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Math Expert V
Joined: 02 Sep 2009
Posts: 64068
Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1 00:00

Difficulty:   65% (hard)

Question Stats: 48% (01:52) correct 52% (02:08) wrong based on 63 sessions

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Competition Mode Question

Is $$|a| > a$$?

(1) $$a^2 < a$$

(2) $$\frac{a}{2}>\frac{2}{a}$$

Are You Up For the Challenge: 700 Level Questions

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Quote:
Is |a| > a?
(1) a^2 > a
(2) a/2 > 2/a

Question: Is |a| > a?
But |a| > a only if a < 0

i.e. Question REPHRASED: Is a < 0?

Statement 1: a^2 > a

i.e. a^2 - a > 0
i.e. a*(a-1) > 0

ie.. a > 1 (NO) or a < 0 (YES)

NOT SUFFICIENT

Statement 2: a/2 > 2/a

if a > 0 then a^2 > 4 i.e. a > 2

if a < 0 then a^2 < 4 i.e. -2 < a < 0

NOT SUFFICIENT

COmbining the statements

the common range of value sis -2 < a < 0 (YES)

SUFFICIENT

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Joined: 21 Mar 2018
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Is |a| > a?

When can this be true? Only when a is < 0, that is when a is negative. Eg: a=-2. |-2| > -2? YES

(A) $$a^{2}$$ < a

We know that $$a^{2}$$ is always positive. This statement basically tells us that a is a positive decimal. Note that a can't be negative because a negative number cant be greater than a positive number. Therefore, this is sufficient to answer our original question -> IS a<0 SUFFICIENT

(B) $$a^{2}$$ > 4

This is not going to be sufficient. Eg: Let's take $$a^{2}$$ to be 16. Therefore, a can be +- 4. If a =4 then |4| IS NOT > 4. However, if a = -4, then |-4| IS > -4.

Hence, A

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Quote:
Is |a| > a? (1) a^2 < a (2) a/2 > 2/a

|a|>a, when a<0

(1) sufic

a^2-a<0
a(a-1)<0
0<a<1

(2) insufic

a/2-2/a>0
a^2-4/2a>0
(a-2)(a+2)>0
a>2 or a<-2

Ans (A)
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Q. |a| > a?
In other words, is a < 0 ?

(1) a^2 > a
a*(a-1) > 0
a<0 or a>1
We don't know which one of the following a<0 (negative) or a>1 is applicable.
NOT SUFFICIENT

(2) a/2 > 2/a
(a^2-4)/2a > 0
-2<a<0 or a>2
We don't know which one of the following -2<a<0 (negative) or a>2 is applicable.
NOT SUFFICIENT

(1)+(2)
Superpositioning all possible values of a, we get -2<a<0 (negative) or a>2. HOWEVER, we still don't know which one is applicable: is it -2<a<0 (negative) or a>2 ?
NOT SUFFICIENT

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Is |a| > a?
It is possible only if a < 0

(1) $$a^2 < a$$
$$a^2 < a$$
$$a^2 - a < 0$$
a(a-1) < 0
0 < a < 1
If a = $$\frac{1}{2}$$ $$|\frac{1}{2}| > \frac{1}{2}$$ NO
If a = $$\frac{1}{3}$$ $$|\frac{1}{3}| > \frac{1}{3}$$ NO

SUFFICIENT.

(2) $$\frac{a}{2} > \frac{2}{a}$$
$$\frac{a}{2} > \frac{2}{a}$$
$$\frac{a}{2} - \frac{2}{a}$$ > 0
$$\frac{a^2 - 4}{2a}$$ > 0
$$a^2 - 4 > 0$$
$$a^2 > 4$$
-2 < x < 2
If a = -1 |-1| > -1 YES
If a = 1 |1| > 1 NO

INSUFFICIENT.

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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from given stmnt we need to determine whether
lal>a or say a is -ve
#1
a2<a
a*(a-1)<0
a<0 or a<1
i.e 0>a<1 ( a can be -ve or fraction from 0 to 1)
insufficient
#2
a/2>2/a
a^2-4/(2a)>0
(a+2)(a-2)/2a>0
possible when either a is -1=a<0 or a >2
insufficient
from 1 &2
we get nothing in common
IMO E

Is |a|>a?

(1) a2<a

(2) a/2>2/a
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
The question is asking whether a is negative or not.

1) a^2 < a, this is true for any value between 0 and 1. For any integer value a^2 will be more than a and for values situated in the range -1 <a< 0 also a^2 > a. So a is not negative. Suff.
2) a/2 > 2/a. When a = 3, this is true. Again when a =-1, this is also true. Not suff
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Is |a| > a? (Two cases)
(1) Is a > a ?(Null and void case)
(11) Is -a > a => 2a <0 => a<0
Question becomes Is a<0? Yes?
Or Is a>=0? No?

