Inequalities 3

$\large \frac{(e^{x}-1)(2x-3)(x^{2}+x+2)}{(\sin{x}-2)(x+1)x}≤0$

If the solution set of $x$ for the inequality above is of the form $(-\infty,a)\cup\left[\frac{b}{c},\infty\right)$ , where $a$ , $b$ and $c$ are integers with $b$ and $c$ being coprime integers, find $|a|+b+c$ .

For more inequality problems,try out my set .