5. Given the system of inequalities about $x$
$$
\left\{\begin{array}{l}
\frac{2 x+5}{3}-t>5 \\
\frac{x+3}{2}-t>x
\end{array}\right.
$$

has exactly three integer solutions. Then the range of $t$ is ( ).
(A) $-\frac{12}{7} \leqslant t<-\frac{8}{7}$
(B) $-\frac{12}{7} \leqslant t<-\frac{3}{2}$
(C) $-\frac{3}{2} \leqslant t<-\frac{4}{3}$
(D) $-\frac{4}{3} \leqslant t<-\frac{8}{7}$
5. C.

Solving the system of inequalities yields $\frac{3}{2} t+5<x<3-2 t$.
Since the system of inequalities has exactly three integer solutions, we have
$$
\begin{array}{l}
2<(3-2 t)-\left(\frac{3}{2} t+5\right) \leqslant 4 \\
\Rightarrow-\frac{12}{7} \leqslant t<-\frac{8}{7} .
\end{array}
$$

When $-\frac{12}{7} \leqslant t<-\frac{3}{2}$,
$$
\frac{17}{7} \leqslant \frac{3}{2} t+5<\frac{11}{4}, 6<3-2 t \leqslant \frac{45}{7} \text {. }
$$

In this case, the integer solutions to the original system of inequalities are $3, 4, 5, 6$, which contradicts the problem statement.
When $-\frac{3}{2} \leqslant t<-\frac{4}{3}$,
$$
\frac{11}{4} \leqslant \frac{3}{2} t+5<3, \frac{17}{3}<3-2 t \leqslant 6 \text {. }
$$

In this case, the integer solutions to the original system of inequalities are $3, 4, 5$, which satisfies the problem statement.
When $-\frac{4}{3} \leqslant t<-\frac{8}{7}$,
$$
3 \leqslant \frac{3}{2} t+5<\frac{23}{7}, \frac{37}{7}<3-2 t \leqslant \frac{17}{3} \text {. }
$$

In this case, the integer solutions to the original system of inequalities are $4, 5$, which contradicts the problem statement.
In summary, the range of values for $t$ is $-\frac{3}{2} \leqslant t<-\frac{4}{3}$.