The function $f(x) = x^4 - 4x^3 + 8x$ has the following x-intercepts (where $f(x) = 0$):

$x = -1.24, \quad x = 0, \quad x = 2, \quad x = 3.24$

5. Determine where the function is positive:

Based on the graph and the intercepts, we can now analyze the intervals. The function is positive (i.e., $f(x) > 0$) in the following intervals:

$(-\infty, -1.24) \cup (0, 2) \cup (3.24, \infty)$

Thus, the solution to the inequality $x^4 - 4x^3 + 8x > 0$ in interval notation is:

$(-\infty, -1.24) \cup (0, 2) \cup (3.24, \infty)$

Let me know if you need any further explanation or have any questions!

Related Questions:

1. How do we handle inequalities with polynomials of degree higher than 2?
2. What are the benefits of graphing functions to solve inequalities?
3. Why are x-intercepts important when solving inequalities?
4. How does the behavior of the function change at critical points?
5. Can all polynomial inequalities be solved using graphing devices?

Tip:

When solving inequalities, always check the behavior of the function at and between critical points to determine where the inequality holds true.