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|x| = 3x - 2

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|x| = 3x - 2 [#permalink] New post 01 Oct 2011, 18:05
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A
B
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|x| = 3x - 2

A. 1
B. 1/2
C. 1 and 1/2
D. -1/2
E. -1

Soln
[Reveal] Spoiler:
When we proceed to solve this we get two eq:
x = 3x - 2 .... (i) when x > 0
-x = 3x - 2 .... (ii) when x < 0

Therefore we get two results x = 1 and x = 1/2 from (i) and (ii) resp.
Lets plug the values back into the eq.

|1| = 3*1 - 2
==> 1 = 1
so (i) holds good.

|1/2| = 3*(1/2) - 2
==> 1/2 = -1/2 this is incorrect, hence soln from eq (ii) does not hold good.

The soln to the eq |x| = 3x - 2 is 1 only.

Quant guru's let me know if there is a flaw in my calculation method.

Thanks!
[Reveal] Spoiler: OA
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Re: Interesting inequality problem. [#permalink] New post 01 Oct 2011, 19:04
Its Both 1 and 1/2

as |x|=3x-2

It cab be x= 3x-2 or x= -(3x-2) so

when x= 3x-2 we get x=1
when x=-(3x-2) we get x= 1/2

so It is both 1 and 1/2
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Re: Interesting inequality problem. [#permalink] New post 01 Oct 2011, 23:09
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There is no big flaws, but it is not fully mathematically correct. When opening the absolute value you should add constraints which you use. In other words, |x|=x if x>=0

So, the precise solution is as follows:

|x|=3x-2

x=3x-2 , x>=0
-x=3x-2, x<0

2x=2, x>=0
4x=2, x<0

x=1, x>=0
x=0.5, x<0

So, the only answer is x=1.
Good luck! Try to solve some more problems with absolute values (both equations and inequalities)
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Re: Interesting inequality problem. [#permalink] New post 02 Oct 2011, 05:54
kotela wrote:
Its Both 1 and 1/2

as |x|=3x-2

It cab be x= 3x-2 or x= -(3x-2) so

when x= 3x-2 we get x=1
when x=-(3x-2) we get x= 1/2

so It is both 1 and 1/2



No x = 1/2 is not the solution, i have proved it, you can also read the message below yours, it too proves the same thing..
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Re: Interesting inequality problem. [#permalink] New post 02 Oct 2011, 23:12
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I think one other point is that visually we should know this has only one solution.

Visualize y = |x|

Image

Visualize y = 3x - 2

Image

These will only intersect once!

Image

On the other hand, consider:

|2x – 3| – 4 = 3
|2x – 3| = 7

We can see straight away this has two solutions. Visualize the left hand side and the right hand side:

Image
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Last edited by pike on 03 Oct 2011, 00:47, edited 1 time in total.
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Re: Interesting inequality problem. [#permalink] New post 02 Oct 2011, 23:33
silly me . i have answered C. I must be burned and buried deeply for such a stupid mistake hehe
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Re: Interesting inequality problem. [#permalink] New post 03 Oct 2011, 03:22
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Expert's post
mave23 wrote:
Hi All,

Came across this rather interesting quant question, and thought of sharing it with folks here:

|x| = 3x - 2

a. 1
b. 1/2
c. 1 and 1/2
d. -1/2
e. -1



I know it's fun to solve when you know what you are doing (and more importantly, why you are doing it) and the graph method is especially satisfying but don't forget a very basic trick in such questions. When you have values for x in the options, and an equation in the question, we could just plug the values in. Since x = 1 satisfies the equation and x = 1/2 doesn't, answer has to be (A).
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Re: Interesting inequality problem. [#permalink] New post 03 Oct 2011, 09:44
VeritasPrepKarishma wrote:

I know it's fun to solve when you know what you are doing (and more importantly, why you are doing it) and the graph method is especially satisfying but don't forget a very basic trick in such questions. When you have values for x in the options, and an equation in the question, we could just plug the values in. Since x = 1 satisfies the equation and x = 1/2 doesn't, answer has to be (A).


Wow thanks, you are correct i had overlooked this option.. do you think such questions will come in the real gmat, where plugging the value may reveal the answer?

