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

Điều kiện $x>-21$.
$\begin{array}{l}
\left[ {{{\log }_3}\left( {{x^2} + 1} \right) - {{\log }_3}(x + 21)} \right]\left( {16 - {2^{x - 1}}} \right) \ge 0 \Leftrightarrow \left\{ \begin{array}{l}
x > - 21\\
\left[ \begin{array}{l}
\left\{ \begin{array}{l}
{\log _3}\left( {{x^2} + 1} \right) - {\log _3}(x + 21) \ge 0\\
16 - {2^{x - 1}} \ge 0
\end{array} \right.\\
\left\{ {\begin{array}{*{20}{l}}
{{{\log }_3}\left( {{x^2} + 1} \right) - {{\log }_3}(x + 21) \le 0}\\
{16 - {2^{x - 1}} \le 0}
\end{array}} \right.
\end{array} \right.
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
x > - 21\\
\left[ \begin{array}{l}
\left\{ \begin{array}{l}
{\log _3}\left( {{x^2} + 1} \right) \ge {\log _3}(x + 21)\\
16 \ge {2^{x - 1}}
\end{array} \right.\\
\left\{ {\begin{array}{*{20}{l}}
{{{\log }_3}\left( {{x^2} + 1} \right) \le {{\log }_3}(x + 21)}\\
{16 \le {2^{x - 1}}}
\end{array}} \right.
\end{array} \right.
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x > - 21\\
\left[ \begin{array}{l}
\left\{ {\begin{array}{*{20}{l}}
{\left( {{x^2} + 1} \right) \ge (x + 21)}\\
{x \le 5}
\end{array}} \right.\\
\left\{ \begin{array}{l}
\left( {{x^2} + 1} \right) \le (x + 21)\\
x \ge 5
\end{array} \right.
\end{array} \right.
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
x > - 21\\
\left[ \begin{array}{l}
\left\{ \begin{array}{l}
\left[ \begin{array}{l}
x \ge 5\\
x \le - 4
\end{array} \right.\\
x \le 5
\end{array} \right.\\
\left\{ \begin{array}{l}
- 4 \le x \le 5\\
x \ge 5
\end{array} \right.
\end{array} \right.
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x > - 21(1)\\
\left[ \begin{array}{l}
\left\{ \begin{array}{l}
\left[ \begin{array}{l}
x \ge 5\\
x \le - 4(2)
\end{array} \right.\\
x \le 5
\end{array} \right.\\
\left\{ \begin{array}{l}
- 4 \le x \le 5\\
x \ge 5
\end{array} \right.(3)
\end{array} \right.
\end{array} \right.
\end{array}$
Từ $(1),(2)$ ta có $\left[\begin{array}{l}x=5 \\ -21<x \leq-4\end{array}\right.$. Do đó số giá trị $x$ nguyên thỏa mãn là $(-4+21)+1=18$.
Từ $(1),(3)$ ta có $x=5$.
Vậy có 18 giá trị nguyên thỏa mãn.