Let's solve the inequality step by step:

The inequality is: $(x - 8)(x - 2)(x + 4) > 0$

Step 1: Find critical points

The critical points occur where the expression equals zero: $(x - 8) = 0 \quad \Rightarrow \quad x = 8$ $(x - 2) = 0 \quad \Rightarrow \quad x = 2$ $(x + 4) = 0 \quad \Rightarrow \quad x = -4$

So, the critical points are $x = -4$, $x = 2$, and $x = 8$.

Step 2: Sign analysis

We divide the number line into intervals using the critical points:

• $(-\infty, -4)$
• $(-4, 2)$
• $(2, 8)$
• $(8, \infty)$

For each interval, we check the sign of the expression $(x - 8)(x - 2)(x + 4)$.

1. Interval $(-\infty, -4)$: Choose $x = -5$: $(-5 - 8)(-5 - 2)(-5 + 4) = (-13)(-7)(-1) = 91 > 0$ The expression is positive.

2. Interval $(-4, 2)$: Choose $x = 0$: $(0 - 8)(0 - 2)(0 + 4) = (-8)(-2)(4) = 64 > 0$ The expression is positive.

3. Interval $(2, 8)$: Choose $x = 5$: $(5 - 8)(5 - 2)(5 + 4) = (-3)(3)(9) = -81 < 0$ The expression is negative.

4. Interval $(8, \infty)$: Choose $x = 9$: $(9 - 8)(9 - 2)(9 + 4) = (1)(7)(13) = 91 > 0$ The expression is positive.

Step 3: Write the solution

We are looking for where the expression is positive, so we take the intervals where the expression is greater than zero:

• $(-\infty, -4)$
• $(-4, 2)$
• $(8, \infty)$

The critical points $x = -4$, $x = 2$, and $x = 8$ make the expression equal zero, so they are not included in the solution.

Final answer:

The solution in interval notation is: $(-\infty, -4) \cup (-4, 2) \cup (8, \infty)$

Let me know if you'd like further clarification or if you have any questions!

Related questions:

1. How do you determine the sign of an expression over an interval?
2. What is the importance of critical points in inequalities?
3. How do you handle inequalities where the expression equals zero at certain points?
4. Can this method be applied to higher-degree polynomials?
5. What happens if you have a "less than or equal to" instead of a "greater than" inequality?

Tip:

Always remember to exclude critical points when dealing with strict inequalities ("greater than" or "less than"), unless specified otherwise by the problem!