Is a < 0 ? (1) a^3 < a^2 + 2a (2) a^2 > a^3

From F.S 1, add 1 on both sides : \(a^2+2a+1>a^3+1 \to (a+1)^2-(a+1)(a^2+1-a)>0 \to (a+1)[(a+1)-(a^2+1-a)]>0 \to (a+1)(2a-a^2)>0\)
Thus, we get a(a+1)(a-2)<0. Either 0<a<2 OR a<-1. Insufficient.

From F.S 2, we can divide by\(a^2\) on both sides and we get a<1. Insufficient.

Taking both together, we know that a<1. Thus, a could be 0<a<1 OR a<-1. Insufficient.

E.
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STMNT 1:

\(a^3<a^2+2a\)
\(a^3-a^2-2a < 0\)
a(a+1)(a-2)<0
when a = -2 expression above is= -8 which is < 0
when a = 1 expression above is = -2 which is <0
hence a can be -ve /+ve insufficient

STMNT 2:
\(a^2\)\(> a^3\)
\(a^2(a-1)<0\)
clearly this satisfies for a<1
hence a can be -ve /+ve insufficient

combining both
still both statement satisfies for a = 0.5 and a =-2
hence insufficient

hence E

st(1) , add 1 to both sides. then a= -1 and a= 1/2. you will have a double case.
st(2), a can be any positive fraction here and a can be any negative integer too such as a= 1/2 and a = -2 . a double case too.
so both statement depict the same answers, which are both a double case.
so Answer is (E)

Even though you have the correct answer, I am sorry but it is not what it means(the highlighted part). It is not an equality, rather an in-equality.Please refer through the above posts, for the correct method.
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The answer is E, but the ranges are not correct.

Is a < 0 ?

(1) a^3 < a^2 + 2a --> \((a+1)a(a-2)<0\) --> \(a<-1\) or \(0<a<2\). Not sufficient.

(2) a^2 > a^3 --> \(a^2(1-a)>0\) --> \(a<0\) or \(0<a<1\). Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is \(a<-1\) or \(0<a<1\). Not sufficient.

Answer: E.


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Bunuel why can't we factorize a^2>a^3 as 1>a or a<1.

a<1 implies that a can be 0. But a=0 does not satisfy a^2>a^3, so the correct ranges for which this inequality holds true is a<0 or 0<a<1 (the same range as you have excluding 0).

Hope it helps.
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Hi,

There's a concept related to inequalities that I fail to understand. Could you please tell me what I'm doing wrong?

For statement 2 we have "a^2 > a^3". Depending on how I solve this, I'm getting two complete different ranges, both of which are incorrect.

First case:

a^2 - a^3 > 0
a^2*(1-a)>0------> this gives me the critical points 0 and 1.

Hence I have 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.

As the inequality has the ">0" sign I took only the 1st and 3rd range and got "a<0 or a>1".

Second case:

0>a^3 - a^2
0>a^2*(a-1)------> this also gives me the critical points 0 and 1.

Hence I have the same 3 ranges: (1st) a<0, (2nd) 0<a<1, and (3rd) a>1.

However, this time we have "0>" sign, so I took only the second range: 0<a<1.

In any case, both are false. What's odd is that I used the same method to find the range for statement 1 but I got the correct answer.

Thanks a lot for your help.

Aurèle

0 is not a critical point. The squared terms (basically even powers) must be ignored because they cannot be negative and hence doesn't affect the sign.
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Great, thanks for the reply. Just to be sure that I understand the concept fully, could you tell me please if I'm solving the following equation correctly (from Manhattan books):

\(x^6 - x^7 > x^5 - x^6\)

I factor the equation and get:

\(x^5*(x-1)^2 < 0\)

Here, If I understand your explanation well, \(x^5\) is raised to an odd power; hence, it should be considered, as it can yield a negative or positive result. Conversely, \((x-1)^2\) is raised to an even power; thus, we ignore it.

The critical point is then : \(0\)

Because we have the \(<0\) sign, we'd get \(x<0\). Would this be correct?

Is a<0?
(1) Insufficient. If a=-5, then -125 < 25-10 --->Yes
If a=1/2, then 1/8 < 1/4+1 --->No

(2) Insufficient. If a=-5, then 25>-125 --->Yes
If a=1/2, then 1/4>1/8 --->No
(1)+(2) Insufficient. We can use the same numbers and we get two different answers

Hello! I was looking through these posts, answering questions, and I came across this one:

Is a<0?

