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If x^3 < x^2, which of the following must be negative?

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If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 15 Apr 2018, 08:58
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A
B
C
D
E

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  25% (medium)

Question Stats:

66% (01:17) correct 34% (01:06) wrong based on 208 sessions

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If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post Updated on: 15 Apr 2018, 09:24
Bunuel wrote:
If x^3 < x^2, which of the following must be negative?

A. x
B. −x
C. x^5
D. x^(−1) --> EDIT to x - 1
E. x^(−1)


Bunuel is there a mistake in the question?
Simplifying gives x^2(x-1)<0 so x<1 and x!=0 so there are no correct answers... also, options (D), (E) are identical


EDIT, to answer corrected question:
As all we're given are equations, we'll work with them using simplification tools.
This is a Precise approach.

Simplifying our stem gives x^3 - x^2 < 0 so x^2(x - 1) < 0. Since x^2 is always non-negative then x - 1 < 0 and x < 1. Also, x cannot be 0, otherwise we would get 0 < 0.
A. could be 1/2 so it is false.
B. could be -(-1)=1 and is false.
C. could be (1/2)^5 > 0 and is false
D. as x < 1 then x - 1 < 0. Yes!

(D) is our answer.
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Originally posted by DavidTutorexamPAL on 15 Apr 2018, 09:10.
Last edited by DavidTutorexamPAL on 15 Apr 2018, 09:24, edited 1 time in total.
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Re: If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 15 Apr 2018, 09:20
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Re: If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 15 Apr 2018, 09:21
Top Contributor
Bunuel wrote:
If x^3 < x^2, which of the following must be negative?

A. x
B. −x
C. x^5
D. x − 1
E. x^(−1)


Given: x³ < x²
This tells us that x ≠ 0. So, we can be certain that x² is POSITIVE.
Since x² is POSITIVE, we can safely divide both sides of the inequality by x²
When we do this, we get: (x³)/(x²) < 1
Simplify: x < 1
Subtract 1 from both sides to get: x - 1 < 0
In other words, x - 1 is NEGATIVE

Answer: D

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Re: If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 15 Apr 2018, 15:46
Top Contributor
Bunuel wrote:
If x^3 < x^2, which of the following must be negative?

A. x
B. −x
C. x^5
D. x − 1
E. x^(−1)


Alternatively, we can TEST some values.

We're told that x^3 < x^2
So, one possible value is x = -1

Now let's test the answer choices. We get:
A. -1, which is negative. KEEP
B. −(-1) =1, which is positive. ELIMINATE
C. (-1)^5 = -1, which is negative. KEEP
D. (-1) − 1 = -2, which is negative. KEEP
E. (-1)^(−1) = 1/(-1) = -1, which is negative. KEEP

Okay, we're still left with A, C, D and E

We need to test another value.
Another possible value is x = 1/2
Test the remaining answer choices...
A. 1/2, which is positive. ELIMINATE
C. (1/2)^5 = 1/32, which is positive. ELIMINATE
D. (1/2) − 1 = -1/2, which is negative. KEEP
E. (1/2)^(−1) = 2/1 = 2, which is positive. ELIMINATE

By the process of elimination, the correct answer is D

Cheers,
Brent
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Re: If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 15 Apr 2018, 20:26
Bunuel wrote:
If x^3 < x^2, which of the following must be negative?

A. x
B. −x
C. x^5
D. x − 1
E. x^(−1)



For x^3 < X^2 to be true, there are 3 cases
Case 1: x< -1, in which case except B all are negative
Case 2: -1<x<0, in which case except B, all are negative
Case 3: 0<x<1, in which case except A,C,E all are negative.

Thus in all the 3 case above for x-1 is negative for which is D is negative.

hence the answer is D
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Re: If x^3 < x^2, which of the following must be negative?  [#permalink]

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New post 29 Apr 2018, 04:43
X^3-x^2<0
x^2(x-1)<0

X^2 is always positive
X-1 is negative

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Re: If x^3 < x^2, which of the following must be negative?   [#permalink] 29 Apr 2018, 04:43
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