Determine the solution set of (x+1)(x-2)/x+3 >= 0Show complete solution

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thilina-g eNotes educator| Certified Educator

`((x+1)(x-2))/(x+3) gt= 0`

The sign of the inequality can be determined by both numerator and denominator. Therefore I would rearrange the expression as below.

`((x+1)(x-2))/(x+3) xx (x+3)/(x+3)gt= 0`

`((x+1)(x-2)(x+3))/(x+3)^2 gt= 0`

Now the denominator is always positive and so we can evaluate the numerator for the inequality.

`(x+1)(x-2)(x+3) >= 0`


We can indentify the ranges for this inequaility as below.


Then, `(x+1)(x-2)(x+3) <= 0`


Then, `(x+1)(x-2)(x+3) >= 0`


Then, `(x+1)(x-2)(x+3) <= 0`

`xgt= 2`

Then, `(x+1)(x-2)(x+3) >= 0`


Therefore the solutions for the given inequality are `-3lt=xlt-1`   and `xgt= 2.`