<s>[INST] Inequality involving conjugate numerator/denominator pairs Question is to solve:

$$\frac{(x-2)(x-4)(x-7)}{(x+2)(x+4)(x+7)} > 1$$

I thought I could negate terms to make them equal (i.e. $-(x-2)$), but that does not happen. I could subtract $1$ from each side but that would be $(very)^2$ heavy work. Is there any simple way of doing this ?
 [/INST] Another way would be to multiply throughout by $(x+2)^2(x+4)^2(x+7)^2$, which is positive ($x =-2,-4,-7$ are not allowable anyway), to get 
$$(x^2-4)(x^2-16)(x^2-49)>(x+2)^2(x+4)^2(x+7)^2$$
Collecting terms in the LHS and factoring, this is equivalent to 
$$-2(x+7)(x+4)(x+2)(13x^2+56)>0$$
and now checking intervals where LHS can change signs gives you the answer $x \in (-\infty, -7) \cup (-4,-2)$. 
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