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Which of the following represents 1<x<9?

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Which of the following represents 1<x<9? [#permalink] New post 31 Jan 2012, 10:22
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Which of the following represents 1<x<9?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

How to do this quickly? The way I m doing it ( i.e. solving each statement possibilities ) takes 3-4 min.
[Reveal] Spoiler: OA
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Re: Absolute Trouble [#permalink] New post 31 Jan 2012, 14:57
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Hi, I'm happy to help with this. :)

Fundamentally, absolute value is about distance. That is something many folks fail to grasp.

|x| is the distance of x from the origin on the number line.

|x - 5| is the distance of x from the point 5 on the number line.

|x + 3| is the distance of x from the point -3 on the number line. (That's because x + 3 = x - (-3))

So, we want to write 1<x<9 as a distance statement.

Step #1: find the middle of that region. The midpoint is at 5. So, this is about distance from 5 ---> |x - 5|

Step #2: how far can we get from 5? Well, up to almost 9, or down to almost 1 -- that is, almost 4. Thus:

distance from 5 < 4

|x - 5| < 4

Once you understand the distance-interpretation of absolute value, you need not spend much time at all on questions like this.

Does that make sense? Please let me know if I can clarify anything here or if you have any more questions.

Here's a challenging absolute value question for practice.

http://gmat.magoosh.com/questions/964

Once you submit your answer, that question should be followed by a video explanation of the answer.

Mike :)
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Re: Absolute Trouble [#permalink] New post 31 Jan 2012, 18:57
Expert's post
docabuzar wrote:
Which of the following represents 1<x<9?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

How to do this quickly? The way I m doing it ( i.e. solving each statement possibilities ) takes 3-4 min.


I'd probably use the distance concept for this question the same way as Mike well explained above. Though one can also use plug-in method:

Given: 1<x<9. Now, try two numbers, one from this range and one out of this range. Correct answer choice must be true for the first number and false for the another. Let's try: x=8 and x=0. So, the correct answer choice must be true for x=8 and false for x=0.

A. |x|<3 --> false for x=8, so no need to try another number. Discard;
B. |x+5|<4 --> false for x=8, so no need to try another number. Discard;
C. |x-1|<9 --> true for x=8, but also true for x=0. Discard;
D. |-5+x|<4 --> true for x=8, and false for x=0. Correct;
E. |3+x|<5 --> false for x=8, so no need to try another number. Discard.

Only one answer choice D works.

Answer: D.

Note that for plug-in method it might happen that for some particular numbers more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only.

You can also check Absolute Value chapter of Math Book, where Walker discusses the same exact question using the distance concept: math-absolute-value-modulus-86462.html

Hope it helps.
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Re: Which of the following represents 1<x<9? [#permalink] New post 04 Dec 2012, 01:10
1<x<9

1) Get the center of the range
CENTER = (9 + 1)/2 = 10/2 = 5
2) Flip the sign and append the variable x
|x - 5|
3) Get the distance of the CENTER from one of the END
DISTANCE = 9 - 5 = 4
4) Decide the inequality sign. WE know that x is within the range so choose "<"

ANsweR: |x-5| < 4
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Re: Which of the following represents 1<x<9? [#permalink] New post 19 Dec 2012, 12:05
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IMO
[Reveal] Spoiler:
D


A. |x|<3 => -3<x<3 => no need to go further its wrong
B. |x+5|<4 => -4<x+5<4 => -9<x..... no need to go further its wrong
C. |x-1|<9 => -9<x+1<9 => -10<x..... no need to go further its wrong
D. |-5+x|<4 => -4<x-5<4 => 1<x<9 - BINGO
E. |3+x|<5 => -5<x+3<5 => -8<x..... no need to go further its wrong

Hence D
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Re: Which of the following represents 1<x<9? [#permalink] New post 19 Dec 2012, 21:21
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docabuzar wrote:
Which of the following represents 1<x<9?

A. |x|<3
B. |x+5|<4
C. |x-1|<9
D. |-5+x|<4
E. |3+x|<5

How to do this quickly? The way I m doing it ( i.e. solving each statement possibilities ) takes 3-4 min.


The below is my approach for any modulus qtn in GMAT. Please read thru the complete explanation to understand the modulus concept.

Remember.
The meaning of |x-y| is "On the number line, the distance between X and +Y"
The meaning of |x+y| is "On the number line, the distance between X and -Y"
The meaning of |x| is "On the number line, the distance between X and 0".

On the # line, Left to 0 are all the -ve #s and right to 0 are all +ve #s

original qtn:
is x between 1 and 9 ==> .....................0..1....................9..... ==> is X in the highlighted area

A) |x| < 3 ---- wrong
on the number line, the distance of x from 0 is < 3 ==> ......-3........0........3....

B) |x+5|<4 --- wrong
on the number line, the distance of x from -5 is < 4 ==> ............-9............-5...........1...0.......3....

C)|x-1|<9 --- wrong
on the number line, the distance of x from 1 is < 9 ==> ..................-8..............................0....1.......................9...10....

D) |-5+x|<4
on the number line, the distance of x from +5 is < 4 ==> ............0..1..............5...............9 ---- This is what is asked in the question, hence, correct answer
E. |3+x|<5
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Re: Which of the following represents 1<x<9? [#permalink] New post 09 Jan 2014, 12:16
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Re: Which of the following represents 1<x<9? [#permalink] New post 25 Mar 2014, 11:41
D. |-5+x|<4



----1--------5---------9-----

(5-4).....origin....(5+4)
Re: Which of the following represents 1<x<9?   [#permalink] 25 Mar 2014, 11:41
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