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!

Learn how to solve the quadratic inequality x² - x < 6 using factoring and sign analysis. Follow the step-by-step solution with detailed testing of intervals, critical points, and key algebraic techniques. Suitable for Grades 9-10 students studying quadratic inequalities.

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