(x-8)%2F2x-7-0.jpeg "(x-6)(x-8)/2x-7 0")
Solving the Inequality: (x-6)(x-8)/(2x-7) > 0
This problem involves finding the values of x that satisfy the inequality (x-6)(x-8)/(2x-7) > 0. To solve this, we'll use a combination of sign analysis and critical points.
1. Finding Critical Points
The critical points are the values of x where the expression equals zero or is undefined.
- Numerator: The numerator is zero when (x-6)(x-8) = 0, which gives us x = 6 and x = 8.
- Denominator: The denominator is zero when 2x-7 = 0, which gives us x = 7/2.
2. Sign Analysis
Now, we'll create a sign table to analyze the behavior of the expression in different intervals defined by the critical points.
| Interval | x < 7/2 | 7/2 < x < 6 | 6 < x < 8 | x > 8 |
|---|---|---|---|---|
| 2x-7 | - | + | + | + |
| x-6 | - | - | + | + |
| x-8 | - | - | - | + |
| (x-6)(x-8)/(2x-7) | + | - | + | + |
3. Interpreting the Results
The sign table shows that the expression (x-6)(x-8)/(2x-7) is positive when:
- x < 7/2
- 6 < x < 8
- x > 8
4. Solution
Therefore, the solution to the inequality (x-6)(x-8)/(2x-7) > 0 is:
x ∈ (-∞, 7/2) ∪ (6, 8) ∪ (8, ∞)
Important Note: The critical point x = 7/2 makes the expression undefined. Therefore, it is not included in the solution set.