Sure, let's solve the inequality [tex]\( x^3 - x^2 \geq 9x - 9 \)[/tex].
First, we rearrange the inequality to have zero on one side:
[tex]\[
x^3 - x^2 - 9x + 9 \geq 0
\][/tex]
Next, we factor the left-hand side if possible. We look for roots of the polynomial equation [tex]\( x^3 - x^2 - 9x + 9 = 0 \)[/tex]. The roots can be found using various methods such as factoring by grouping, synthetic division, or other polynomial root-finding methods.
After finding the roots, we can determine the intervals where the polynomial is non-negative.
The roots of the polynomial [tex]\( x^3 - x^2 - 9x + 9 = 0 \)[/tex] are [tex]\( x = -3, x = 1, x = 3 \)[/tex].
We use these roots to divide the number line into intervals and test the sign of the polynomial in each interval. The intervals to consider are:
- [tex]\( (-\infty, -3) \)[/tex]
- [tex]\( (-3, 1) \)[/tex]
- [tex]\( (1, 3) \)[/tex]
- [tex]\( (3, \infty) \)[/tex]
By testing values within each interval, we can determine where the polynomial is greater than or equal to zero.
- In [tex]\( (-\infty, -3) \)[/tex], the polynomial is negative.
- In [tex]\( (-3, 1) \)[/tex], the polynomial is non-negative.
- In [tex]\( (1, 3) \)[/tex], the polynomial is negative.
- In [tex]\( (3, \infty) \)[/tex], the polynomial is non-negative.
Combining these observations, the intervals where the polynomial is non-negative are [tex]\( [-3, 1] \)[/tex] and [tex]\( [3, \infty) \)[/tex].
Therefore, the solution to the inequality [tex]\( x^3 - x^2 - 9x + 9 \geq 0 \)[/tex] is:
[tex]\[
[-3, 1] \cup [3, \infty)
\][/tex]
Expressing this in interval notation:
[tex]\[
\boxed{(-3 \leq x \leq 1) \, \cup \, (3 \leq x < \infty)}
\][/tex]
