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The inequality provided is:

\[
x(x + 6)(x - 6) \leq 0
\]

To solve this inequality, let's break it down into steps:

### Step 1: Find the roots
The expression equals zero when any of the factors is zero. Setting each factor to zero:

- \( x = 0 \)
- \( x + 6 = 0 \Rightarrow x = -6 \)
- \( x - 6 = 0 \Rightarrow x = 6 \)

So, the critical points (roots) are \( x = -6 \), \( x = 0 \), and \( x = 6 \).

### Step 2: Determine the sign of the expression in each interval
Now, we need to test the sign of the expression \( x(x + 6)(x - 6) \) in the intervals determined by the roots:

- **Interval 1:** \( (-\infty, -6) \)
- **Interval 2:** \( (-6, 0) \)
- **Interval 3:** \( (0, 6) \)
- **Interval 4:** \( (6, \infty) \)

We can choose test points in each interval (e.g., \( x = -7 \), \( x = -3 \), \( x = 3 \), \( x = 7 \)) and check the sign of the product \( x(x + 6)(x - 6) \).

- For \( x = -7 \), all factors are negative: \( (-7)(-7 + 6)(-7 - 6) = (-7)(-1)(-13) = -91 \). So, the product is negative.
- For \( x = -3 \), the first factor is negative, the second is positive, and the third is negative: \( (-3)(-3 + 6)(-3 - 6) = (-3)(3)(-9) = 81 \). So, the product is positive.
- For \( x = 3 \), the first factor is positive, the second is positive, and the third is negative: \( (3)(3 + 6)(3 - 6) = (3)(9)(-3) = -81 \). So, the product is negative.
- For \( x = 7 \), all factors are positive: \( (7)(7 + 6)(7 - 6) = (7)(13)(1) = 91 \). So, the product is positive.

### Step 3: Solve the inequality
We are interested in where the expression is less than or equal to zero. From the test points:

- The expression is **negative** in the intervals \( (-\infty, -6) \) and \( (0, 6) \).
- The expression is **zero** at \( x = -6 \), \( x = 0 \), and \( x = 6 \).
- The expression is **positive** in the intervals \( (-6, 0) \) and \( (6, \infty) \).

### Step 4: Write the solution
The solution to the inequality is where the expression is less than or equal to zero:

\[
x \in (-\infty, -6] \cup [0, 6]
\]

This means that the values of \( x \) that satisfy the inequality are in the intervals \( (-\infty, -6] \) and \( [0, 6] \).

Would you like more details or have any questions? Here are five related questions to expand on this:

1. How do we solve similar inequalities with more factors?
2. What is the importance of test points in inequality solving?
3. Can this inequality be solved graphically?
4. How do we handle inequalities with strict inequality signs (e.g., \(<\) instead of \(\leq\))?
5. What happens if the product of factors is positive instead of negative?

**Tip:** When solving inequalities involving factors, always pay attention to critical points and check the sign of the product in each interval carefully.

Solve the inequality x(x + 6)(x - 6) <= 0.

Learn how to solve the inequality x(x + 6)(x - 6) <= 0 step by step. This problem covers algebraic concepts including polynomial factorization and sign analysis, using the Zero Product Property to find critical points. Suitable for students in Grades 9-11.

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