Answer: We can begin by factoring the quadratic expression in the numerator and denominator of the left-hand side of the inequality:
x^2 - 1 = (x + 1)(x - 1)
x^2 + 5x + 4 = (x + 1)(x + 4)
Substituting these expressions into the inequality, we get:
(x + 1)(x - 1)/(x + 1)(x + 4) ≤ 0
We can simplify this expression by canceling out the common factor of (x + 1) from both the numerator and the denominator:
(x - 1)/(x + 4) ≤ 0
To solve this inequality, we can use a sign chart or test values. Here's a sign chart:
x x - 1 x + 4 (x - 1)/(x + 4)
-4 -5 0 +
-1 -2 3 -
1 0 5 0
4 3 8 +
The inequality is satisfied when (x - 1)/(x + 4) is less than or equal to 0, which occurs when x is between -4 and -1, or when x is equal to 1. Therefore, the solution to the inequality is:
-4 ≤ x < -1 or x = 1
Explanation: