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# Is P + Q > 1/P + 1/Q? 1. P < Q < 1 2. P*Q < 1

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Is P + Q > 1/P + 1/Q? 1. P < Q < 1 2. P*Q < 1 [#permalink]

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17 Oct 2007, 12:06
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Is P + Q > 1/P + 1/Q?

1. P < Q < 1
2. P*Q < 1

Can someone point out a way to approach these questions?
SVP
Joined: 01 May 2006
Posts: 1798
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17 Oct 2007, 12:47
(C) for me

To me, u should try to work on signs of expressions

P + Q > 1/P + 1/Q
<=> P - 1/P > 1/Q - Q
<=> (P^2 - 1)/P > - (Q^2 - 1)/Q

Cases:
o If P > 1, then P^2 - 1 > 0 and P > 0, making (P^2 - 1)/P > 0
o If 0 < P < 1, then P^2 - 1 < 0 and P > 0, making (P^2 - 1)/P < 0
o If -1 < P < 0, then P^2 - 1 < 0 and P < 0, making (P^2 - 1)/P > 0
o If P < -1, then P^2 - 1 > 0 and P < 0, making (P^2 - 1)/P < 0

It's similar for Q...

So what happens with statments?

Stat1
P < Q < 1

We could easily conclude that under 1, P & Q expressions have 3 intervals on which we flip the sign.

INSUFF.

Stat2
P*Q < 1

implies
o P > 0
o Q > 0

but:
o P > 1 and Q < 1
o P < 1 and Q > 1
o P < 1 and Q < 1

All of these possibilities imply different signs for expressions of P & Q.

INSUFF.

Both (1) and (2)
We have
o 0 < P < 1
o 0 < Q < 1

and as we have,
o If 0 < P < 1, then P^2 - 1 < 0 and P > 0, making (P^2 - 1)/P < 0
o If 0 < Q < 1, then Q^2 - 1 < 0 and Q > 0, making -(Q^2 - 1)/Q > 0

we arrive to
(P^2 - 1)/P < -(Q^2 - 1)/Q

SUFF.
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17 Oct 2007, 15:19
I get E... we dont know if P and Q are positive or not, or integers or fractions??

whats the OA
SVP
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17 Oct 2007, 23:05
fresinha12 wrote:
I get E... we dont know if P and Q are positive or not, or integers or fractions??

whats the OA

In my explanation, the analysis of stat 2 is wrong. We could have P*Q < 1 when P > 0 and Q < 0 for instance.

That means at end, yes, both statments together, we still have P & Q on several intervals that make flipping the signs of Q & P expressions.

(E) it is
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Re: DS Inequalities & Variables - Challenge 20 #19 [#permalink]

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18 Oct 2007, 02:37
bmwhype2 wrote:
Is P + Q > 1/P + 1/Q?

1. P < Q < 1
2. P*Q < 1

Can someone point out a way to approach these questions?

E for me.

Let's rewrite the inequation:
P + Q > 1/P + 1/Q
P + Q > (P + Q) / PQ
PQ > (P + Q)/(P + Q)
this is equal (1) if P is not equal to -Q
In such case PQ > 1

Stat.1 not suff: we dont know if P is not equal to -Q and if PQ>1
Stat.2 not suff: we know that PQ<1 but not if P is not equal to -Q

If we combine both of the stat. we still have no answer, because in stat P coud be 0.5 and Q=-0.5

Thus Ans. is E
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Re: DS Inequalities & Variables - Challenge 20 #19 [#permalink]

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18 Oct 2007, 20:35
bmwhype2 wrote:
Is P + Q > 1/P + 1/Q?

1. P < Q < 1
2. P*Q < 1

Can someone point out a way to approach these questions?

agree E it is

I just tried diff. numbers: negative intengers, positive/negative fractions..
Re: DS Inequalities & Variables - Challenge 20 #19   [#permalink] 18 Oct 2007, 20:35
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