Is x^3 > x^2?

Thanks so much Bunuel.

However I did not understand how you arrived at

\(x^2(x-1)>0\)--> is \(x>1?\)

Is \(x^3>x^2\) --> is \(x^3-x^2>0\) --> is \(x^2(x-1)>0\) --> as \(x^2\) is non negative, for \(x^2(x-1)\) to be positive \(x\) must not be zero (first multiple) and \(x-1\) must be more than zero (second multiple) --> \(x\neq{0}\) and \(x-1>0\) --> \(x>1\).

Thanks a bunch. Makes sense now.

Another way of looking at the problem can be
X^2 > X means two things
1) X cannot be a fraction.
2) X can be a negative no.

Once it is ensured that X is non Negative number (as done by option A)
it will ensue that X^3 > X^2.

I Hope it helps

answer should be C, you need to have positive number that is > 1.

1) will fail for fractions like 1/2 , 1/4 ...

2) will fail for -ve numbers

combine both you will get the number that satisfy the base question

I'm confused because MGMAT starts off the answer this way:

X^3 > X^2 equals:
X^3 - X^2 > 0
X^2 (X-1) > 0
Therefore we have an inequality in the form xy>0. After this they start heading to statement 1 and 2.

But on page 89 (Chapter 6 - Inequalities strategy of the same book) it says "Note that we should never subtract or divide two inequalities."

This is how the answer is presented in MGMAT:
Question: X^3>X^2?
Rephrase: X^3 - X^2>0?
Rephrase again: X^2(X-1)>0?

Both terms must be positive or negative, because we have an inequality in the form of xy>0. X^2 can never be negative, so we just need to know that both X^2>0 (meaning X is non zero) and X>1. Since every number greater than 1 is non-zero, we can rephrase the question to:
X>1?

Statement 1 tells us: X is positive. Insufficient.
Statement 2 tells us: X^2>X, so X<0 OR X>1. Insufficient.

Combining the statements, X>1. SOLVED...

I'm confused. Please help.

Is X^3 > X^2?
1) X>0
2) X^2 > X

I have seen addition and subtraction of two inequalities. It helps us solve many inequality question. Usually, the inequalities can be subtracted when two inequalities have opposite sign;

1<2 ---A
100>50 ----B

Subtract B from A because the two inequalities have opposite signs; first inequality has "<" sign and second inequality has ">".
1-100<2-50 ; The sign "<" will remain that of A's
-99<-48

Perhaps you will need to read gmatclub's Number properties.

Coming to the question:
Is X^3 > X^2?
1) X>0
2) X^2 > X

X^3>X^2
X^3-X^2>0
X^2(X-1)>0
X>1

Question is reduced to:
Is X>1?

1. X>0.
So, X can be greater than 1 or smaller than 1.
Not sufficient.

2. X^2 > X
X^2-X>0
X(X-1)>0
X>1 or X<0
Not Sufficient.

Combining both;
X>1.
Sufficient.

Ans: "C"

X^3 - x^2 > 0

x^2(x-1) > 0

So either x^2 > 0 and x-1 > 0, because x^2 > 0 is always true therefore x > 1 has to be true

Note that the other part is not possible, i.e, x^2 < 0 and x-1 < 0

Now, (1) is insuff as we can't infer if x > 1 from this.


From (2)

x^2 - x > 0

x(x-1) > 0

For this to be true, either x > 0 and x-1 > 0, (which cannot be inferred)

or x < 0 and x < 1, which is contrary to what should happen, hence insufficient

But on taking (1) and (2) together, x > 0, so x > 1, hence sufficient, answer is C.

Step 1) The first thing I notice is that one exponent is odd and the other is even. So I immediately know what it's testing.

Step 2) I know I need to check positive and negative...but I also need to check fractions as well.

Check positive/negative:
If x = 2, then of course x^3 > x^2 [GOOD]
If x= -2, then x^3 is NOT > x^2 [NO GOOD]

So (A) and (D) are no good since there's inconsistency here with GOOD/NO GOOD.
For now, I don't need to check fractions. Just evaluate (2) for now.

Step 3) Evaluating (2): x^2 > x
Genearlly, squaring a number means it gets bigger. BUT--I know there's an exception--FRACTIONS!
If x = 1/2, then (1/2)^2 = (1/4)
So when you square a fraction, it actually gets smaller!
OK, so (2) is telling us: "Let's use only non-fractions."

Obviously, if we plug in a normal number like 3, it's gonna be a [TRUE] statement.
What happens if we plugin a negative number to x^3 > x^2?
Try x=-2. Well, then you'd get -8>+4 [FALSE].
So with x=+2, you get TRUE, but with x=-2, you get FALSE. .

So since you sometimes get FALSE and sometimes get TRUE---then you know this conflict means (B) is NO GOOD!

Step 4) But what about if you combine (1) and (2) together.

Well, (1) basically tells us we can restrict x to be positive.
(2) basically tells us we can restrict x to be only non-fractions.

So what if we only use x values that are positive AND non-fractions. That means x>=1.

Can we answer this original question definitively?

Is x^3 > X^2?

Well, when we only use x>1, then this statement is ALWAYS true. So voila!

When we combine (1) and (2) we can restrict the scope of possible X's to only those that are POSITIVE and NON-FRACTIONS--which means x>1.

Turns out this works out great. So we can choose (C) is the final answer.

https://www.gmatpill.com/practice-questi ... exponents/

Thank you all for the awesome explanations. It makes more sense now.

"You can only apply subtraction when their signs are in the opposite directions:

If a>b and c<d (signs in opposite direction: > and <) --> a-c>b-d (take the sign of the inequality you subtract from).
Example: 3<4 and 5>1 --> 3-5<4-1."

This was definitely news to me. Glad to know...

