To solve the inequality $x^2 - x < 6$, we start by rewriting it as a standard quadratic inequality:

$x^2 - x - 6 < 0$

Step 1: Factor the Quadratic Expression

First, factor the expression on the left:

$x^2 - x - 6 = (x - 3)(x + 2)$

Step 2: Determine the Critical Points

Set each factor equal to zero to find the critical points:

$x - 3 = 0 \implies x = 3$

$x + 2 = 0 \implies x = -2$

Step 3: Test Intervals Around the Critical Points

The critical points divide the number line into three intervals:

1. $(-\infty, -2)$
2. $(-2, 3)$
3. $(3, \infty)$

We need to test each interval to determine where the inequality holds.

Interval 1: $(-\infty, -2)$
Choose a test point, say $x = -3$:

$(x - 3)(x + 2) = (-3 - 3)(-3 + 2) = (-6)(-1) = 6$

Since $6 > 0$, this interval does not satisfy the inequality.

Interval 2: $(-2, 3)$
Choose a test point, say $x = 0$:

$(x - 3)(x + 2) = (0 - 3)(0 + 2) = (-3)(2) = -6$

Since $-6 < 0$, this interval satisfies the inequality.

Interval 3: $(3, \infty)$
Choose a test point, say $x = 4$:

$(x - 3)(x + 2) = (4 - 3)(4 + 2) = (1)(6) = 6$

Since $6 > 0$, this interval does not satisfy the inequality.

Step 4: Combine the Results

The solution to the inequality is:

$x \in (-2, 3)$

Final Answer

$\boxed{x \in (-2, 3)}$

Would you like more details or have any questions? Here are some related questions:

1. How do you solve quadratic inequalities using a number line?
2. What happens if the inequality is $x^2 - x \leq 6$?
3. How can you verify the solution graphically?
4. What other methods can be used to solve quadratic inequalities?
5. How does the sign of the inequality affect the solution?

Tip: Always check your critical points to ensure they are correctly placed on the number line and test each interval to confirm the inequality's behavior!