If x^3

Question

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)

BrentGMATPrepNow · Accepted Answer

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

RELATED VIDEO
 https://www.youtube.com/watch?v=Od98BOicIq4

DavidTutorexamPAL · Answer

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.

Bunuel · Answer

______________
Edited. Thank you.

BrentGMATPrepNow · Answer

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

nayanparikh · Answer

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

push12345 · Answer

X^3-x^2<0
x^2(x-1)<0

X^2 is always positive
X-1 is negative

Posted from my mobile device

ScottTargetTestPrep · Answer

Since x^3 is less than x^2, there are two cases for x: 1) x could be a negative number, or 2) x could be a (positive) number whose value is between 0 and 1.

We see that, in either case, x - 1 will always be negative.

Answer: D

rishab0507 · Answer

Good way is to test standard values 0 ,2,1/2 ,-1/2 ,-2. These values always help in these kind of questions , so that we don't miss any case as now , x^3

Sr1994 · Answer

Here, it is given that  X^3  <  X^2 
 X^3 - X^2  < 0
 X^2  (X-1) < 0

No matter the value of X (either + or -),  X^2  will always be Positive.
So, for  X^2 (X-1) < 0, (X-1) should be negative.

Answer D

Fdambro294 · Answer

(X)^3 < (X)^2

there are 2 Cases in which (X)^3 will be LESS THAN < (X)^2

Case 1: when X is a (+)Pos. Proper Fraction from 0 < X < 1 ------> Raising a (+)Pos. Proper Fraction to a HIGHER Integer Exponent will result in a DECREASE in Value

or

Case 2: when X is ANY (-)Negative Value EXCEPT (-)1

-A- X

does NOT have to be (-)Neg. if X = 1/2

-B- (-)X

does NOT have to be (-)Neg. if X = -5

(-)-5 = +5

-C- X^5

if X = +1/2 -----> does NOT have to be (-)Negative

-E- (X)^ -1

Rule of (-)Negative Integer Exponents: you take the Reciprocal of the Base and change the Sign of the (-)Neg. Integer Exponent

(X)^ -1 = (1/X)^1 = 1/X

if X = +1/2 -----> 1/X = 2, which is NOT Negative

-D- X - 1

case 1: if X is a +Pos. Proper Fraction, the result will be (-)Negative

and

case 2: any (-)Negative Value combined with another (-)Negative Value will result in a (-)Negative Value

-D- is the correct answer

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards