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This question already has an answer here:

I want to find the solutions for

$$(4k+1)(2k)!\ge 10^6$$

I tried the expressions

Solve[(4k+1) Factorial[2k]>=10^6 && k\[Element]Integers && k>0,k]

Solve[(4x+1) Gamma[2x+1]>=10^6 && x>0,x,Reals]

and the same two using also NSolve and Reduce instead of Solve and for the first expression (using factorials) the system hang on (more than 5 minutes of evaluation!) and for the second the software says that this cannot be solved by the methods used. Im doing something wrong?

How I can solve this inequality?

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marked as duplicate by Artes, Community Jun 19 '17 at 12:39

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ MinValue[{k, (4 k + 1) Factorial[2 k] >= 10^6, k \[Element] Integers, 0 < k < 20}, k] $\endgroup$ – Bob Hanlon Jun 19 '17 at 16:13
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You can use FindRoot to find the value of $k$ for which the inequality is true:

FindRoot[(4 k + 1) Factorial[2 k] == 10^6, {k, 10}]

{k -> 4.08331}

As your function is monotonic in $k$, you then know $k\geq 5$. Not sure what you should do for non-monotonic equations.

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