%2F(x%5E2%2Bx%2B5)(x%5E2-4x-5)-0.jpeg "(x-5)/(x^2+x+5)(x^2-4x-5) 0")
Solving the Inequality: (x-5) / (x^2 + x + 5)(x^2 - 4x - 5) < 0
This problem involves solving a rational inequality. Let's break it down step by step:
1. Factor the denominator
The denominator factors into:
(x^2 + x + 5)(x^2 - 4x - 5) = (x^2 + x + 5)(x + 1)(x - 5)
2. Identify Critical Points
Critical points are the values of x where the expression equals zero or is undefined.
- Zeroes: The expression equals zero when the numerator is zero, so x = 5.
- Undefined points: The expression is undefined when the denominator is zero. This occurs when:
- x^2 + x + 5 = 0 (This quadratic has no real roots, so it doesn't contribute to our critical points)
- x + 1 = 0 => x = -1
- x - 5 = 0 => x = 5
Therefore, our critical points are x = -1 and x = 5.
3. Create a Sign Chart
We'll use a sign chart to determine the sign of the expression in different intervals:
| Interval | x - 5 | x + 1 | x^2 + x + 5 | (x-5) / (x^2+x+5)(x^2-4x-5) |
|---|---|---|---|---|
| x < -1 | - | - | + | + |
| -1 < x < 5 | - | + | + | - |
| x > 5 | + | + | + | + |
Explanation:
- x - 5: This factor is negative when x < 5 and positive when x > 5.
- x + 1: This factor is negative when x < -1 and positive when x > -1.
- x^2 + x + 5: This quadratic is always positive, as its discriminant is negative.
- (x-5) / (x^2+x+5)(x^2-4x-5): The sign of the whole expression is determined by the signs of the individual factors.
4. Determine the Solution
We want to find where the expression is less than zero. From our sign chart, this occurs when -1 < x < 5.
5. Express the Solution in Interval Notation
The solution to the inequality (x-5) / (x^2 + x + 5)(x^2 - 4x - 5) < 0 is: (-1, 5).