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Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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06 Feb 2005, 17:09

MA wrote:

banerjeea_98 wrote:

MA wrote:

banerjeea_98 wrote:

"B".

x/|x|<x-------x<x|x|.....for +ve x...|x| > 1...for -ve x .....|x| < 1....that means x can be -1<x<0....or x > 1....in any case it has to be > -1

lets suppose x be 0.5 which is more than -1. 0.5/l0.5l=1 which is not less than 0.5.

ya but in no case x can be < -1.....it has to be > -1....but not bet 0 and 1.....x > 1 is wrong.....as the eqn also satisfies when x = -1/2

baner, unclear to me.... pls be specific.

Let's imagine a number line with points -1, 0 and 1.....as u can see x can't be <-1...e.g. x = -2......-1 < -2....will not satisfy....

Now let's see -1<x<0......x = -1/2.....-1 < -1/2.....satisfies
Now for 0<x<1.....x = 1/2.....1 < 1/2.....will not satisfy....
For x > 1....x = 2.....1 < 2....satisfies

As u see the soln lies in the following categories:
-1<x<0
x > 1

In all case x > -1 even tho it can't be 0<x<1.....altho I think the ques is lil weird.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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06 Feb 2005, 21:45

HongHu wrote:

x/|x|<x if x>0, then 1/x<1 ie, x>1 if x<0, then 1/(-x)>1, ie, x>-1

So combine these two, we know that x must be greater than -1.

baner is right, (B).

i think we need AkamaiBrah.

if so, then why 0<x>1 does not fit in the inequality x/|x|<x? i mean why all the values for x>-1 donot satisfy the inequality x/|x|<x? pls do explain........................

Last edited by MA on 06 Feb 2005, 21:50, edited 1 time in total.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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06 Feb 2005, 21:50

1

This post was BOOKMARKED

It doesn't say that all x greater than -1 would satisfy the equation. We get two range of x that satisfies the equation (-1,0) and (1,infinite). All that the answer says is that every x in our solution ranges are greater than -1, which is true.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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06 Feb 2005, 23:10

I believe we're debating solution B, which says x>-1 must be always true.

I can pick positive 1 as an integer that is greater than -1, but does not satisfy x/|x|<x.

If x = +1,
x/|x| = 1 which is not < 1. (1=1)

Now, if there's just one value of x > -1 that does not satisfy the inequality, then we can't say x>-1 is always true.
If you look at my previous post, I've not said I'm solving for x, i'm just stating values of x that hold true for the inequality. I came up with x<-1, and x>1.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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07 Feb 2005, 07:26

(A) cannot be correct, people.

Let me give you another example. Lets say we have two sets of integers.
S1=(0), S2=(2,3,4)
Now we know that x belongs to S1 or S2. Ask what must be true for x:
(A) x>1
(B) x>-1

(A) is obviously wrong
Because x could be an element of S1 or S2. If x belongs to S1, then it would be 0, and it is NOT greater than 1.
(B) is, however, correct
No matter which set x belongs, x is greater than -1.
You cannot say that well what about x=1? It satisfies x>-1, but is not in either S1 or S2. The mistake you are having here, is instead of A->B, you are trying to say B->A.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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07 Feb 2005, 07:38

HongHu wrote:

(A) cannot be correct, people.

Let me give you another example. Lets say we have two sets of integers. S1=(0), S2=(2,3,4) Now we know that x belongs to S1 or S2. Ask what must be true for x: (A) x>1 (B) x>-1

(A) is obviously wrong Because x could be an element of S1 or S2. If x belongs to S1, then it would be 0, and it is NOT greater than 1. (B) is, however, correct No matter which set x belongs, x is greater than -1. You cannot say that well what about x=1? It satisfies x>-1, but is not in either S1 or S2. The mistake you are having here, is instead of A->B, you are trying to say B->A.

Glad to see that I am not the only one in minority here, I am sticking with my ans "B" till we get the OA.

Re: x/|x|<x. which of the following must be true about x ? [#permalink]

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07 Feb 2005, 08:53

My example is completely parallel to the question, only in the integer and limited form.

Look:

HongHu wrote:

(A) cannot be correct, people.

Let me give you another example. Lets say we have two sets of integers. S1=(0), S2=(2,3,4) Now we know that x belongs to S1 or S2. Ask what must be true for x: (A) x>1 (B) x>-1

Compare it with:
S1=(-1,0), S2=(1,infinity)
We know x belongs to S1 or S2. What must be true for x?

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.

Answer: B.

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 Contradiction.

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

Can you please clarify?

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

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