Answer:
Fourth answer choice.
Step-by-step explanation:
Start by factoring the numerator and the denominator:
(x - 1)(x + 1)
-----------------
(x + 1)(x + 4)
Note that x can be neither -1 nor -4, since either results in an undefined quotient. These two x-values are critical values because of this. If we cancel the (x + 1) terms, we obtain the result
(x - 1)
--------- for x ≠ -1 and x ≠ - 4
(x + 4)
The next step is to evaluate the given quotient on the three intervals defined by {-4, -1}: (-∞, -4), (-4, -1), (-1, ∞ ). We choose an x-value from within each interval and evaluate the given function at each. Suitable test values include {-10, -3, 0}:
At x = -10, the reduced given quotient (x - 1) / (x + 4) takes on the value (-10 - 1) / (-10 + 4) = -11/(-6), which is positive. Reject this interval, as we want and expect the quotient value to be 0 or less.
At x = -3, we get (-3 - 1) / (-3 + 4), which is negative. The given inequality is true on the interval (-4, -1) (or -4 < x < -1).
At x = 0, we get (0 - 1) / (0 + 4), which is negative, so the inequality is true on (-1, ∞ ).
So the fourth answer choice is the correct one.
