what can x not equal for (x^2 -9) / (x^2) X (5x) / (x^2-x-6)

We are given the expression:

\[
\frac{x^2 - 9}{x^2} \times \frac{5x}{x^2 - x - 6}
\]

To determine what values \( x \) cannot equal (i.e., restrictions on the domain), we need to focus on when the denominators become zero, as division by zero is undefined.

### Step 1: Factor the expressions

1. Factor \( x^2 - 9 \). This is a difference of squares:
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]

2. Factor \( x^2 - x - 6 \). Find two numbers that multiply to \(-6\) and add to \(-1\). The factors are \((x - 3)(x + 2)\).

So the expression becomes:

\[
\frac{(x - 3)(x + 3)}{x^2} \times \frac{5x}{(x - 3)(x + 2)}
\]

### Step 2: Identify the restrictions

We now examine the denominators:

1. \( x^2 \) cannot be zero, so \( x \neq 0 \).

2. \( (x - 3)(x + 2) \) also cannot be zero. Thus, \( x - 3 \neq 0 \) and \( x + 2 \neq 0 \). Solving these:
   \[
   x \neq 3 \quad \text{and} \quad x \neq -2
   \]

### Final Answer

The values that \( x \) cannot equal are:

\[
x \neq 0, \, x \neq 3, \, x \neq -2
\]

These are the values that would make the expression undefined.

---

Would you like a detailed explanation of how to simplify the expression further, or do you have any other questions?

Here are 5 related questions:

1. How do you factor quadratic expressions like \( x^2 - x - 6 \)?
2. Why is division by zero undefined in mathematics?
3. How can you identify restrictions in rational expressions?
4. What is the difference of squares and how is it factored?
5. How does simplifying rational expressions help in determining restrictions?

**Tip:** Always check the denominator for zeros before simplifying any rational expression to avoid undefined values.

Explore how to determine the values x cannot equal for the rational expression (x^2 - 9) / (x^2) x (5x) / (x^2 - x - 6). This problem covers factoring techniques, division by zero, and identifying restrictions in rational expressions. Suitable for students in Grades 9-10.

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