What is the solution of x2-1/x2+5x+4 less than or equal to 0?

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.

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step-by-step explanation: