Example 5 Solve the system of inequalities $\left\{\begin{array}{l}x+2<3+2 x, \\ 4 x-3<3 x-1, \\ 8+5 x \geqslant 6 x+7 .\end{array}\right.$
The original system of inequalities can be transformed into:
$$
\left\{\begin{array}{l}
x>-1, \\
x<2, \\
x \leqslant 1 .
\end{array}\right.
$$

Therefore, the solution set of the system of inequalities is $(-1,1]$.