Thanks for the detailed post. I apologize for posting this in the wrong forum - duly noted.

Now while trying to follow your logic, I seem to come to a dead end. Please help me out here -
You solved (2) x^2 > x --> x(x-1)>0 --> x<0 or x>1. Not sufficient.
What did you do to arrive at this conclusion. The way I see it, x(x-1)>0 is an xy inequality...where x can be -ve and y +ve OR vice versa
Therefore
X>0 and X<1
OR
X<0 and X>1

What's wrong with my logic here?

Finally, how do we get (1)+(2) Intersection of the ranges from (1) and (2) gives: x>1. Sufficient.

Thanks for the answer, I have a question here too...how did you arrive at this conclusion:
X^3>X^2
X^3-X^2>0
X^2(X-1)>0
X>1

The way I see it: X^2(X-1)>0 is an xy>0 solution where either x>0 and y<0 OR x<0 and y>0
Therefore in this case, X^2>0 and X-1<0 which means X>0 and X<1 ----- (A)
OR
X^2<0 and X-1>0 which means X<0 and X>1 ----- (B)

When we combine statements 1 and 2 we know that X>0 in (1)
Therefore (B) is not possible because X cannot be less than 0...what am I doing wrong here...

I'll stay away from the top part of your question. As for the bottom:
[quote2gmatpill]Well, (1) basically tells us we can restrict x to be positive. (x>0)
(2) basically tells us we can restrict x to be only values outside of the -1 to 1 range.

So what if we only use x values that are positive AND non-fractions. That means x>1.[/quote2]



Hope that helps! By the way, be careful with all these symbols and calculations you are doing. The GMAT tests your REASONING skills and your ability to "make sense" of it. I may be wrong--but I get the sense you're getting too bogged down on the nitty gritty details yet don't really understand how to translate the math symbols into logic and reasoning.

X^2(X-1)>0 is an xy>0 solution where either x>0 and y<0 OR x<0 and y>0

This is not correct.

If a*b > 0; It means either a and b are both +ve OR a and b are both -ve.
If a*b < 0; It means either a is +ve and b is -ve OR a is -ve and b is +ve.

X^2(X-1)>0

Here; either X^2 and (X-1) are both negative OR X^2 and (X-1) are both +ve

X^2 can't be -ve.
Note:
A square can never be a negative value. It can be 0 or +ve.
X^2 is also not 0. Because if X=0; then X^2=0; That will make X^2(X-1) = 0; but X^2(X-1)>0. Thus, X is not 0.

So, X^2 is +ve and X-1 is also +ve.
X-1>0
X>1 and X>0
Combing both; X>1.

So; the ultimate question becomes;
Is X>1

1. This statement tells us that X>0.
X can be 0.5<1 or 2>1.
Thus not sufficient.

2. X^2>X
X^2-X>0
X(X-1)>0

Again a*b>0; a +ve and b+ve OR a-ve and b-ve

X(X-1)>0
X>0 and X-1>0 i.e. X>1
Combing both; X>1
OR
X<0 and X-1<0 i.e. X<1
Combing both; X<0

We have X>1 or X<0
Not sufficient.

Combing both;
Statement 1 tells us that X>0
Statement 2 tells us that X could be less than 0 or greater than 1. Since St 1 tells us that X>0; we can ignore the St2 condition where it says X<0.
Thus, X>0 and X>1. Means X>1

C.
***************************************************

These questions become easier if you knew how to find proper ranges of X in the inequality. Get yourself acquainted with the following trick:
https://gmatclub.com/forum/inequalities-trick-91482.html

This was little cumbersome for me in the beginning. But, with adequate practice and guidance from Bunuel, I am getting hold of it. Yet to become an exponent though. It's really proven fruitful for me.

Good luck!!

X^2(X-1)>0 is an xy>0 solution where either x>0 and y<0 OR x<0 and y>0

This is not correct.

If a*b > 0; It means either a and b are both +ve OR a and b are both -ve.
If a*b < 0; It means either a is +ve and b is -ve OR a is -ve and b is +ve.

X^2(X-1)>0

Here; either X^2 and (X-1) are both negative OR X^2 and (X-1) are both +ve

X^2 can't be -ve.
Note:
A square can never be a negative value. It can be 0 or +ve.
X^2 is also not 0. Because if X=0; then X^2=0; That will make X^2(X-1) = 0; but X^2(X-1)>0. Thus, X is not 0.

So, X^2 is +ve and X-1 is also +ve.
X-1>0
X>1 and X>0
Combing both; X>1.

So; the ultimate question becomes;
Is X>1

1. This statement tells us that X>0.
X can be 0.5<1 or 2>1.
Thus not sufficient.

2. X^2>X
X^2-X>0
X(X-1)>0

Again a*b>0; a +ve and b+ve OR a-ve and b-ve

X(X-1)>0
X>0 and X-1>0 i.e. X>1
Combing both; X>1
OR
X<0 and X-1<0 i.e. X<1
Combing both; X<0

We have X>1 or X<0
Not sufficient.

Combing both;
Statement 1 tells us that X>0
Statement 2 tells us that X could be less than 0 or greater than 1. Since St 1 tells us that X>0; we can ignore the St2 condition where it says X<0.
Thus, X>0 and X>1. Means X>1

C.
***************************************************

These questions become easier if you knew how to find proper ranges of X in the inequality. Get yourself acquainted with the following trick:
https://gmatclub.com/forum/inequalities-trick-91482.html

This was little cumbersome for me in the beginning. But, with adequate practice and guidance from Bunuel, I am getting hold of it. Yet to become an exponent though. It's really proven fruitful for me.

Good luck!!

Thanks for the post. You're right, I might be getting bogged down in the details. I need to see through the haze...