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Inequality

This topic has 1 expert reply and 1 member reply

Inequality

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If 'a' not equal to 0, is 1/a> a/(b^4+3)?

1. a^2=b^2
2. a^2=b^4

Please help.

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Thanks,
Bharat.

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pareekbharat86 wrote:
If 'a' not equal to 0, is 1/a> a/(b^4+3)?

1. a^2=b^2
2. a^2=b^4

Please help.
To find: 1/a > a/(b^4 + 3)

Statement 1:
a^2=b^2
==> either a = b or a = -b
If a = b, then
1/b > b/(b^4 + 3)
b^4 + 3 > b^2

If a = -b
1/-b > -b/(b^4+3)
b/(b^4 + 3) > 1/b
b^2 > b^4 + 3
INSUFFICIENT

Statement 2:
a = b^2 or -a = b^2
if a = b^2
1/a > a/(b^4 + 3)
1/b^2 > b^2/(b^4 + 3)
b^4 + 3 > b^4
3 > 0
YES

if -a = b^2
-a/(b^4+3) > 1/-a
b^2/(b^4+3) > 1/b^2
b^4 > b^4 + 3
0 > 3
NO

Combining...
a = b or a = -b
a = b^2 or -a = b^2
We cannot deduce anything

{E}????

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R A H U L

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Quote:
If 'a' not equal to 0, is 1/a> a/(b^4+3)?

1. a^2=b^2
2. a^2=b^4
There is no obvious way to rephrase this question. We don't want to cross-multiply, since we don't know whether a or b^4 + 3 is negative. So, all we can do is consider the statements.

1) As Rahul said, we want to translate this question as a=b or a=-b. In other words, |a|=|b|.
We can quickly test numbers to prove insufficiency:

If a=2 and b=2, then:
(1/2) > 2/(16 + 3)
yes

If a=-2 and b=2, then:
(1/-2) > -2/(16 + 3)
no
Insufficient

2) If a^2=b^4, then |a|=b^2
Again, test values:

If a=4 and b=2, then:
1/4 > 4/(16 + 3)
yes

If a=-4 and b=2, then:
1/-4 > -4/(16 + 3)
no
Insufficient

1&2) Together, the statements tell us something very significant. The statements both have to be true, and thus can't contradict each other. If a^2=b^2 and a^2=b^4, then a^2 has to be equal to 1 or 0. The question stem tells us that a can't be 0, so a^2 has to be 1.

Thus, a=1 or a=-1, and b=1 or b=-1. Test values again:

If a=1 and b=1, then:
1/1 > 1/(1 +3)
yes

If a=-1 and b=1, then:
1/-1 > -1/(1 + 3)
no
Insufficient

The answer is E.[/i]

_________________


Ceilidh Erickson
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EdM in Mind, Brain, and Education
Harvard Graduate School of Education


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