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Retired Moderator B
Joined: 05 Jul 2006
Posts: 1375
Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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3
34 00:00

Difficulty:   65% (hard)

Question Stats: 60% (02:13) correct 40% (02:18) wrong based on 854 sessions

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Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

Originally posted by yezz on 16 Aug 2009, 00:44.
Last edited by Bunuel on 04 Dec 2012, 03:22, edited 2 times in total.
Renamed the topic and edited the question.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India

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5
6
puneetj wrote:
Got to the correct answer but took too much time...E

If you get messed up in mods, such a question can be done in under a minute by trying out some values. Now, I am no fan of plugging in numbers, especially not in DS questions, but such questions are perfect for plugging in if you are not comfortable with algebra. Why? Because they have asked for the range of x. If there is even one value in the given range that doesn't satisfy the inequality, it is not the answer and if there is even one value outside the given range that does satisfy the inequality, it is not the answer.

0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

Say, consider option A. If x = -1, |x|-4x = 5 which doesn't satisfy the inequality so A is out.
If x = 1/2, |x|-4x is negative so B and C are out.
If x = -4/5, |x|-4x = 4 so D is out and E is the answer.
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Karishma
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##### General Discussion
Manager  Joined: 28 Jul 2009
Posts: 84
Location: India
Schools: NUS, NTU, SMU, AGSM, Melbourne School of Business

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2
2
I am getting E as an answer if x is -ve.
If x is -ve, we get

0 < -5x < 5

Dividing both sides by -5, we flip both the sides and we land up on

0 > x > -1
ie
-1 < x < 0.

Is that the correct method to solve this problem? Please explain.
Manager  Joined: 27 May 2009
Posts: 156

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4
IMO E:

5 + 4x > |x|
5 + 4x > x first cndtn
5 > - 3x
x>-5/3

or

5 + 4x > -x
x > -1

using

4x < |x|
or 4x < x
x<0

or 4x < -x
x<0

thus range lies between -1 to 0

correct me if i m wrong!!
Retired Moderator B
Joined: 05 Jul 2006
Posts: 1375

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0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

i did it this way

split the inquality into 2

a) /x/-4x<5 ie: -5<x-4x<5 ie: -5<-3x<5 ie: 5/3>x>-5/3....1

b) /x/-4x>0 thus /x/>4x thus eithe x>4x ie: -3x>0 ie: x <0 or -x<4x ie: x>-4x ie: x>0

and i get lost here??
Director  Joined: 01 Feb 2011
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0<-5x<5

=> -1<x<0

Retired Moderator B
Joined: 16 Nov 2010
Posts: 1222
Location: United States (IN)
Concentration: Strategy, Technology

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I'm also getting E.

2 cases x > 0 or x < 0

0< -x - 4x < 5

=> 0< -5x < 5

=> x > -1 and x < 0

x - 4x < 5

=> -3x < 5

=> x > -5/3

so x > 0 as x is +ve

So -1 < x < 0

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Intern  Joined: 14 Jun 2011
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hi,
apply definition absolute value obtain two possibilites:
A. if x≥0 then 0<|x|-4x<5 go 0<x-4x<5 go 0<-3x<5, go 0>x>-5/3. The intersection is empty
B. if x<0 then 0<|x|-4x<5 go 0<-x-4x<5 go 0<-5x<5 go 0>x>-1. the intersection is -1<x<0

Bye...
Senior Manager  Joined: 13 Aug 2012
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Concentration: Marketing, Finance
GPA: 3.23

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0<|x|-4x<5

A. x<0
Test: x=-5
$$|-5|-4(-5) = 5 + 20$$ is not less than 5
FALSE!

B. 0<x<1
Test: x=$$\frac{1}{4}$$
$$\frac{1}{4}-4(\frac{1}{4})=-\frac{3}{4} is not greater than 0$$
FALSE!

C. -3/5<x<1
x<1 as tested with B
FALSE!

D. -3/5<x<0
E. -1<x<0

We see that D and E are almost the same except for E. covers -3/5 unlike D.
Let x=-3/5
$$|-\frac{3}{5}|-4(-\frac{3}{5})=\frac{15}{5}=3$$

