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If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1

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If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1 [#permalink]

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If r and s are variables such that \(\frac{1}{r^2} − \frac{1}{s^2} > −9\) and \(\frac{2}{r^2} – \frac{1}{s^2} < 7\), which of the segments best represents the overlap zone for values of r and s?

1. \((-∞, − \frac{1}{4} ), (\frac{1}{4} , ∞)\)
2. \((− \frac{1}{4} , \frac{1}{4} )\)
3. \((-∞, − \frac{1}{5} ), (\frac{1}{5} , ∞)\)
4. \((− \frac{1}{4} , \frac{1}{4} )\)
5. \((-∞, − \frac{1}{5} ), (\frac{1}{4} , ∞)\)
[Reveal] Spoiler: OA

Last edited by chetan2u on 20 Nov 2017, 06:14, edited 1 time in total.
formatted the Q

Kudos [?]: 17 [0], given: 333

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Kudos [?]: 6130 [2], given: 121

Re: If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1 [#permalink]

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New post 20 Nov 2017, 06:06
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rahul16singh28 wrote:
If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1/s^2 < 7, which of the segments best represents the overlap zone for values of r and s?

1. (-∞, − 1/ 4 ), (1/4 , ∞)
2. (− 1/4 , 1/4 )
3. (-∞, − 1/5 ), (1/5 , ∞)
4. (- 1/5 , 1/5 )
5. (-∞, - 1/5 ), (1/4 , ∞)



We have TWO variables and two inequalities..
solve for one variable and you will get your answer..


\(\frac{1}{r^2} − \frac{1}{s^2} > −9\)....(i)

\(\frac{2}{r^2} – \frac{1}{s^2} < 7\)....(ii)
multiply (ii) by -
\(\frac{1}{s^2} – \frac{2}{r^2} > -7\)

now add (i) to it..
\(\frac{1}{s^2} – \frac{2}{r^2}+\frac{1}{r^2} − \frac{1}{s^2} > -7-9.....................\frac{-1}{r^2}>-16........\frac{1}{r^2}<16....r^2>\frac{1}{16}\)
so \(r>\frac{1}{4}\)or \(r<-\frac{1}{4}\)


Now for second variable

\(\frac{1}{r^2} − \frac{1}{s^2} > −9\)....(i)
multiply (i) by 2
\(\frac{2}{r^2} – \frac{2}{s^2} > -18\)

\(\frac{2}{r^2} – \frac{1}{s^2} < 7\)....(ii)
multiply (ii) by -
\(\frac{1}{s^2} – \frac{2}{r^2} > -7\)

now add (i) to it..
\(\frac{1}{s^2} – \frac{2}{r^2}+\frac{2}{r^2} − \frac{2}{s^2} > -7-18.....................-\frac{1}{s^2}>-25........\frac{1}{r^2}<25....r^2>\frac{1}{25}\)
so \(r>\frac{1}{5}\)or \(r<-\frac{1}{5}\)

so our answer in NEGATIVE : \(r<-\frac{1}{4}\) and \(s<-\frac{1}{5}\)...... so overlap will be from LOWER of -1/4 and -1/5 till infinity so (-∞, − 1/ 4 )
and our answer in POSITIVE : \(r>\frac{1}{4}\) and \(s>\frac{1}{5}\)...... so overlap will be from HIGHER of 1/4 and 1/5 till infinity so \((\frac{1}{4},∞)\)

ans A

NOTE - rahul16singh28, pl use proper mathematical formula, it will be much easier to comprehend. Doing it 4 u.
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 6130 [2], given: 121

Re: If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1   [#permalink] 20 Nov 2017, 06:06
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