(x-1)(x-3)-0.jpeg "(x+2)(x-1)(x-3) 0")
Solving the Inequality: (x+2)(x-1)(x-3) > 0
This article will guide you through the process of solving the inequality (x+2)(x-1)(x-3) > 0. We will use a combination of algebraic manipulation and sign analysis to determine the solution set.
Understanding the Inequality
The inequality (x+2)(x-1)(x-3) > 0 asks us to find the values of 'x' that make the product of three factors positive. To do this, we need to consider the signs of each factor individually.
Finding the Critical Points
The critical points are the values of 'x' that make the expression equal to zero. To find them, we set each factor equal to zero and solve:
- x + 2 = 0 => x = -2
- x - 1 = 0 => x = 1
- x - 3 = 0 => x = 3
These critical points divide the number line into four intervals:
- x < -2
- -2 < x < 1
- 1 < x < 3
- x > 3
Sign Analysis
Now, we analyze the sign of each factor within each interval:
| Interval | x + 2 | x - 1 | x - 3 | (x+2)(x-1)(x-3) |
|---|---|---|---|---|
| x < -2 | - | - | - | - |
| -2 < x < 1 | + | - | - | + |
| 1 < x < 3 | + | + | - | - |
| x > 3 | + | + | + | + |
Explanation:
- x + 2: This factor is negative when x < -2 and positive when x > -2.
- x - 1: This factor is negative when x < 1 and positive when x > 1.
- x - 3: This factor is negative when x < 3 and positive when x > 3.
The sign of the entire product is determined by the product of the signs of each factor.
Determining the Solution Set
We are looking for the intervals where (x+2)(x-1)(x-3) is greater than zero. From our sign analysis, we see that this occurs when:
- -2 < x < 1 or
- x > 3
Therefore, the solution set to the inequality (x+2)(x-1)(x-3) > 0 is (-2, 1) U (3, ∞).
Conclusion
By finding the critical points, analyzing the signs of each factor, and combining those signs, we have determined the solution set for the inequality (x+2)(x-1)(x-3) > 0. The solution includes all values of x greater than -2 but less than 1, and all values of x greater than 3.