Retired Moderator B
Joined: 05 Jul 2006
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VeritasPrepKarishma wrote:
puneetj wrote:
Got to the correct answer but took too much time...E

If you get messed up in mods, such a question can be done in under a minute by trying out some values. Now, I am no fan of plugging in numbers, especially not in DS questions, but such questions are perfect for plugging in if you are not comfortable with algebra. Why? Because they have asked for the range of x. If there is even one value in the given range that doesn't satisfy the inequality, it is not the answer and if there is even one value outside the given range that does satisfy the inequality, it is not the answer.

0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

Say, consider option A. If x = -1, |x|-4x = 5 which doesn't satisfy the inequality so A is out.
If x = 1/2, |x|-4x is negative so B and C are out.
If x = -4/5, |x|-4x = 4 so D is out and E is the answer.

I believe the choice of -4/5 to execlude D is wrong -4/5 is not in the range of -3/5<x<0 ????? accordingly i think both D and E could solve as the right range in my opinion is -5/3 < x < 0??? am i right or wrong plz advise!
Verbal Forum Moderator B
Joined: 10 Oct 2012
Posts: 578
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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yezz wrote:
Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

|x|-4x>0 = |x|>4x =$$\frac{x}{|x|}$$<1 --> x<0. Again, |x|-4x<5 = -x-4x<5 = -5x<5-->x>-1.
E.
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Veritas Prep GMAT Instructor V
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yezz wrote:
VeritasPrepKarishma wrote:
puneetj wrote:
Got to the correct answer but took too much time...E

If you get messed up in mods, such a question can be done in under a minute by trying out some values. Now, I am no fan of plugging in numbers, especially not in DS questions, but such questions are perfect for plugging in if you are not comfortable with algebra. Why? Because they have asked for the range of x. If there is even one value in the given range that doesn't satisfy the inequality, it is not the answer and if there is even one value outside the given range that does satisfy the inequality, it is not the answer.

0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

Say, consider option A. If x = -1, |x|-4x = 5 which doesn't satisfy the inequality so A is out.
If x = 1/2, |x|-4x is negative so B and C are out.
If x = -4/5, |x|-4x = 4 so D is out and E is the answer.

I believe the choice of -4/5 to execlude D is wrong -4/5 is not in the range of -3/5<x<0 ????? accordingly i think both D and E could solve as the right range in my opinion is -5/3 < x < 0??? am i right or wrong plz advise!

Focus on
"and if there is even one value outside the given range that does satisfy the inequality, it is not the answer."
given above.

-4/5 is a value which satisfies 0 < |x|-4x < 5 since |-4/5|-4(-4/5) = 4.
Since -4/5 does not lie in the range -3/5<x<0 so (D) cannot be the answer. The correct range needs to cover all possible values of x.
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Karishma
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Retired Moderator B
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Posts: 1375
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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@Karishma ... U r absolutely right , been away for years from GMAT ... E is the perfect answer
Retired Moderator B
Joined: 05 Jul 2006
Posts: 1375
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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1
we ve got 3 critical values that are ( 0 from modulus , -5/3& -1 from /x/ = 5+4x) , draw on the number line and test values

.....-5/3........-1...........0....................

only -1<x<0 is the range where all values of x satisfy the compound inequality 0</x/-4x<5
Senior Manager  Joined: 13 May 2013
Posts: 396
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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1
Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

There are two options here - plugging in values given to us in the answer choices or simplifying the inequality.