(1) a3<a2+2a
(2) a2>a3

from: forum/is-a-157421.html

In the topic above, the answer is that both statements together are insufficient to answer the problem but surely the only case in which \(a^{2}\) is greater than\(a^{3}\) is if a is negative?
My answer is that both statements alone can answer the question, as the only way \(a^{3}\) is smaller than what it is compared to is if a is negative. What am I overlooking?

Thank you.

Hi cmcmcm,

Always be very careful on how you are finding out the range of an inequality. Let me help you out with finding the range of a for both the inequalities.

Statement-I
\(a^3<a^2+2a\) can be simplified to \(a(a + 1)(a - 2) < 0\). Using the wavy line method to find the range of \(a\) with the zero points being 2, 0 and -1.



We can see from the wavy line diagram that the inequality is negative in the range where \(a < -1\) or \(0 < a < 2\). Hence, you can't say for sure if \(a < 0\) using statement-I alone

Statement-II
\(a^2>a^3\) can be simplified to \(a^2(a - 1) < 0\). Since \(a^2\) is always non-negative, for \(a^2(a - 1) < 0\), \((a - 1) < 0\) i.e. \(a < 1\).
So \(a < 0\) or \(a > 0\). Hence using statement-II alone you cant' say for sure if \(a < 0\).

Combining statement-I & II
Combining statements-I & II will give us the range as \(a < -1\) or \(0 <a <1\). Hence, it is not sufficient to tell if \(a < 0\). Therefore the answer is E.

You can read more about the Wavy line method here.

Hope it's clear .Let me know if you have any doubt in any part of the explanation.

Regards
Harsh
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Hi cmcmcm,

The prompt does NOT state that "A" has to be an integer, so you have to consider the possibility that it's NOT an integer (meaning "A" could be a fraction).

While that level of 'thoroughness' isn't going to be required on that many DS questions, Test Takers who score at the higher levels in the Quant section are more likely to see questions in which fractional answers have to be considered.

GMAT assassins aren't born, they're made,
Rich

E
(1) take a = -1, -2 ---- If a=-1 then eqn gives -1<-1 which is not true and if a=-2 then eqn gives -8<0 which is true ....thats y insuff
(2) a^2>a^3 implies a can be -ve or 0<a<1 thus insuff....
Combined also is insuff because of above reasons
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(1) a^3 < a^2 + 2a - --- a=-1 then -1<-1 not true ; a=-2 then -8<0 true insuff
(2) a^2 > a^3 then a<0 or 0<a<1 insuff

If we combine both we cannot answer bc of above reasons
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Target question: Is a < 0 ?

Statement 1: a³ < a² + 2a
Subtract a² and 2a from both sides to get: a³ - a² - 2a < 0
Factor: a(a² - a - 2) < 0
Factor more: a(a - 2)(a + 1) < 0
There are several values of a that satisfy this inequality. Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a² > a³
There are several values of a that satisfy this inequality. Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
There are several values of a that satisfy BOTH statements Here are two:
Case a: a = 0.5. In this case, a > 0
Case b: a = -10. In this case, a < 0
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

Is a < 0 ?
(1) a^3 < a^2 + 2a
\(a^3-a^2-2a<0......a( a^2-a-2)<0......a(a-2)(a+1)<0\)
So a <-1 will give ans as YES..
a=0 or 1 will also be true and ans will be NO
Insufficient

(2) a^2 > a^3
a^2-a^3>0......\(a^2(1-a)>0\)..
So 1-a>0..a<1...
So a can be -1 or 0 or 0.5
Insuff..

Combined
Again a as 0, 0.5 or -3 etc still remains..
Insufficient

E
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Is a < 0?

Statement 1: a^3 < a^2 +2a --> a(a+1)(a-2) < 0 Therefore, if this equals zero, then a can be -1, 0, or 2.
Try a = 1, the expression is negative. We can fill in the number line with signs because signs will switch back and forth.
Therefore <----(-1)++++(0)-----(2)+++++>. Statement 1 is correct when a is less than -1 or between 0 and 2. Insufficient.

Statement 2: a^2 > a^3 --> a(a)(a-1) > 0 Therefore, if this equals zero, then a can be 0 or 1.
Try a = 2, the expression is positive. We can fill in the number line with signs because signs will switch back and forth.
Therefore <++++(0)------(1)+++++>. Statement 2 is correct when a is less than 0 or greater than 1. Insufficient.

Combined:
From statement 1, a can be less than -1 or between 0 and 2.
From statement 2, a can be less than 0 or greater than 1.
Therefore, a must be less than -1 or between 1 and 2.
Is a < 0? Yes and no. Insufficient.