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.