0<|x|-4x<5

x>0: 0<x-4x<5 0<-3x<5 0<x<-5/3 -5/3<x<0 INVALID as x does not fall within the range of x>0
OR
x<0: 0<(-x)-4x<5 0<-5x<5 0<x<-1 -1<x<0 VALID as x falls within the range of x<0

There is only one valid solution: -1<x<0.
(E)
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Concentration: Finance
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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Would be happy to hear some comments on whether this approach is correct

|x|-4x>0

So we have two cases

If x>0 then x-4x>0
-3x>0
x<0, this contradicts and hence is not a valid solution

If x<0 then -5x>0
x<0, this solution is valid

So we get that -1<x<0 replacing in the original inequality

E
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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2
1
jlgdr wrote:
Would be happy to hear some comments on whether this approach is correct

|x|-4x>0

So we have two cases

If x>0 then x-4x>0
-3x>0
x<0, this contradicts and hence is not a valid solution

If x<0 then -5x>0
x<0, this solution is valid

So we get that -1<x<0 replacing in the original inequality

E

Knowing only x < 0, how do you choose between (A), (D) and (E)?
You need to consider |x| - 4x < 5 too
When x < 0, -x -4x < 5
-5x < 5
x > -1
That's how you get -1 < x < 0

Or work on the whole inequality in one go
0 < |x| - 4x < 5
When x < 0,
0< -x - 4x < 5
0 < -5x < 5
0 < -x < 1
0 > x > -1

which is the same as -1 < x < 0
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Karishma
Veritas Prep GMAT Instructor

Intern  Joined: 07 Aug 2012
Posts: 3
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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Hey,
So I had a doubt. For the equaltiy: 0<|x|-4x<5 if I try and solve it algebraically, i first take x<0
In that case won't this equality be: 0<-x-4(-x)<5. I can't just substitute mod x with -x and leave the other x be can I? Please help!
Thanks!
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 9983
Location: Pune, India
Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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1
1
natashakumar91 wrote:
Hey,
So I had a doubt. For the equaltiy: 0<|x|-4x<5 if I try and solve it algebraically, i first take x<0
In that case won't this equality be: 0<-x-4(-x)<5. I can't just substitute mod x with -x and leave the other x be can I? Please help!
Thanks!

Say you have an inequality: 4x < 5
and you know that x must be negative. How will you solve the inequality?
Will you say that the inequality becomes -4x < 5? No. You are given that 4x < 5. Without changing the inequality, you can write this as x < 5/4.
x needs to be negative. All negative values will be less than 5/4.

Why do you substitute -x in place of |x|? You cannot solve an equation/inequality with |x| in it. You need to remove the absolute value sign.

You know that |x| = x if x is positive and |x| = -x if x is negative.

Since you know that x is negative, you can write -x in place of |x| without changing the inequality.
If you change the simple x to -x in the inequality, the inequality changes.

Check out this post for a more detailed explanation: http://www.veritasprep.com/blog/2014/06 ... -the-gmat/
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Karishma
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Manager  Joined: 02 Nov 2014
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Re: Solve for x: 0<|x|-4x<5 = ?  [#permalink]

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Solve for x: 0<|x|-4x<5 = ?

A. x<0
B. 0<x<1
C. -3/5<x<1
D. -3/5<x<0
E. -1<x<0

Soln: We'll consider two cases: x>0 and $$x<0$$ where $$|x|$$ will become $$x$$ and $$-x$$ respectively.
x>0: 0 < x - 4x < 5, or, 0 < -3x < 5, or -5/3 < x < 0 (dividing by -3 and thereby changing the direction of the inequality)
BUT, if $$x>0$$ then x cannot lie in $$-5/3 < x < 0$$. No solution here.

$$x<0: 0 < -x - 4x < 5,$$ or, $$0 < -5x < 5,$$ or $$-1 < x < 0$$ (dividing by -3 and thereby changing the direction of the inequality)
This range is GOOD. Anything in this range will be permissible.

Choice E is the same range: $$-1 < x < 0$$ Re: Solve for x: 0<|x|-4x<5 = ?   [#permalink] 07 Dec 2015, 00:33

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