Có bao nhiêu số nguyên $x$ thỏa mãn $\left[ {{\log }_{3}}\left(...

Điều kiện $x>-21$.
$\begin{aligned}
& \left[ {{\log }_{3}}\left( {{x}^{2}}+1 \right)-{{\log }_{3}}\left( x+21 \right) \right]\left( 16-{{2}^{x-1}} \right)\ge 0\Leftrightarrow \left\{ \begin{aligned}
& x>-21 \\
& \left[ \begin{aligned}
& \left\{ \begin{aligned}
& lo{{g}_{3}}\left( {{x}^{2}}+1 \right)-{{\log }_{3}}\left( x+21 \right)\ge 0 \\
& 16-{{2}^{x-1}}\ge 0 \\
\end{aligned} \right. \\
& \left\{ \begin{aligned}
& lo{{g}_{3}}\left( {{x}^{2}}+1 \right)-{{\log }_{3}}\left( x+21 \right)\le 0 \\
& 16-{{2}^{x-1}}\le 0 \\
\end{aligned} \right. \\
\end{aligned} \right. \\
\end{aligned} \right. \\
& \Leftrightarrow \left\{ \begin{aligned}
& x>-21 \\
& \left[ \begin{aligned}
& \left\{ \begin{aligned}
& lo{{g}_{3}}\left( {{x}^{2}}+1 \right)\ge {{\log }_{3}}\left( x+21 \right) \\
& 16\ge {{2}^{x-1}} \\
\end{aligned} \right. \\
& \left\{ \begin{aligned}
& lo{{g}_{3}}\left( {{x}^{2}}+1 \right)\le {{\log }_{3}}\left( x+21 \right) \\
& 16\le {{2}^{x-1}} \\
\end{aligned} \right. \\
\end{aligned} \right. \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& x>-21 \\
& \left[ \begin{aligned}
& \left\{ \begin{aligned}
& \left( {{x}^{2}}+1 \right)\ge \left( x+21 \right) \\
& x\le 5 \\
\end{aligned} \right. \\
& \left\{ \begin{aligned}
& \left( {{x}^{2}}+1 \right)\le \left( x+21 \right) \\
& x\ge 5 \\
\end{aligned} \right. \\
\end{aligned} \right. \\
\end{aligned} \right. \\
& \Leftrightarrow \left\{ \begin{aligned}
& x>-21 \\
& \left[ \begin{aligned}
& \left\{ \begin{aligned}
& \left[ \begin{aligned}
& x\ge 5 \\
& x\le -4 \\
\end{aligned} \right. \\
& x\le 5 \\
\end{aligned} \right. \\
& \left\{ \begin{aligned}
& -4\le x\le 5 \\
& x\ge 5 \\
\end{aligned} \right. \\
\end{aligned} \right. \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& x>-21\ \left( 1 \right) \\
& \left[ \begin{aligned}
& \left\{ \begin{aligned}
& \left[ \begin{aligned}
& x\ge 5 \\
& x\le -4 \\
\end{aligned} \right. \\
& x\le 5 \\
\end{aligned} \right.\ \left( 2 \right) \\
& \left\{ \begin{aligned}
& -4\le x\le 5 \\
& x\ge 5 \\
\end{aligned} \right.\ \left( 3 \right) \\
\end{aligned} \right. \\
\end{aligned} \right. \\
\end{aligned}$
Từ $\left( 1 \right),\left( 2 \right)$ ta có $\left[ \begin{aligned}
& x=5 \\
& -21<x\le -4 \\
\end{aligned} \right. $. Do đó số giá trị $ x $ nguyên thỏa mãn là $ \left( -4+21 \right)+1=18$.
Từ $\left( 1 \right),\left( 3 \right)$ ta có $x=5$.
Vậy có $18$ giá trị nguyên thỏa mãn.