$解：(1) 根据 “异号两数相乘, 积为负” 可得：$
$①\ \left\{\begin{array}{l}2 x-3\lt 0 \\ x+1\gt 0\end{array}\right.\ 或②\ \left\{\begin{array}{l}2 x-3\gt 0 \\ x+1\lt 0\end{array}\right. ,$
$解①得\ -1\lt x\lt \frac{3}{2} , ②无解,$
$\ \therefore\ 不等式的解集为\ -1\lt x\lt \frac{3}{2} ;$
$(2)当\ 4 x-1 \neq 0 时,$
$根据 “同号两数相除, 商为正” 可得:$
$①\ \left\{\begin{array}{l}4 x-1\gt 0 \\ x+2\gt 0\end{array}\right.\ 或② \left\{\begin{array}{l}4 x-1\lt 0 \\ x+2\lt 0\end{array}\right. ,$
$解①得\ x\gt \frac{1}{4} , 解②得\ x\lt -2 ,$
$\ \therefore\ 不等式的解集为\ x\lt -2\ 或\ x\gt \frac{1}{4} ;$
$当\ 4 x-1=0 , 即\ x=\frac{1}{4}\ 时, 原不等式也成立;$
$综上所述,\ x\lt -2\ 或\ x \geqslant\ \frac{1}{4} .$
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