14. (15 points) Solve the equation: $[x] \times\{x\}+x=2\{x\}+9$, where $[x]$ represents the integer part of $x$, and $\{x\}$ represents the fractional part of $x$. For example, $[3.14]=3,\{3.14\}=0.14$. (List all solutions)
【Solution】Solution: According to the analysis, let the integer part of $x$ be $a, a \geqslant 1 ;$ the decimal part of $x$ be $b, 0 \leqslant b<1$, according to the problem: $a b+a+b=2 b+9$,

Rearranging gives: $(a-1)(b+1)=8, \because 1 \leqslant b+1<2, \therefore 4<a-1 \leqslant 8$, and $a-1$ is an integer.
(1) When $a-1=8$, i.e., $a=9, b=0, x=9$;
(2) When $a-1=7, a=8, b=\frac{1}{7}, x=8 \frac{1}{7}$;
(3) When $a-1=6$, i.e., $a=7, b=\frac{1}{3}, x=7 \frac{1}{3}$;
(4) When $a-1=5$, i.e., $a=6, b=\frac{3}{5}, x=6 \frac{3}{5}$.

In summary, the solutions to the equation are: $x=9 ; x=8 \frac{1}{7} ; x=7 \frac{1}{3} ; x=6 \frac{3}{5}$.
Therefore, the answers are: $x=9 ; x=8 \frac{1}{7} ; x=7 \frac{1}{3} ; x=6 \frac{3}{5}$.