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Is |X| < 1 ?

This topic has 4 expert replies and 1 member reply
kamalakarthi Senior | Next Rank: 100 Posts Default Avatar
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Is |X| < 1 ?

Post Sun Aug 06, 2017 2:50 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Hi, This is from MCAT Question.

    Can you please help me on solving this question.

    Steps Followed :: I tried to rephrase the question , Essentially the question is -1<X<1 ?

    St 1:: If X is +ve, X < 1
    if X is _ve, -x/X < -x
    so 1/x <-1 (Dividing -x on both the sides) Is this correct ?
    -1< x or X > -1 NS

    St 2:: X is +ve, It cannot happen |X| >x
    but if x is negative , x < -x ..I got stuck here. and I thought This is also not sufficient

    After this, I thought C is also not sufficient so I chose E . Can you please help., What is my mistake in Statement 2.


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    Post Sun Aug 06, 2017 4:27 pm
    Hi kamalakarthi,

    This question can be solved by TESTing VALUES and Number Properties.

    We're told that X cannot equal 0. We're asked if |X| is less than 1. This is a YES/NO question.

    1) X/|X| < X

    From a Number Property standpoint, there are specific values of X that will 'fit' this inequality:
    -Any positive number greater than 1.
    -Any negative fraction (re: -1 < X < 0)

    For example, IF....
    X = 2.... 2/|2| = 1 < 2 and the answer to the question is NO.
    X = -1/2.... (-1/2)/|-1/2| = -1 < -1/2 and the answer to the question is YES.
    Fact 1 is INSUFFICIENT

    2) |X| > X

    From a Number Property standpoint, there are specific values of X that will 'fit' this inequality:
    -Any negative value

    For example, IF....
    X = -1/2.... |-1/2| > -1/2 and the answer to the question is YES.
    X = -2... |-2| > -2 and the answer to the question is NO.
    Fact 2 is INSUFFICIENT

    Combined, there is only one specific type of value that will fit BOTH Facts: negative fractions (-1 < X < 0), so the answer to the question will be ALWAYS YES.
    Combined, SUFFICIENT

    Final Answer: C

    GMAT assassins aren't born, they're made,
    Rich

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    Thanked by: kamalakarthi

    GMAT/MBA Expert

    Post Sun Aug 06, 2017 9:48 pm
    Here's one way:

    S1:

    Since |x| must be > 0, we can safely multiply both sides by it:

    x < x * |x|

    Now subtract x from both sides:

    0 < x * |x| - x

    Now factor out x:

    0 < x * (|x| - 1)

    This gives us two possibilities:

    (i) x AND |x| - 1 are both positive
    OR
    (ii) x and |x| - 1 are both negative

    (i) reduces to |x| - 1 > 0, or |x| > 1, or x > 1
    (ii) reduces to |x| - 1 < 0, or |x| < 1, or -1 < x < 0

    So IF x is negative, then |x| < 1. This helps, but isn't sufficient on its own. (It does reduce the question to "Is x < 0?", though!)

    S2:

    |x| > x

    This obviously won't be true if x ≥ 0 (since, if x ≥ 0, |x| = x). So x < 0. This is great ... but not on its own!

    Together, though, we're set. S1 reduces the prompt to "Is x < 0?" and S2 answers that question in the affirmative.

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    Post Sun Aug 06, 2017 9:51 pm
    Another way, if the first is too technical:

    S1:

    x > x / |x|

    |x| * x > x

    Now take two cases. If x > 0, then we divide both sides to get |x| > 1.

    If x < 0, then we divide both sides to get |x| < 1.

    So |x| will be < 1 if and only if x < 0.

    S2 I'd treat the same way I did in my post above, and much like that one, we'll arrive at the same conclusion: S1 simplifies the question and S2 answers that simplified question, leaving us with an answer of C.

    Thanked by: kamalakarthi
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    kamalakarthi Senior | Next Rank: 100 Posts Default Avatar
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    Post Mon Aug 07, 2017 8:48 am
    Thank you for the help.

    GMAT/MBA Expert

    Post Fri Aug 18, 2017 1:19 pm
    kamalakarthi wrote:
    Thank you for the help.
    My pleasure!

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