(x+19)(x-7)(x+17)>0=-19<x<-17 or x>7
is this answer correct
Yes, the answer is correct. To understand why, let's break it down step by step.
Given the inequality: (x+19)(x-7)(x+17) > 0
1. First, we can use the zero-product property, which states that if the product of two factors is equal to zero, then at least one of the factors must be zero. In this case, we want the product to be greater than zero, so none of the factors can be zero.
2. So we have three cases to consider:
a) If all three factors are positive (+), then the product will be positive (+).
b) If two factors are negative (-) and one factor is positive (+), then the product will be negative (-).
c) If one factor is negative (-) and two factors are positive (+), then the product will be negative (-).
3. Now, let's analyze the sign of each factor:
- For the factor (x+19): It becomes positive (+) when x > -19 and negative (-) when x < -19.
- For the factor (x-7): It becomes positive (+) when x > 7 and negative (-) when x < 7.
- For the factor (x+17): It becomes positive (+) when x > -17 and negative (-) when x < -17.
4. Based on the analysis of the signs, we can conclude:
- For the product to be positive (+), we need either all three factors to be positive (+) or one factor to be positive (+) and the other two factors to be negative (-).
- Therefore, we have two intervals where the product is positive (+):
i) -19 < x < -17 (All factors are positive)
ii) x > 7 (One factor is positive, while the other two are negative)
5. In conclusion, the correct answer to the inequality is -19 < x < -17 or x > 7. Any values of x within these intervals will satisfy the inequality and make the product greater than zero.