(x%2B3)(2-x)(1-x)2-x*(x%2B6)(x-9)(2x2%2B4x%2B9)-0.jpeg "(2x-1)(x+3)(2-x)(1-x)2 X*(x+6)(x-9)(2x2+4x+9) 0")
Solving the Polynomial Inequality: (2x-1)(x+3)(2-x)(1-x)2 * x*(x+6)(x-9)(2x2+4x+9) > 0
This problem involves solving a polynomial inequality where the expression on the left-hand side is a product of several factors. To find the solution, we'll use the following steps:
1. Finding the Critical Points
The critical points are the values of x where the expression on the left-hand side equals zero. These points divide the number line into intervals.
- Factor 1: (2x-1) = 0 => x = 1/2
- Factor 2: (x+3) = 0 => x = -3
- Factor 3: (2-x) = 0 => x = 2
- Factor 4: (1-x) = 0 => x = 1
- Factor 5: x = 0
- Factor 6: (x+6) = 0 => x = -6
- Factor 7: (x-9) = 0 => x = 9
- Factor 8: (2x2+4x+9) = 0 This quadratic has no real roots, as its discriminant is negative.
The critical points are: -6, -3, 0, 1/2, 1, 2, and 9.
2. Creating a Sign Chart
We'll create a sign chart to analyze the sign of the expression in each interval created by the critical points.
| Interval | (2x-1) | (x+3) | (2-x) | (1-x)2 | x | (x+6) | (x-9) | (2x2+4x+9) | Product |
|---|---|---|---|---|---|---|---|---|---|
| x < -6 | - | - | + | + | - | - | - | + | + |
| -6 < x < -3 | - | - | + | + | - | + | - | + | - |
| -3 < x < 0 | - | + | + | + | - | + | - | + | + |
| 0 < x < 1/2 | - | + | + | + | + | + | - | + | - |
| 1/2 < x < 1 | + | + | + | + | + | + | - | + | + |
| 1 < x < 2 | + | + | + | + | + | + | - | + | + |
| 2 < x < 9 | + | + | - | + | + | + | - | + | - |
| x > 9 | + | + | - | + | + | + | + | + | + |
3. Analyzing the Sign Chart
The expression is greater than zero (positive) in the intervals where the product column has a plus sign.
Therefore, the solution to the inequality is:
x ∈ (-∞, -6) ∪ (-3, 0) ∪ (1/2, 1) ∪ (1, 2) ∪ (9, ∞)
Note: The quadratic factor (2x2+4x+9) is always positive because its discriminant is negative, meaning it has no real roots. Therefore, it does not affect the sign of the product.