I2.4 $[a]$ represents the largest integer not greater than $a$. For example, $\left[2 \frac{1}{3}\right]=2$. Given that the sum of the roots of the equation $[3 x+R]=2 x+\frac{3}{2}$ is $S$, find the value of $S$.
$\begin{array}{l}{[3 x+1]=2 x+\frac{3}{2} \Rightarrow 3 x+1=2 x+\frac{3}{2}+a, \text { where } 0 \leq a<1} \\ a=x-\frac{1}{2} \Rightarrow 0 \leq x-\frac{1}{2}<1 \Rightarrow 2.5 \leq 2 x+\frac{3}{2}<4.5 \\ \because 2 x+\frac{3}{2} \text { is an integer } \therefore 2 x+\frac{3}{2}=4 \text { or } 3 \\ x=0.75 \text { or } 1.25 \\ S=0.75+1.25=2\end{array}$