(x+20)(x-9)(x+10)>0
Please help? I'm stuck...need to find the solution set.
so far I have
x^3+21x^2-70x-1800>0
am I correct so far?
What do I need to do next?
-20<-10<x<9
is this correct?
Why did you reverse the carat for all the values? That should only happen when multiplying/dividing by a negative number.
x>9
x>-10
x>-20
Yes, you are correct so far. To find the solution set for the inequality (x+20)(x-9)(x+10) > 0, you first need to solve the corresponding equation x^3 + 21x^2 - 70x - 1800 = 0.
To proceed, you can use a graphing calculator or computer software to graph the equation y = x^3 + 21x^2 - 70x - 1800 and determine the x-axis intercepts. These x-values represent the possible solutions to the equation.
Alternatively, you can use a method called "sign analysis" to solve this equation by hand. Here's how you can do it:
1. Factor the equation as much as possible: x^3 + 21x^2 - 70x - 1800 = 0.
This equation can be factored to (x + 20)(x - 9)(x + 10) = 0.
2. Determine the sign of each factor for different intervals of x:
a) When x < -20: All three factors (x + 20), (x - 9), and (x + 10) are negative.
b) When -20 < x < -10: Only (x + 20) is positive, while (x - 9) and (x + 10) are negative.
c) When -10 < x < 9: Only (x + 20) and (x + 10) are positive, while (x - 9) is negative.
d) When x > 9: All three factors are positive.
3. Determine the intervals where the product of the factors is positive:
From the sign analysis, you can see that the product of the factors is positive when either all three factors are positive or only (x + 20) and (x + 10) are positive.
4. Write the solution set in interval notation:
The solution set for the inequality (x+20)(x-9)(x+10) > 0 is:
(-∞, -20) U (-10, -9) U (9, ∞).
Therefore, the solution set for this inequality is all real numbers except for the interval (-20, -10) and the point x = 9.