(x%2B2)(x-9)-0.jpeg "(x-4)(x+2)(x-9) 0")
Solving the Inequality: (x-4)(x+2)(x-9) < 0
This problem involves finding the values of 'x' that satisfy the inequality (x-4)(x+2)(x-9) < 0. To solve this, we'll use the concept of sign analysis.
1. Finding Critical Points:
The critical points are the values of 'x' that make the expression equal to zero. Therefore, we set each factor to zero and solve:
- x - 4 = 0 => x = 4
- x + 2 = 0 => x = -2
- x - 9 = 0 => x = 9
These critical points divide the number line into four intervals:
- Interval 1: x < -2
- Interval 2: -2 < x < 4
- Interval 3: 4 < x < 9
- Interval 4: x > 9
2. Sign Analysis:
We now analyze the sign of the expression (x-4)(x+2)(x-9) in each interval:
- Interval 1 (x < -2): All three factors are negative, resulting in a negative product.
- Interval 2 (-2 < x < 4): The factor (x+2) is positive, while the other two are negative, resulting in a positive product.
- Interval 3 (4 < x < 9): The factors (x-4) and (x+2) are positive, while (x-9) is negative, resulting in a negative product.
- Interval 4 (x > 9): All three factors are positive, resulting in a positive product.
3. Solution:
We are looking for the intervals where the expression is less than zero. Based on our sign analysis, the solution is:
x < -2 or 4 < x < 9
This is the solution to the inequality (x-4)(x+2)(x-9) < 0. It represents all the values of 'x' that satisfy the given condition.