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To solve the inequality \( x^2 - x < 0 \), follow these steps:

### Step 1: Factor the expression
We begin by factoring the quadratic expression:
\[
x^2 - x = x(x - 1)
\]
So the inequality becomes:
\[
x(x - 1) < 0
\]

### Step 2: Determine critical points
The critical points of this inequality occur when \( x(x - 1) = 0 \), which gives:
\[
x = 0 \quad \text{or} \quad x = 1
\]
These critical points divide the number line into three intervals: \( (-\infty, 0) \), \( (0, 1) \), and \( (1, \infty) \).

### Step 3: Analyze the sign of the product in each interval
We test the sign of \( x(x - 1) \) in each interval:

- **Interval \( (-\infty, 0) \)**: Let \( x = -1 \). Substituting into \( x(x - 1) \):
  \[
  (-1)((-1) - 1) = (-1)(-2) = 2 > 0
  \]
  So, in this interval, \( x(x - 1) > 0 \).

- **Interval \( (0, 1) \)**: Let \( x = \frac{1}{2} \). Substituting into \( x(x - 1) \):
  \[
  \left(\frac{1}{2}\right)\left(\frac{1}{2} - 1\right) = \left(\frac{1}{2}\right)\left(-\frac{1}{2}\right) = -\frac{1}{4} < 0
  \]
  So, in this interval, \( x(x - 1) < 0 \).

- **Interval \( (1, \infty) \)**: Let \( x = 2 \). Substituting into \( x(x - 1) \):
  \[
  (2)(2 - 1) = (2)(1) = 2 > 0
  \]
  So, in this interval, \( x(x - 1) > 0 \).

### Step 4: Solution
We are looking for where \( x(x - 1) < 0 \), which occurs in the interval \( (0, 1) \).

Thus, the solution to the inequality is:
\[
\boxed{(0, 1)}
\]

Would you like further details or explanations on any of the steps?

Here are some related questions:
1. How do you solve quadratic inequalities graphically?
2. What is the difference between solving \( x^2 - x \leq 0 \) and \( x^2 - x < 0 \)?
3. How do critical points help in solving inequalities?
4. Can you solve similar inequalities with different factorable quadratics?
5. How would the solution change if the inequality were \( x^2 - x > 0 \)?

**Tip:** Always consider the sign changes at critical points when solving polynomial inequalities.

Learn how to solve the quadratic inequality x^2 - x < 0 step-by-step. This solution uses factoring and sign analysis to determine the intervals where the inequality holds. Suitable for high school students (Grades 9-12).

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