Do not be so modest !
It is true for any $x > x_0$
$$x_0=3.6956333556853889395248451976513568330892774031006\cdots$$
Using series for the logarithm of the lhs gives
$$\log(\text{lhs})=\frac{80-11 \pi ^2}{11520 \pi ^2 x^4}+\frac{13}{13440
x^5}+\frac{11}{2073600 x^6}+O\left(\frac{1}{x^7}\right)$$
The positive root of the above quadratic equation is
$$\frac{\pi \left(234 \pi +\sqrt{\frac{1}{5} \left(279709 \pi
^2-43120\right)}\right)}{42 \left(11 \pi ^2-80\right)}=3.85540$$
Using the series expansion to $O\left(\frac{1}{x^9}\right)$ gives $3.67853$.
Using the series expansion to $O\left(\frac{1}{x^{20}}\right)$ gives $3.69561$.
Edit
If you use the extended Ramanujan approximation as given in the edit of my answer here, the difference with Stirling approximation is
$$\frac{8521}{34406400 x^8}\left(1-\frac{1}{2 x}+\frac{1}{8
x^2}+O\left(\frac{1}{x^3}\right) \right)$$
This does not affect the above results in any manner.
Update
In order to be fully consistent and not use at any point Stirling approximation, I repeated the calculation using the extended Ramanujan approximation
$$n!\sim \sqrt{\pi}\left(\frac ne\right)^n\sqrt [6]{8 n^3+4
n^2+n+\frac{1}{30} \left(1+\sum _{i=0}^m \frac{a_i} {n^{i}}\right)}$$ the first $a_i$ being
$$\left\{0,-\frac{11}{8},\frac{79}{112},\frac{3539}{6720},-\frac{9511}{13440},-\frac{30
153}{71680},\frac{233934691}{212889600},\frac{3595113569}{5960908800},\cdots\right\}$$
For consistency, the argument of the first logarithm has been replaced by
$$8 n^3+4
n^2+n+\frac{1}{30} \left(1+\sum _{i=0}^m \frac{a_i} {n^{i}}\right)$$
Now, the lhs is grater than $1$ for any $n$ ! In fact, the expansion of the lhs stabilizes very quickly to
$$1+\frac{1}{144 \pi ^2 n^4} \large A$$ where
$${\large A}=1+\frac{\pi ^2}{1800 n^2}+\frac{30+\pi ^2}{360 \pi ^2
n^3}+\frac{\left(\frac{175}{\pi ^4}-8\right) \pi ^2}{25200
n^4}+\frac{\left(7-\frac{5400}{\pi ^2}\right) \pi ^2}{6804000
n^5}+\frac{29400+980 \pi ^2+49 \pi ^4+1200 \pi ^6}{4233600 \pi
^4 n^6}+O\left(\frac{1}{n^7}\right)$$
Why is that ?
$$\Delta=(n!)_{\text{approximate}}-\log(\Gamma(n+1))=\frac{8521}{34406400 n^8}\large B$$
where
$${\large B}=1-\frac{1}{2 n}+\frac{1}{8 n^2}+\frac{201648777301}{27635648040
n^3}+\frac{52815059449}{255098289600
n^4}+O\left(\frac{1}{n^5}\right)$$
$\forall n >0$, $\Delta >0$ and the inequality should not hold while with expansion of $\Gamma(n+1)$ is does !
