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Is x^2 > 1/x ?

This topic has 3 expert replies and 0 member replies

Is x^2 > 1/x ?

Post Thu Sep 14, 2017 1:48 pm
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    Is x^2 > 1/x ?

    (1) x^2 > x

    (2) 1 > 1/x

    The OA is D.

    Really each statement alone is sufficient? Why?

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    Post Fri Sep 15, 2017 12:41 am
    Vincen wrote:
    Is x^2 > 1/x ?

    (1) x^2 > x

    (2) 1 > 1/x

    The OA is D.

    Really each statement alone is sufficient? Why?
    Statement 1: x^2 > x

    We can consider three important ranges.

    1. If x is negative, then x^2 is always a positive number and x is always a negative number, thus, x^2 is greater than x.
    2. If x > 1, then x^2 is always greater than x.

    3. If 0 < x < 1, the inequality x^2 > x will not hold true. For example, say x = 1/2, then x^2 = 1/4 < 1/2 (= x). So, this is not a valid case.

    From #1 and #2, we get either x < 0 or x > 1.

    1. If x < 0, the ineuqality x^2 (= positive) > 1/x (= negative). The asnwer is Yes.
    2. If x > 1, say x = 2, then x^2 (= 4) > x (= 1/2). The asnwer is Yes.

    Statement 2: 1 > 1/x

    Again, we can consider three important ranges.

    1. If x is negative, then 1/x is always a negative number, thus, 1 is greater than 1/x.
    2. If x > 1, then 1 is always greater than 1/x.

    3. If 0 < x < 1, the inequality 1 > 1/x will not hold true. For example, say x = 1/2, then 1 < 1/(1/2) => 1 < 2. So, this is not a valid case.

    From #1 and #2, we get either x < 0 or x >1.

    This is the same result that we got from Statement 1. Thus, each statement itself is sufficient.

    The correct answer: D

    Hope this helps!

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    Thanked by: Vincen
    Post Fri Sep 15, 2017 9:43 am
    Hi Vincen,

    We're asked if X^2 is greater than 1/X. This is a YES/NO question. You can solve it by TESTing VALUES or using Number Properties.

    1) X^2 > X

    With the information in Fact 1, there are 2 'groups' of numbers that fit this inequality:
    -ANY negative number
    -Positive numbers that are GREATER than 1

    IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
    IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
    Thus, the answer to the question is ALWAYS YES.
    Fact 1 is SUFFICIENT

    2) 1 > 1/X

    With the information in Fact 1, there are 2 'groups' of numbers that fit this inequality:
    -ANY negative number
    -Positive numbers that are GREATER than 1

    These are the SAME 2 groups of numbers that fit Fact 1 - so we already know the answer to the question:
    IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
    IF... X = a negative, then X^2 is greater than 1/X and the answer to the question is YES.
    Thus, the answer to the question is ALWAYS YES.
    Fact 2 is SUFFICIENT

    Final Answer: D

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

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    Post Fri Sep 15, 2017 10:49 am
    Vincen wrote:
    Is x² > 1/x ?

    (1) x² > x
    (2) 1 > 1/x
    Target question: Is x² > 1/x ?

    Statement 1: x² > x
    First off, this inequality tells us that x ≠ 0
    Second, we can conclude that x² is POSITIVE.
    So, we can safely divide both sides of the inequality by x² to get: 1 > 1/x
    If 1 > 1/x, then there are two possible cases:
    Case a: x > 1. If x is a positive number greater than 1, then 1/x will definitely be less than 1.
    Case b: x is negative. If x is negative, then 1/x will definitely be less than 1.

    IMPORTANT: So how do these two cases affect the answer to the target question? Let's find out.
    Case a: If x > 1, then x² is greater than 1, AND 1/x is less than 1. This means x² > 1/x
    Case b: If x is negative, then x² is positive, AND 1/x is negative. This means x² > 1/x
    Perfect - in both cases, we get the SAME answer to the target question
    Since we can answer the target question with certainty, statement 1 is SUFFICIENT

    Statement 2: 1 > 1/x
    Notice that this inequality is the SAME as the inequality derived from statement 1 (we got 1 > 1/x)
    Since we already saw that statement 1 is sufficient, it must be the case that statement 2 is also SUFFICIENT

    Answer: D

    Cheers,
    Brent

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    Thanked by: Vincen
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