(x-4)(x-7)-%D0%B1%D0%BE%D0%BB%D1%8C%D1%88%D0%B5-0.jpeg "(x+8)(x-4)(x-7) Больше 0")
Solving the Inequality: (x+8)(x-4)(x-7) > 0
This inequality involves a product of three linear expressions. To solve it, we need to find the intervals where the product is positive. Here's a step-by-step approach:
1. Find the Critical Points
The critical points are the values of 'x' where the expression equals zero. Set each factor to zero and solve:
- x + 8 = 0 => x = -8
- x - 4 = 0 => x = 4
- x - 7 = 0 => x = 7
These critical points divide the number line into four intervals:
- Interval 1: x < -8
- Interval 2: -8 < x < 4
- Interval 3: 4 < x < 7
- Interval 4: x > 7
2. Test Each Interval
We'll choose a test value within each interval and evaluate the sign of the expression:
| Interval | Test Value | (x+8)(x-4)(x-7) | Sign |
|---|---|---|---|
| x < -8 | x = -9 | (-1)(-13)(-16) | Negative |
| -8 < x < 4 | x = 0 | (8)(-4)(-7) | Positive |
| 4 < x < 7 | x = 5 | (13)(1)(-2) | Negative |
| x > 7 | x = 8 | (16)(4)(1) | Positive |
3. Determine the Solution
The inequality (x+8)(x-4)(x-7) > 0 is true when the expression is positive. Therefore, the solution is the union of the intervals where the sign is positive:
Solution: -8 < x < 4 or x > 7
Conclusion
The solution to the inequality (x+8)(x-4)(x-7) > 0 is -8 < x < 4 or x > 7. This means that the inequality holds true for all values of 'x' within these intervals.