(x-7)(x-12)-0.jpeg "(x-3)(x-7)(x-12) 0")
Solving the Inequality: (x-3)(x-7)(x-12) > 0
This article will guide you through solving the inequality (x-3)(x-7)(x-12) > 0. This inequality involves a product of linear factors, and we'll use the concept of sign analysis to find the solution.
Understanding Sign Analysis
Sign analysis helps us understand the behavior of an expression by analyzing the sign of each factor at different intervals. We need to identify the critical points, which are the values of x that make the expression equal to zero.
In our case, the critical points are:
- x = 3
- x = 7
- x = 12
These critical points divide the number line into four intervals:
- Interval 1: x < 3
- Interval 2: 3 < x < 7
- Interval 3: 7 < x < 12
- Interval 4: x > 12
Analyzing the Sign of Each Factor
Let's analyze the sign of each factor in each interval:
| Interval | x - 3 | x - 7 | x - 12 | (x-3)(x-7)(x-12) |
|---|---|---|---|---|
| x < 3 | - | - | - | - |
| 3 < x < 7 | + | - | - | + |
| 7 < x < 12 | + | + | - | - |
| x > 12 | + | + | + | + |
Explanation:
- (x - 3): This factor is negative when x is less than 3 and positive when x is greater than 3.
- (x - 7): This factor is negative when x is less than 7 and positive when x is greater than 7.
- (x - 12): This factor is negative when x is less than 12 and positive when x is greater than 12.
Determining the Solution
We are looking for the intervals where the expression (x-3)(x-7)(x-12) is greater than zero (positive). From the sign analysis table, we can see that the expression is positive in the following intervals:
- 3 < x < 7
- x > 12
Therefore, the solution to the inequality (x-3)(x-7)(x-12) > 0 is:
x ∈ (3, 7) ∪ (12, ∞)
This means the inequality is true for all values of x greater than 3 but less than 7, and for all values of x greater than 12.