For f(x) equal to (x − 1)/(x + 5) to be less than 0, either (x − 1) must be negative and (x + 5) positive or (x − 1) must be positive and (x + 5) negative.
Case 1 If (x − 1) < 0, then x < 1. If (x + 5) > 0, then x > -5. The domain of x that meets both conditions is written as -5 < x < 1.
Case 2 If (x − 1) > 0, then x > 1. If (x + 5) < 0, then x < -5. There is no value of x that can be both greater than 1 and less than -5.
It then follows that the result of Case 1, -5 < x < 1, is the only valid solution of the rational inequality, (x − 1)/(x + 5) < 0.
Also note that neither f(1) = 0 nor f(-5) (which is undefined) will satisfy (x − 1)/(x + 5) < 0.
