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# Which of the follwoing represents the range for all x which

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Manager
Joined: 23 May 2007
Posts: 109
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Which of the follwoing represents the range for all x which [#permalink]  15 Jun 2007, 16:57
Which of the follwoing represents the range for all x which satisfy the inequality |2-x| < 2?

1.(-2,2)
2.(-2,4)
3.(0,2)
4.(0,4)
5.(2,4)

Intern
Joined: 11 Feb 2007
Posts: 30
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i would go with C.

2-2 = 0 < 2
2-0 = 2 < 2??

this was the only that agreed even half way.
Manager
Joined: 23 May 2007
Posts: 109
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Kudos [?]: 2 [0], given: 0

I also selected C .But the answer is (0,4) can someone throw some light
Manager
Joined: 23 Dec 2006
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Kudos [?]: 11 [0], given: 0

4:

We have | 2-x| <2, so:

1) | 2-x| <2> 0.

2) Because we are dealing with a modulus, we must account for when it is all negative, so suppose -2+x<2: we end up with x<4.

hence, 4: (0,4)
Manager
Joined: 07 May 2007
Posts: 181
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|x-2| <2> A

By definition of modulus, when x-2 > 0, |x-2| = x-2

Substituting in A gives x-2 <2> x <4> B

By definition of modulus, when x-2 < 0, |x-2| = -(x-2) = 2-x

Substituting in A gives 2-x <2> 0 <x> C

Combine B and C to get 0 < x < 4
Manager
Joined: 18 Apr 2007
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The way I deal with absolute values is to always rewrite the equation to account for the positive and the negative possibilities. In this case,

|2-x|<2 can be either of the following:

1) 2-x<2
2) x-2<2 (this is the same as -(2-x)<2)

Then solve each equation for x:
1) (-x)<0>0 (the inequality reverses when dividing by -1)
2) x<4

Therefore, the inequality reads 0<x<4 and so (0,4)

Director
Joined: 26 Feb 2006
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Re: Challenge inequality [#permalink]  15 Jun 2007, 19:11
dreamgmat1 wrote:
Which of the follwoing represents the range for all x which satisfy the inequality |2-x| < 2?

1.(-2,2)
2.(-2,4)
3.(0,2)
4.(0,4)
5.(2,4)

the question seems poorly structured cuz x cannot range up to 4. if it is 4, then l2-xl is equal to 2 which is not less than 2. the same applies to 0 too.

therefore D should be "0 < x < 4" not "0 to 4".
SVP
Joined: 01 May 2006
Posts: 1798
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(D) for me too

So, fully detailed :
|2-x| < 2
<=> |(-1)*(x-2)| < 2
<=> |(-1)|*|x-2| < 2
<=> 1*|x-2| < 2
<=> |x-2| < 2
<=> -2 < x-2 < 2
<=> 0 < x < 4
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