解:由$x^2=\frac{1}{4}$得$x=\frac{1}{2}$或$x=-\frac{1}{2}$
当$x=\frac{1}{2}$时,$\frac{x+1}{3-2x}=\frac{\frac{1}{2}+1}{3-2×\frac{1}{2}}=\frac{\frac{3}{2}}{2}=\frac{3}{4}$
当$x=-\frac{1}{2}$时,$\frac{x+1}{3-2x}=\frac{-\frac{1}{2}+1}{3-2×(-\frac{1}{2})}=\frac{\frac{1}{2}}{4}=\frac{1}{8}$
综上,分式的值为$\frac{3}{4}$或$\frac{1}{8}$