(1) a^2< a
Let a=1/2 , we have (1/2)^2<1/2
a = +ve here
Let a= -2, we have (-2)^2 <2 (Not possible) there’s no way a can
be negative
.: a>0 (Sufficient)
Or
a <a (Nope!) and -a <a —> 2a>0 —> a>0 (Sufficient)

(2) a/2 >2/a
Let a=8, 8/2 >2/8 a= +ve
Let a=-1, -1/2 > -2 here a=-ve
(Not sufficient)
A like Actuary

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Is |a|>a?

So by asking if the absolute value of a is larger than a what the problem is really asking is if a is negative.

(1) a2<a

This condition tells us that a is not 0, not negative and not positive greater than 1; so a will have to be any value between 0 & 1.
By result this tells as that A IS POSITIVE.
It is sufficient to find out that a is not smaller than its absolute value. Possible solutions A or D.

(2) a/2>2/a

With the 2nd option we have just to test if it is sufficient by itself, so we have that A has to be positive & larger than 2 OR negative and smaller than 1.
As what we seek is to know if it is positive or negative, this question does not provide any more clarity. Not sufficient.

So A is the right answer.

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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Question stem: |a|>a?
this is same as is a < 0?

statement 1:
$$a2<a$$; $$a(a-1)<0$$
0<a<1.
this is sufficient as a is always positive.

statement 2:
$$\frac{a}{2}>\frac{2}{a}$$
$$\frac{a^2-4}{2a}>0$$
$$\frac{(a-2)(a+2)}{2a}>0$$
a>2 or -2<a<0
a can be positive or negative
not sufficient

Ans: A
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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for |a|>a.......a must be < 0

(1) a^2<a.....this will be satisfied only if 0<a<1.....sufficient to that |a| not < a.........SUFFICIENT

(2) a/2>2/a.......a>2 and -2<a<0....this is insufficient

OA:A
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Is |a|>a?

(1) a2<a

(2) a/2>2/a

stem:
|a| gives positive val,a has to be negative
1) cannot find sign
2) proves a is negative

so b)
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Is |a|>a|a|>a?

ST1: (1) a2<a --------> Solving we get 0<a<1 ------------> a>0 therefore mod(a) = a .... the question will be false .... hence sufficient

(2) a/2>2/a --------------> Solving we get -2<a<0 OR a>2 ------------> If -2<a<0 then mod(a) > 0 and a is negative Hencethe question will be TRUE ... IF a>2 ... the question will FALSE .... Hence the statement is INSUFICIENT ..

ANS B
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Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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1
Is $$|a|>a?$$
--> $$a < 0$$???

(Statement1): $$a^{2} < a$$
$$a (a-1) <0$$
$$0 < a < 1$$
Always NO
Sufficient

(Statement2): $$\frac{a}{2} >\frac{2}{a}$$
$$\frac{(a^{2}-4 )}{2a} >0$$
$$\frac{(a-2)(a+2)}{2a} > 0$$
--> $$-2 <a < 0$$
--> $$a >2$$
Insufficient

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Posts: 2
Re: Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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The general question can be rephrased into "Is a<0 ?"
(1) a^2<a => a(a-1)<0 => 0<a<1 The answer is NO (sufficient)
(2) a/2>2/a => a/2-2/a>0 => (a^2-4)/2a>0 => (a-2)(a+2)/2a>0 => -2<a<0 The answer is YES (sufficient)
Hence: D

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Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  [#permalink]

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Sir, please with your statement ( 2) cross multiplication of inequalities is not allowed ,we don’t even know the sign of (a) GMATinsight wrote:
Quote:
Is |a| > a?
(1) a^2 > a
(2) a/2 > 2/a

Question: Is |a| > a?
But |a| > a only if a < 0

i.e. Question REPHRASED: Is a < 0?

Statement 1: a^2 > a

i.e. a^2 - a > 0
i.e. a*(a-1) > 0

ie.. a > 1 (NO) or a < 0 (YES)

NOT SUFFICIENT

Statement 2: a/2 > 2/a

if a > 0 then a^2 > 4 i.e. a > 2

if a < 0 then a^2 < 4 i.e. -2 < a < 0

NOT SUFFICIENT

COmbining the statements

the common range of value sis -2 < a < 0 (YES)

SUFFICIENT

Posted from my mobile device
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Veritas valet et vincet Is |a| > a? (1) a^2 > a (2) a/2 > 2/a   [#permalink] 05 Apr 2020, 23:43

# Is |a| > a? (1) a^2 > a (2) a/2 > 2/a  