5. If the hyperbola
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)
$$

has a point with the abscissa $\frac{3}{2} a$ whose distance to the right focus is greater than its distance to the left directrix, then the range of the eccentricity $e$ of the hyperbola is $\qquad$.
5. $(2,+\infty)$.

Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola, respectively. Then
$$
\begin{array}{l}
\left|P F_{2}\right|>\frac{3 a}{2}+\frac{a^{2}}{c} \\
\Rightarrow e\left(\frac{3 a}{2}-\frac{a}{c}\right)^{2}>\frac{3 a}{2}+\frac{a^{2}}{c} \\
\Rightarrow e\left(\frac{3}{2}-\frac{1}{e}\right)>\frac{3}{2}+\frac{1}{e} \\
\Rightarrow e>2
\end{array}
$$