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Understanding Resolve[Exists[...]] type command

Posted 10 years ago
Hi everyone,
I'd be most thankful for any and all inputs into understanding what I describe below. I thank you for your patience and time in hearing out my question.
I frequently use
where inq1,...,inqM and eq1,...,eqN are all real multivariable polynomials in x1,...,xL (and x1,..,xL are all positive and so real).
I use the command above to check for the existence of a solution satisfying all the inequalities and equalities as represented by the expressions inq1,...,inqM>0 and eq1==...=eqN==0. The command works perfectly fine.
But now there is one additional inequality
(Polynomial 1 in {x1,...,xL}) - (Polynomial 2 in {x1,...,xL})*Log[(1+x1)/(1-x1*x2)] > 0
that I'd like to introduce into the command above, call it INQ.
In generally, introducing INQ into the above appears to bottleneck Mathematica. In fact, leaving the conditions inq1,...,inqM>0 and eq1==...=eqM==0 aside,
which is obviously true in my case does not terminate. My first question is, is there any reason as to why the commands Resolve[] and Log[] are not compatible?
I tried Resolve[Exists[...],WorkingPrecision->number (say, 10)] to potentially quicken the calculation.
I am not sure why, but regardless of introducing INQ>0, the original command
updated to
does not converge. For example, the original command takes less than 10 seconds, but the new command simply doesn't converge.
My second question is, isn't WorkingPrecision->number supposed to fasten the calculations? Why is this happening?
Finally, my third question is, is there any way of quickening Resolve[Exists[...]] type of command?

Thank you so much in advance for all your help!!

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