Inequality question (when fraction < zero)

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Homework Statement

Solve for t:

[-2(t²+1) / 9(t²-1)] < 0

The Attempt at a Solution

I know that the answer is -inf<t<-1 and 1<t<inf, but how do I show the calculation to get that answer? When I tried, I narrowed it down to t<root-1, but that's not possible (without complex numbers) and doesn't match the answer?

 

[LCKurtz]

Try factoring the denominator and analyzing the signs of the factors.

 

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So 9(t-1)(t+1), but what do I deduce from that?

 

[LCKurtz]

So 9(t-1)(t+1), but what do I deduce from that?

The sign of a fraction is determined by the signs of its factors. You have a - in front of the fraction and the t² + 1is always positive. The only places where the denominator changes signs are at 1 and -1. So figure out the signs everywhere else. Wherever you have an even number of negative signs your fraction is negative and an odd number makes it positive.

 

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Ah OK. So it's more by inspection. I would factor as we have done, and then I'd choose for example -2, 0 and 2 and determine the sign giving me the interval values around 1 and -1, correct?

 

[LCKurtz]

That's the idea. Since those factors can only change sign at their roots, if you check the values at a point on each subinterval you will know the signs on the intervals.

 

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Awesome. Thanks. Now I can sketch this parametric.