Answer
$S$ is linearly independent.
Work Step by Step
Assume the combination
$$a(\frac{3}{4},\frac{5}{2},\frac{3}{2})+b(3,4,\frac{7}{2})+c(-\frac{3}{2},6,2)=(0,0,0).$$
For simplicity, we multiply both sides of the above equation by $4$ and get
$$a(3,10,6)+b(12,16,14)+c(-6,24,8)=(0,0,0).$$
We have the system
\begin{align*}
3a+12b-6c&=0\\
10a+16b+24c&=0\\
6a+14b+8c&=0.
\end{align*}
Since the determinant of the coefficient matrix is given by
$$\left| \begin{array} {ccc} 3&12&-6\\ 10&16&24
\\ 6&14&8
\end{array} \right|=-120\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$
Consequently, $S$ is linearly independent.