To solve the inequality \( x^2 - x - 6 > 0 \), we can follow these steps:

### 1. Factor the quadratic expression:
First, we factor the quadratic equation \( x^2 - x - 6 \).

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

So, the inequality becomes:

\[
(x - 3)(x + 2) > 0
\]

### 2. Determine the critical points:
The critical points are the values of \( x \) that make the expression equal to zero. Set each factor equal to zero:

\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\]

These critical points divide the number line into three intervals: \( (-\infty, -2) \), \( (-2, 3) \), and \( (3, \infty) \).

### 3. Test the intervals:
To determine where the expression is positive, test a point from each interval:

- For \( x \in (-\infty, -2) \), choose \( x = -3 \):
  \[
  (-3 - 3)(-3 + 2) = (-6)(-1) = 6 > 0
  \]

- For \( x \in (-2, 3) \), choose \( x = 0 \):
  \[
  (0 - 3)(0 + 2) = (-3)(2) = -6 < 0
  \]

- For \( x \in (3, \infty) \), choose \( x = 4 \):
  \[
  (4 - 3)(4 + 2) = (1)(6) = 6 > 0
  \]

### 4. Write the solution:
The expression is positive in the intervals \( (-\infty, -2) \) and \( (3, \infty) \). Therefore, the solution to the inequality is:

\[
x \in (-\infty, -2) \cup (3, \infty)
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you solve inequalities involving absolute values?
2. What are the steps to solve a quadratic equation by completing the square?
3. How can you graph the solution set of \( x^2 - x - 6 > 0 \)?
4. What is the significance of the discriminant in quadratic equations?
5. How do you solve a system of quadratic inequalities?

### Tip:
Always check the sign of the product in each interval to ensure the correct solution set for inequalities involving products or quotients.

Discover how to solve the quadratic inequality x^2 - x - 6 > 0 using factoring, critical points, and sign analysis. This guide is perfect for students in Grades 9-11, covering key algebraic concepts and methods.

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