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Solving the Inequality: (x+1)(x-3)² > 0
This problem involves solving a polynomial inequality. Let's break it down step by step:
1. Finding the Critical Points
The critical points are the values of x where the expression equals zero or is undefined.
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Setting the expression equal to zero: (x + 1)(x - 3)² = 0 This gives us x = -1 and x = 3.
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Checking for undefined values: The expression is defined for all real values of x.
2. Creating a Sign Chart
We'll use a sign chart to determine the intervals where the expression is positive or negative.
| Interval | x + 1 | (x - 3)² | (x + 1)(x - 3)² |
|---|---|---|---|
| x < -1 | - | + | - |
| -1 < x < 3 | + | + | + |
| x > 3 | + | + | + |
Explanation:
- x + 1: This factor is negative for x < -1 and positive for x > -1.
- (x - 3)²: This factor is always positive because it is squared.
- (x + 1)(x - 3)²: The sign of the product is determined by the signs of the individual factors.
3. Interpreting the Results
From the sign chart, we can see that:
- (x + 1)(x - 3)² > 0 when -1 < x < 3 or x > 3
Solution
Therefore, the solution to the inequality (x + 1)(x - 3)² > 0 is:
x ∈ (-1, 3) ∪ (3, ∞)
In interval notation, the solution is x belongs to the open interval from -1 to 3, excluding 3, and the open interval from 3 to positive infinity.