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.

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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