , when X>0

Why don't we replace these values in actual X i.e. why do we open only |X| if X<0. Like in this question why don't we substitute
X/|X| , if X<0 would lead to -X/|X| which will give -1. In this question the result is same but I have seen questions where these substitution leads to different result

Thanks

Posted from my mobile device

Hi..
When you say x is negative, the negative sign is already there in the variable.
So if x<0, then -x>0..
Let us take an example..
Say x is -2..
So X/|X| will be -2/|-2|=-2/2=-1..
But if you change the sign that is x/|X|=-x/|X|=-(-1)/|-1|=1/1=1
_________________ Re: If x/|x|<x which of the following must be true about x?   [#permalink] 23 Feb 2019, 01:41

Go to page   Previous    1   2   3   4    Next  [ 67 posts ]

Display posts from previous: Sort by

# If x/|x|<x which of the following must be true about x?   