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If |x| < x^2, which of the following must be true?

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If |x| < x^2, which of the following must be true? [#permalink] New post   27 Nov 2021, 04:28
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If |x|<x2, which of the following must be true?

I. x2>1

II. x>0

III. x<−1


(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only
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A
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Re: If |x| < x^2, which of the following must be true? [#permalink] New post   01 Dec 2021, 03:20
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Given: |x|<x2;

Reduce by |x|: 1<|x| (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too, as if x would be zero inequality wouldn't hold true, so |x|>0);

So we have that x<−1 or x>1.

I. x2>1 --> always true;

II. x>0 --> may or may not be true;

III. x<−1 --> may or may not be true.

Answer: A (I only).
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Re: If |x| < x^2, which of the following must be true? [#permalink] New post   01 Dec 2021, 09:00
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To answer this, I would suggest first to consider that |x| and x^2 are both ALWAYS POSITIVE.

Next, notice that this is a MUST question, so finding any good NO case (counterexample) will prove it wrong.

I. Given that |x| is defined as sqrt(x^2), then it follows that the number x must be greater than 1 -- that is, the only cases where the square is smaller than the number itself are fractions between 0 and 1. This MUST BE TRUE.

II.
>NO case make x = -2 and we see |-2| = 2 while (-2)^2 = 4

This one doesn't have to be true.

III.
>NO case x = 2 and we see |2| = 2 while 2^2 = 4

This one doesn't have to be true.

The Answer is A.
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Re: If |x| < x^2, which of the following must be true? [#permalink] New post   29 Sep 2024, 19:17
Carcass wrote:
Given: |x|<x2;

Reduce by |x|: 1<|x| (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too, as if x would be zero inequality wouldn't hold true, so |x|>0);

So we have that x<−1 or x>1.

I. x2>1 --> always true;

II. x>0 --> may or may not be true;

III. x<−1 --> may or may not be true.

Answer: A (I only).


Could you please explain the case for III part? below in the reply the case for third part has been explained but it takes x as 2 whereas x is supposed to be a negative number here
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Re: If |x| < x^2, which of the following must be true? [#permalink] New post   29 Sep 2024, 23:06
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|x|<x2 is given as fact and then we asked to determine which of the following statements MUST be true.

|x|<x2 means that either x<−1 or x>1, x can be ANY value from these two ranges, (I think in your own solution you've reached this conclusion: when x<−1 the graph of |x| is below (less than) the graph of x2 and

when x>1 again the graph of |x| is below the graph of x2).

Now, III says x<−1 this statement is not always true as x can be for example 3 and in this case x<−1 doesn't hold true.

Hope it's clear.
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Re: If |x| < x^2, which of the following must be true? [#permalink]
29 Sep 2024, 23:06
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