Thanks.
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Re: Interesting inequality problem. [#permalink] New post 03 Oct 2011, 18:35
mave23 wrote:
VeritasPrepKarishma wrote:

I know it's fun to solve when you know what you are doing (and more importantly, why you are doing it) and the graph method is especially satisfying but don't forget a very basic trick in such questions. When you have values for x in the options, and an equation in the question, we could just plug the values in. Since x = 1 satisfies the equation and x = 1/2 doesn't, answer has to be (A).


Wow thanks, you are correct i had overlooked this option.. do you think such questions will come in the real gmat, where plugging the value may reveal the answer?

Thanks.


depends on how you are doing in the exam :)
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Re: Interesting inequality problem. [#permalink] New post 03 Oct 2011, 20:41
Just another simple approach. Since |x| is always positive so 3x - 2 should always be positive for this equation to be true. You can simply rule out options (D) and (E), as any negative values of x will make 3x-2 negative. For 1/2, again 3x-2 becomes negative. So only option left is (A).
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Re: Interesting inequality problem. [#permalink] New post 04 Oct 2011, 03:03
Expert's post
mave23 wrote:
Wow thanks, you are correct i had overlooked this option.. do you think such questions will come in the real gmat, where plugging the value may reveal the answer?

Thanks.


Sure it is possible. That is why it is good to keep this in mind since a relatively tricky question can be easily solved this way. Though, GMAC is wise to such tricks and since these questions don't involve much work on your part, don't expect the options to help you often (nevertheless, you should always keep one eye on the options), at least not right in the beginning. If you do get such a question, it may not be a very high level question.
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Re: |x| = 3x - 2 Interesting inequality problem [#permalink] New post 06 Oct 2011, 02:53
Answer: A

Approach:
Substituted option A i.e x=1. Inequality satisfied.

This eliminates all options except option C. This is because only Option A & C carries x=1 which satisfies the inequality.

Substituted x=1/2. Inequality not satisfied. Hence Option C eliminated. Only remaining option A.
Selected Option A.

----------------------------------------------
P.S. - Thanks pike for the graphical solution. I loved your approach.
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Re: |x| = 3x - 2 [#permalink] New post 18 Sep 2013, 06:23
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Re: |x| = 3x - 2 [#permalink] New post 21 Sep 2013, 00:06
Absolute values are easy. The hard part is remembering to substitute back into the original question to check if it satisfies the equation.

Here are some rules:
|x|>=0
|x| = sqrt(x)^2
|0| = 0
|-x|=|x|
|x|+|y|>= |x+y|

Absolute values can also be thought of as the distance between 2 points.

Remember to always always always check whether your solution satisfies the equation!!
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Re: |x| = 3x - 2 [#permalink] New post 26 Mar 2014, 08:40
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Here is the sol. I have attached a screenshot. HTH
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Re: |x| = 3x - 2 [#permalink] New post 18 Jun 2014, 20:51
email2vm wrote:
Here is the sol. I have attached a screenshot. HTH


As (3x-2)>0, x>2/3>0, then we do not need to consider the case x<0. Reduce the amount of caculation.
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Re: |x| = 3x - 2 [#permalink] New post 30 Aug 2014, 08:30
so what is the question here? I am not able to see it. is it to find value of X?


mave23 wrote:
|x| = 3x - 2

A. 1
B. 1/2
C. 1 and 1/2
D. -1/2
E. -1

Soln
[Reveal] Spoiler:
When we proceed to solve this we get two eq:
x = 3x - 2 .... (i) when x > 0
-x = 3x - 2 .... (ii) when x < 0

Therefore we get two results x = 1 and x = 1/2 from (i) and (ii) resp.
Lets plug the values back into the eq.

|1| = 3*1 - 2
==> 1 = 1
so (i) holds good.

|1/2| = 3*(1/2) - 2
==> 1/2 = -1/2 this is incorrect, hence soln from eq (ii) does not hold good.

The soln to the eq |x| = 3x - 2 is 1 only.

Quant guru's let me know if there is a flaw in my calculation method.

Thanks!

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Re: |x| = 3x - 2   [#permalink] 30 Aug 2014, 08:30
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