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Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0

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Bunuel
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Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink] Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0   06 Oct 2021, 01:00
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Is |x| < 1 ?

(1) \(x^4 - 1 < 0\)

(2) \(\frac{1}{1-|x|}> 0\)


M36-02

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Re: Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink] Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0   09 Oct 2021, 08:30
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Official Solution:


Is \(|x| < 1\) ?

(1) \(x^4 - 1 < 0\):

\(x^4 < 1\).

Take the fourth root: \(|x| < 1 \). Sufficient.

(2) \(\frac{1}{1-|x|} > 0\):

The numerator is positive, so for the fraction to be positive the denominator must also be positive: \(1 - |x| > 0 \). So, \(|x| < 1 \). Sufficient.


Answer: D
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Re: Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink] Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0   06 Oct 2021, 06:00
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Is |x| < 1 ?

(1) \(x^4 - 1 < 0………..(x^2-1)(x^2+1)<0\)
\(x^2+1\) is always positive, so \(x^2-1<0\) or \frac{x^2<1[}{fraction], that is |x|<1.
Sufficient

(2) \([fraction]1/1-|x|}> 0\)
Since 1>0, 1-|x|>0 or |x|<1.
Sufficient


D
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Re: Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink] Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0   06 Oct 2021, 06:40
Bunuel wrote:
Is |x| < 1 ?


(1) \(x^4 - 1 < 0\)
A number can decrease if it's negative that scenario is ruled out since modulus istaken
other possibility being that x is a fraction <1 since the value goes down with power

Clearly sufficient

(2) \(\frac{1}{1-|x|}> 0\)
=> 1>1-|x| >0
=>0> -|x|>1
=> 0< |x| <1 sign changes with multiplication of negative

Clearly sufficient

Therefore IMO D
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Re: Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink] Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0   09 Oct 2021, 08:15
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Bunuel wrote:
Is |x| < 1 ?

(1) \(x^4 - 1 < 0\)

(2) \(\frac{1}{1-|x|}> 0\)


Is |x| < 1 ?

(1) \(x^4-1< 0\) --> \(x^4<1\) --> since both side are non-negative we can take the fourth root from both: \(|x|<1\). Sufficient.

(2) \(\frac{1}{1-|x|}> 0\) --> since the numerator is positive, then the fraction to be positive denominator must also be positive: \(1-|x|>0\) --> \(|x|<1\). Sufficient.

Answer: D.
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Re: Is |x| < 1 ? (1) x^4 - 1 > 0 (2) 1/(1 - |x|) > 0 [#permalink]
09 Oct 2021, 08:15
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