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$\begingroup$

$$\frac{x+6}{x-6}\left(\frac{x-4}{x+4}\right)^2+\frac{x-6}{x+6}\left(\frac{x+9}{x-9}\right)^2<\frac{2x^2+72}{x^2-36}$$

I'm quite lost here, can't spot anything to start working from.

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    I would try to rewrite the inequality using that $$\frac{x+a}{x-a}=\frac{x-a}{x-a}+\frac{2a}{x-a}=1+\frac{2a}{x-a}$$ – 2017-02-12
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    What have you tried so far for this inequality? Please provide the method, so we can assist. – 2017-02-12
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    Not much, actually, tried using intervals to determine maybe one side would be positive, then the next one negative, but no luck, sadly. – 2017-02-12
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    Yeah, then if i expand the LHS similar way i can move both to one side and factor out common multiples, but still struggling, any more help? – 2017-02-12
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    There is a saying: "If you are going thru hell - keep going". – 2017-02-12

1 Answers 1

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$$\frac{x+6}{x-6}\left(\frac{x-4}{x+4}\right)^2+\frac{x-6}{x+6}\left(\frac{x+9}{x-9}\right)^2=\frac{2(x^2+36)(x^4-71x^2+1296)}{(x-9)^2(x+4)^2(x-6)(x+6)}$$

Thus, we need to solve $$\frac{x^4-71x^2+1296}{(x-9)^2(x+4)^2(x-6)(x+6)}<\frac{1}{(x-6)(x+6)}$$ or $$\frac{x(5x^2-12x-180)}{(x-9)^2(x+4)^2(x-6)(x+6)}<0,$$ which gives the answer:

$x<-6$ or $\frac{6-6\sqrt{26}}{5}

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    A question - how do you calculate the first expression quickly? Because reducing to a common denominator and multiplying everything seems a bit too much work, maybe there is a fast trick to spot something? – 2017-02-12
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    @John Robbers For example: $(x+6)^2(x-4)^2(x-9)^2+(x-6)^2(x+4)^2(x+9)^2=(x^3-7x^2-42x+216)^2+(x^3+7x^2-42x-216)^2=(2x^3-84x)^2-2((x^3-42x)^2-(7x^2-216)^2)=2(x^3-42x)^2+2(7x^2-216)^2...$ – 2017-02-12