Question

Solving Rational Inequalities and use sign diagram to sketch the graph. Image attached for better understanding.


`(x+2)/((x-1)(x+1)) \ \textless \ 0`

Answer 1

x ∈ (-∞ , -2) ∪ (-1, 1)

Explanation:

The expression is already factored. Note that for the polynomial that appears in the numerator
`(x + 2)` there is 1 root:

`x = -2`

For the polynomial that appears in the denominator there are 2 roots:

`x = 1\\x=-1`

Note that
`x = 1` and
`x = -1` does not belong to the domain of f(x) because it zeroes the denominator of the function and the division between zero is not defined.

With these three roots we do the study of signs to find out when
`f(x) <0`

Observe the attached image

Note that:

`(x+2) <0` when
`x <-2`

`(x-1) <0` when
`x <1`

`(x+1) <0` when
`x <-1`

Finally, we have the solution:

x ∈ (-∞ , -2) ∪ (-1, 1)