Answer:
x ∈ (-∞ , -2) ∪ (1, 3)
Step-by-step explanation:
The expression is already factored. Note that for the polynomial that appears in the numerator [tex](x-3)(x + 2)[/tex] there are 2 roots:
[tex]x = 3\\x = -2[/tex]
For the polynomial that appears in the denominator there is 1 root:
[tex]x = 1[/tex]
Note that [tex]x = 1[/tex] 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 [tex]f(x) <0[/tex]
Observe the attached image
Note that:
[tex](x-3) <0[/tex] when [tex]x <3[/tex]
[tex](x + 2) <0[/tex] when [tex]x <-2[/tex]
[tex](x-1) <0[/tex] when [tex]x <1[/tex]
Finally, we have the solution:
x ∈ (-∞ , -2) ∪ (1, 3)
