(2x%5E2%2B1)(2-x)-0.jpeg "(3x+10)(2x^2+1)(2-x) 0")
Solving the Inequality: (3x + 10)(2x² + 1)(2 - x) > 0
This inequality involves a product of three factors, and we need to determine the intervals where the product is positive. Here's a step-by-step approach:
1. Find the Critical Points
The critical points are the values of x that make one or more of the factors equal to zero.
- 3x + 10 = 0 => x = -10/3
- 2x² + 1 = 0 => This equation has no real solutions because 2x² is always non-negative and adding 1 makes it always positive.
- 2 - x = 0 => x = 2
Therefore, our critical points are x = -10/3 and x = 2.
2. Create a Sign Chart
The critical points divide the number line into three intervals:
- x < -10/3
- -10/3 < x < 2
- x > 2
We'll create a sign chart to determine the sign of each factor in each interval:
| Interval | 3x + 10 | 2x² + 1 | 2 - x | (3x + 10)(2x² + 1)(2 - x) |
|---|---|---|---|---|
| x < -10/3 | - | + | + | - |
| -10/3 < x < 2 | + | + | + | + |
| x > 2 | + | + | - | - |
Explanation:
- 3x + 10: This factor is negative when x < -10/3 and positive when x > -10/3.
- 2x² + 1: This factor is always positive.
- 2 - x: This factor is positive when x < 2 and negative when x > 2.
3. Determine the Solution
We want to find the intervals where the product of the factors is positive. Looking at the sign chart, the product is positive in the interval:
-10/3 < x < 2
4. Final Answer
The solution to the inequality (3x + 10)(2x² + 1)(2 - x) > 0 is -10/3 < x < 2.