The solution set of inequality
`((e^(x)-1)(2x-3)(x^(2)+x+2))/((sinx-2)(x+1)x) le 0`

Options

A. `[(3)/(2),oo)`
B. `(-oo,-1) uu [(3)/(2),oo)`
C. `(-1,0) uu [(3)/(2),oo)`
D. `R-{0,-1}`

Answer

B

Explanation

`(b)` We have `((e^(x)-1)(2x-3)(x^(2)+x+2))/((sinx-2)(x+1)x) le 0`
`implies ((e^(x)-1)(2x-3))/((x+1)x) ge =0`
(as `x^(2)+x+2 gt 0` and `sinx-2 lt 0` for all real `x`)
Sign scheme of `E` is as shown below.

At `x=0`, sign of the expression does not change as two factors `'x'` and `'e^(x)-1'` are becoming zero at this point.
`:.` from the sign scheme, `x in(-oo,-1)uu[(3)/(2),oo)`.