If x 0, is x^2

Question

If x > 0, is x^2 < x ?

(1) 0.1 < x < 0.4
(2) x^3 < x^2

BrentGMATPrepNow · Accepted Answer

Target question :     Is x² ＜ x ?

This is a great candidate for  rephrasing the target question . 
 Aside: Here’s a video with tips on rephrasing the target question:  https://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

Given : x ＞ 0  
Since we're told that x is POSITIVE, we can safely take  x² ＜ x  and divide both sides by x to get:  x ＜ 1 
  REPHRASED target question :     Is x ＜ 1 ?

Statement 1 : 0.1 ＜ x ＜ 0.4   
If x is BETWEEN 0.1 and 0.4, then we can be certain that   x ＜ 1 
Since we can answer the  REPHRASED target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : x³ ＜ x²  
If x is POSITIVE, then we know that x² is also POSITIVE
This means we can safely take x³ ＜ x² and divide both sides by x² to get  x ＜ 1 
Aha! This is exactly what our REPHRASED target question is asking! 
Since we can answer the  REPHRASED target question  with certainty, statement 2 is SUFFICIENT

Answer:  D

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smartyman · Answer

My confusion is on statement 2.

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

Therefore
x< -1 or 0<x<1

So, how can the answer be D?

acegmat123 · Answer

Statement 2:

x^3 < x^2

x^3 - x^2 < 0

x^2(x-1) < 0

x < 1 (since x^2 will always be positive)

From question stem we have to find whether 0<x<1.

Question stem says x>0.

So x lies between 0 and 1.

0<x<1

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sobby · Answer

If x ＞ 0, is x^2 ＜ x ?
(1) 0.1 ＜ x ＜ 0.4
(2) x^3 ＜ x^2

lets take x^2-x <0 --We can rewrite it as: x(x-1)<0, if we plot it on number line , we will get a range between {0-1}(0 and 1 not included) that will satisfy x^2-x <0.
so any solution having range 0-1 will satisfy the given equation.

A. Suff.

B.x^2(x-1)<0---if plot in a number line , x can only have value between {0-1}(excluding 0 and 1) and can have x<0 as solution..as in the question stem it is given as x can take only positive value.only possible range is {0-1}.this range satisfy x^2-x <0 also .so suff.

Answer : D

awdxzs · Answer

You did the factorisation wrong.
x^3<x^2
x^3-x^2<0
x^2(x-1)<0
Since x^2 is always >0 so x-1<0 and which leads to x<1
Hope it helps

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malavika1 · Answer

This is strange. The original post says: (2) x^3 > x^2

However, it seems that all the solutions have used (2) x^3 < x^2

What's the actual question?

Bunuel · Answer

Edited. It's (2) x^3 < x^2

malavika1 · Answer

Thanks Bunuel. This is good to know.

GMATWhizTeam · Answer

Step 1: Analyse Question Stem

The question tells us that x is a positive number.

We have to evaluate if \(x^2\) < x.

Using the number line is an effective approach in solving these kind of questions.
On the number line, 4 intervals can be marked and the relationship between different functions can be easily compared.

Attachment:

20th Apr 2022 - Post 1.png [ 1.49 KiB | Viewed 4200 times ]

Of the four intervals, \(x^2\) < x is satisfied only when 0 < x < 1. Therefore, the question can be rephrased as, “ Is 0 < x < 1?”

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: 0.1 < x < 0.4

This information answers the question directly and tells us that 0 < x < 1.

The data in statement 1 is sufficient to answer the question with a definite YES.
Statement 1 alone is sufficient. Answer options B, C and E can be eliminated.

Statement 2: \(x^3\) <\( x^2\)

Picking values from the 4 different intervals on the number line, we see that \(x^3\) < \(x^2\) is satisfied when x lies in the following ranges:

However, it is important to note that the question mentions that x > 0. Therefore, the first range can be ruled out and hence, the only range which satisfies the given inequality is 0 < x < 1.

The data in statement 2 is sufficient to answer the question with a definite YES.
Statement 2 alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

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If x > 0, is x^2 < x ? (1) 0.1 < x < 0.4 (2) x^3 < x^2

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