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The given inequality is:

\[
x^4 - 9x^2 \geq 0
\]

To solve this inequality, let's factor the expression.

1. **Factoring the expression:**

   \[
   x^4 - 9x^2 = (x^2)(x^2) - (3)(3)(x^2) = x^2(x^2 - 9)
   \]

   We can further factor the expression \(x^2 - 9\) as a difference of squares:

   \[
   x^2(x^2 - 9) = x^2(x - 3)(x + 3)
   \]

   So the inequality becomes:

   \[
   x^2(x - 3)(x + 3) \geq 0
   \]

2. **Finding the critical points:**

   The critical points occur where the expression equals zero. These points are:
   - \(x^2 = 0 \) gives \( x = 0 \)
   - \(x - 3 = 0 \) gives \( x = 3 \)
   - \(x + 3 = 0 \) gives \( x = -3 \)

   So, the critical points are \(x = -3\), \(x = 0\), and \(x = 3\).

3. **Testing the intervals:**

   We need to test the sign of the expression in each interval created by the critical points. The intervals are:

   - \((- \infty, -3)\)
   - \((-3, 0)\)
   - \((0, 3)\)
   - \((3, \infty)\)

   Let's choose a test point from each interval and determine the sign of the expression \(x^2(x - 3)(x + 3)\):

   - For \(x \in (-\infty, -3)\), choose \(x = -4\):
     \[
     x^2(-4 - 3)(-4 + 3) = 16 \times (-7) \times (-1) = 112 > 0
     \]
   
   - For \(x \in (-3, 0)\), choose \(x = -1\):
     \[
     x^2(-1 - 3)(-1 + 3) = 1 \times (-4) \times 2 = -8 < 0
     \]
   
   - For \(x \in (0, 3)\), choose \(x = 1\):
     \[
     x^2(1 - 3)(1 + 3) = 1 \times (-2) \times 4 = -8 < 0
     \]
   
   - For \(x \in (3, \infty)\), choose \(x = 4\):
     \[
     x^2(4 - 3)(4 + 3) = 16 \times 1 \times 7 = 112 > 0
     \]

4. **Considering the inequality:**

   The expression \(x^2(x - 3)(x + 3) \geq 0\) is true where the expression is positive or zero. This happens in the intervals:

   \[
   (-\infty, -3] \cup [0] \cup [3, \infty)
   \]

   So the solution to the inequality is:

   \[
   x \in (-\infty, -3] \cup [0] \cup [3, \infty)
   \]

Would you like a further explanation or have any questions? Here are some related questions:

1. How do you solve polynomial inequalities in general?
2. What is the difference between strict and non-strict inequalities?
3. Can inequalities be solved graphically?
4. What are critical points in the context of polynomial inequalities?
5. How do you determine the sign of an expression over an interval?
6. How do you factor higher-degree polynomials?
7. What is the significance of the interval notation in inequalities?
8. Can you solve inequalities with rational expressions?

**Tip:** When solving polynomial inequalities, always factor the polynomial first and then test the sign of each factor over the relevant intervals.

Explore how to solve the polynomial inequality x^4 - 9x^2 ≥ 0 step-by-step. This problem involves factoring, interval testing, and sign analysis to determine the solution intervals. Suitable for advanced high school students and college-level mathematics enthusiasts.

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