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

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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  02 Jun 2014, 20:11
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  03 Jun 2014, 00:18
Expert's post
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????

x=0 does not satisfy x/|x| < x, so x cannot be 0.

x/|x| < x, means that -1<x<0 or x>1. ANY x from these ranges will be greater than -1.
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  02 Aug 2015, 10:06
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  31 Aug 2015, 09:24
Bunuel wrote:
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

We are given that $$\frac{x}{|x|}<x$$ (this is a true inequality), so first of all we should find the ranges of $$x$$ for which this inequality holds true.

$$\frac{x}{|x|}< x$$ multiply both sides of inequality by $$|x|$$ (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> $$x<x|x|$$ --> $$x(|x|-1)>0$$:
Either $$x>0$$ and $$|x|-1>0$$, so $$x>1$$ or $$x<-1$$ --> $$x>1$$;
Or $$x<0$$ and $$|x|-1<0$$, so $$-1<x<1$$ --> $$-1<x<0$$.

So we have that: $$-1<x<0$$ or $$x>1$$. Note $$x$$ is ONLY from these ranges.

Option B says: $$x>-1$$ --> ANY $$x$$ from above two ranges would be more than -1, so B is always true.

nonameee wrote:
Quote:
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

Bunuel, I think there's a mistake in the question or in the answer choices:

Here's my solution:
1) x<0:
-x/x < x
-1<x<0

2) x>=0
x/x<x
x>1

The solution of the inequality is then:
-1<x<0 union x>1

The answer can't be B, since let x=1/2. We get:
(1/2)/(1/2) < (1/2)
1< 1/2

I think the answer should be A since it satisfies all the scenarios.

The options are not supposed to be the solutions of inequality $$\frac{x}{|x|}<x$$.

Hope it's clear.

how did you came up −1<x<0 or x>1. i can not understand this. i am sorry if i so silly for you guys. but plz help me
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  31 Aug 2015, 09:38
Bunuel wrote:
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

We are given that $$\frac{x}{|x|}<x$$ (this is a true inequality), so first of all we should find the ranges of $$x$$ for which this inequality holds true.

$$\frac{x}{|x|}< x$$ multiply both sides of inequality by $$|x|$$ (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> $$x<x|x|$$ --> $$x(|x|-1)>0$$:
Either $$x>0$$ and $$|x|-1>0$$, so $$x>1$$ or $$x<-1$$ --> $$x>1$$;
Or $$x<0$$ and $$|x|-1<0$$, so $$-1<x<1$$ --> $$-1<x<0$$.

So we have that: $$-1<x<0$$ or $$x>1$$. Note $$x$$ is ONLY from these ranges.

Option B says: $$x>-1$$ --> ANY $$x$$ from above two ranges would be more than -1, so B is always true.

nonameee wrote:
Quote:
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

Bunuel, I think there's a mistake in the question or in the answer choices:

Here's my solution:
1) x<0:
-x/x < x
-1<x<0

2) x>=0
x/x<x
x>1

The solution of the inequality is then:
-1<x<0 union x>1

The answer can't be B, since let x=1/2. We get:
(1/2)/(1/2) < (1/2)
1< 1/2

I think the answer should be A since it satisfies all the scenarios.

The options are not supposed to be the solutions of inequality $$\frac{x}{|x|}<x$$.

Hope it's clear.

x<x|x| --> x(|x|−1)>0:
Either x>0 and |x|−1>0, so x>1 or x<−1 --> x>1;
Or x<0 and |x|−1<0, so −1<x<1 --> −1<x<0.
i can not undersdant the coloured portion from where we can derive this?
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  31 Aug 2015, 10:14
1
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fluke wrote:
syh244 wrote:
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

Answer is A. I hope the following explanation permanently sets A as the right answer for this problem. I see that some people tried simplifying the formula first, and some people performed a common mistake while doing so. Lets look at what many people tried to do.

Step 1: multiply |x| to both sides, resulting in x<x*|x| (this step is correct as |x| will always be positive and thus multiplying |x| to both sides will not cause ambiguity in the direction of the inequality)
Step 2: divide x to both sides, resulting in 1<|x| (this step is incorrect as x could be positive or negative, and such ambiguity restricts such operation. very common mistake that should be avoided at all costs as GMAT question makers will use this against you)

So.... lets go back to Step 1, x<x*|x|
From here, you can further simplify the equation to 0<x(|x|-1) as many people above did, but simplifying the equation this will only create more brain damage than simplify the problem.

The best course of action after Step 1 is to plug in numbers (or even plug in numbers straight to the original equation without performing Step 1).

-2 does not work because -2<-2*|-2| = -2<-4, which is not true
-1 does not work because -1<-1*|-1| = -1<-1, which is not true
-1/2 works because -1/2<-1/2*|-1/2| = -1/2<-1/4, which is true
0 does not work because 0<0*|0| = 0<0, which is not true
1/2 does not work because 1/2<1/2*|1/2| = 1/2<1/4, which is not true
1 does not work because 1<1*|1| = 1<1, which is not true
2 works because 2 < 2*|2| = 2<4, which is true
3 works because 3<3|3 = 3<9, which is true

So we have a situation in which 0>x>-1, and x>1.

Now here is where many people are getting REALLY confused.
Many people say that the right answer is B because, yes, x>-1, BUT it has certain restrictions. The statement x>-1 should also include 0 and 1/2 as the right solutions, but the original equation fails when these numbers are plugged in, making x>1 the only right answer for the equation. Yes, I understand that the answer A disregards the portion where 0>x>-1, but who cares, the question is asking "which of the following must be true about x" and not "what are the solutions for x"

Your analysis is amazing, but the summing up is not entirely correct.

As you yourself said:
the question is asking "which of the following must be true about x" and not "what are the solutions for x".

When the condition is given:
x/|x|<x

And we get the range for x:
-1<x<0 OR x>1.

It is imperative that "x" MUST BE in that range. So, x just can't be less than equal to -1, OR Anything between 0 AND 1, inclusive, such as "1/2".

If there is a value of x, it must be in the range specified above.

So,
(A) x>1: It is one of the possible ranges. But, we don't know what will x bear. What if x=-0.5. Not necessarily true.
(B) x>-1: No matter which value "x" bears, it will always be >-1. Must be true.
(C) |x|<1: This CAN be true, but what if x = 2. Not necessarily true.
(D) |x|=1: This cannot be true. Both -1 AND +1 are outside the range.
(E) |x|^2>1: x<-1 OR x>1. What if x=-0.5. Not necessarily true.

Ans: "B"

hi bunnel, i can not understand how did you make the range
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Re: x/|x|<x. which of the following must be true about x ? [#permalink]  15 Sep 2015, 00:18
I hope this helps.

From x/|x|<x, we know that :

1. x cannot be 0, since we can't divide a number by 0
2. x cannot be greater than 1, since we are dividing the number by the same number
3. x cannot be less than -1, since we are dividing the number by then same number

So we get, -1<x<1
From here, if we want x/|x|<x to be true, pick some numbers and you will see that only negative fraction works.
So it means,
-1<x<0

a) x>1 --> false
b) x>-1 --> correct
c) |x|<1 -->false, since x can be positive fraction
d) |x|=1 --> false, since x cannot be greater than 0
e) |x|^2>1 --> false, since squaring fraction gives you less than 0
Re: x/|x|<x. which of the following must be true about x ?   [#permalink] 15 Sep 2015, 00:18

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