Example 13. Given that the characteristic of $\lg x^{\frac{3}{2}}$ is the solution to the inequality
$$
a^{2}-8 a+15<0
$$

and the mantissa is the sum of the solutions to the equation
$$
\frac{1}{4}\left|y+\frac{1}{2}\right|+\frac{6}{121} \sqrt{z-1}=0
$$

find the value.
Solution: Let $\lg x^{\frac{3}{2}}=p+q$
(where $p$ is the characteristic, and $q$ is the mantissa). Solving the inequality $a^{2}-8 a+15<0$, we get
$$
3<a<5 \text {. }
$$
Since the characteristic of a logarithm of a number must be an integer,
$$
\therefore \quad p=a=4 \text {. }
$$

From equation (2), we get
$$
y=-\frac{1}{2}, \quad \text { and } \quad z=1 .
$$

According to the problem, we have
$$
q=y+z=-\frac{1}{2}+1=\frac{1}{2} .
$$

Substituting $p$ and $q$ into the formula, we get
$$
\begin{aligned}
& \lg x^{\frac{4}{2}}=1+\frac{1}{2}=\frac{9}{2}, \\
\therefore \quad & \lg x=\frac{9}{2} \div \frac{3}{2}=3 .
\end{aligned}
$$

Therefore, $x=1000$.