Question

How do you solve `(-x-3)/(x+2)<=0`?

Answer 1

`color(green)(x > -2) or color(green)(x <= -3)`

Explanation:

Given
`color(white)("XXX")(-x-3)/(x+2) <= 0`

Provided `x!=-2` we need to consider two possibilities:

Case 1. If `color(black)(x < -2)`
Since `(x+2) < 0` we need to reverse the inequality when dividing by `(x+2)`
`color(white)("XXX")-x-3 >= 0`

`color(white)("XXX")-x >= 3`

`color(white)("XXX")x <= -3`
`color(white)("XXX")`Considering the two restrictions `x < -2` and `x <= -3`
`color(white)("XXX")`we employ the more restrictive:
`color(white)("XXXXXX")color(green)(x <=-3)`

Case 2. If `color(black)( x > -2)`
Since `(x+2) > 0` we can divide by `(x+2)` without effecting the inequality
`color(white)("XXX")`then -x-3 <= 0#

`color(white)("XXX")-x <= 3`

`color(white)("XXX")x >= -3`
`color(white)("XXX")`Again considering the two restrictions `x > -2` and `x >= -3`
`color(white)("XXX")`we employ the more restrictive
`color(white)("XXXXXX")color(green)(x > -2)`

Here is a graph of the `(-x-3)/(x+2)` to help verify this result: