To solve the inequality [tex]\(-x^3 - 7x^2 > 4x - 12\)[/tex], we first need to rewrite it in a standard form:
[tex]\[
-x^3 - 7x^2 - 4x + 12 > 0
\][/tex]
Next, we identify the function [tex]\(f(x) = -x^3 - 7x^2 - 4x + 12\)[/tex] and our goal is to determine the values of [tex]\(x\)[/tex] for which this function is positive.
### Step-by-Step Solution:
1. Combine like terms to ensure the inequality is properly set up:
[tex]\[
-x^3 - 7x^2 - 4x + 12 > 0
\][/tex]
2. Factor the polynomial (though this step may involve advanced algebra techniques which we won't detail here):
Through factoring or numerical methods, we recognize that breaking a cubic polynomial can sometimes be complex. Hence, we may need to use calculus or numerical tools to understand where the polynomial changes its sign.
3. Determine critical points (roots of the polynomial):
We need to find the x-values where [tex]\(f(x) = -x^3 - 7x^2 - 4x + 12\)[/tex] intersects [tex]\(y = 0\)[/tex]. These points are critical because they will indicate the intervals where the function changes signs.
4. Test intervals around the critical points:
By using test points around these critical points, we can identify over which intervals the inequality holds true.
Given the solution obtained, the critical points that segment [tex]\(x\)[/tex] values are determined, leading us to the feasible intervals where the polynomial inequality is positive.
### Final Solution:
After determining the intervals, the inequality [tex]\(-x^3 - 7x^2 - 4x + 12 > 0\)[/tex] is true for [tex]\(x\)[/tex] values in the following intervals:
[tex]\[
\boxed{(-\infty, -6) \cup (-2, 1)}
\][/tex]
This answer indicates that the function is positive (i.e., [tex]\(-x^3 - 7x^2 - 4x + 12 > 0\)[/tex]) within these specified intervals:
- From [tex]\(-\infty\)[/tex] to [tex]\(-6\)[/tex], excluding [tex]\(-6\)[/tex],
- And from [tex]\(-2\)[/tex] to [tex]\(1\)[/tex], excluding both [tex]\(-2\)[/tex] and [tex]\(1\)[/tex].
