(x-3)(x-5)-0.jpeg "(x-1)(x-3)(x-5) 0")
Solving the Inequality: (x-1)(x-3)(x-5) > 0
This article will guide you through solving the inequality (x-1)(x-3)(x-5) > 0. This type of inequality involves a polynomial and can be solved using a sign table method.
Understanding the Concept
The inequality (x-1)(x-3)(x-5) > 0 asks for the values of x where the product of the three factors is positive. This means we need to find the intervals where the expression is greater than zero.
Step-by-Step Solution
-
Find the Critical Points:
- The critical points are the values of x where the expression equals zero.
- Set each factor equal to zero and solve:
- x - 1 = 0 => x = 1
- x - 3 = 0 => x = 3
- x - 5 = 0 => x = 5
-
Create a Sign Table:
- Draw a number line and mark the critical points (1, 3, and 5).
- These points divide the number line into four intervals:
- x < 1
- 1 < x < 3
- 3 < x < 5
- x > 5
- Choose a test value within each interval and evaluate the sign of the expression (x-1)(x-3)(x-5) at that test value.
| Interval | Test Value (x) | (x-1) | (x-3) | (x-5) | (x-1)(x-3)(x-5) | Sign |
|---|---|---|---|---|---|---|
| x < 1 | x = 0 | - | - | - | - | Negative |
| 1 < x < 3 | x = 2 | + | - | - | + | Positive |
| 3 < x < 5 | x = 4 | + | + | - | - | Negative |
| x > 5 | x = 6 | + | + | + | + | Positive |
- Determine the Solution:
- The inequality asks for the values of x where the expression is greater than zero, meaning the intervals with a positive sign.
- Therefore, the solution to the inequality (x-1)(x-3)(x-5) > 0 is:
- 1 < x < 3 or x > 5
Conclusion
The solution to the inequality (x-1)(x-3)(x-5) > 0 is 1 < x < 3 or x > 5. This means that the expression is positive for all values of x within those intervals. The sign table method provides a systematic way to find the solution by examining the sign of the expression in each interval.