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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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Updated on: 20 Nov 2017, 07:14
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45% (medium)

Question Stats:

58% (00:45) correct 42% (00:37) wrong based on 12 sessions

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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. $$(-∞, − \frac{1}{5} ), (\frac{1}{4} , ∞)$$

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Originally posted by rahul16singh28 on 19 Nov 2017, 23:42.
Last edited by chetan2u on 20 Nov 2017, 07:14, edited 1 time in total.
formatted the Q
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Joined: 02 Aug 2009
Posts: 5779
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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20 Nov 2017, 07:06
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Expert's post
[quote="rahul16singh28"]If r and s are variables such that 1 /r^2 − 1 /s^2 > −9 and 2 /r^2 – 1/s^2 −9[/m]....(i)

$$\frac{2}{r^2} – \frac{1}{s^2} -7$$

$$\frac{1}{s^2} – \frac{2}{r^2}+\frac{1}{r^2} − \frac{1}{s^2} > -7-9.....................\frac{-1}{r^2}>-16........\frac{1}{r^2}\frac{1}{16}$$
so $$r>\frac{1}{4}$$or $$r −9$$....(i)
multiply (i) by 2
$$\frac{2}{r^2} – \frac{2}{s^2} > -18$$

$$\frac{2}{r^2} – \frac{1}{s^2} -7$$

$$\frac{1}{s^2} – \frac{2}{r^2}+\frac{2}{r^2} − \frac{2}{s^2} > -7-18.....................-\frac{1}{s^2}>-25........\frac{1}{r^2}\frac{1}{25}$$
so $$r>\frac{1}{5}$$or $$r\frac{1}{4}$$ 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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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, 07:06
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