To solve the inequality \( 2x(x - 1) < 0 \), let's follow these steps:

### Step 1: Identify the roots of the related equation
First, find the roots by solving the related equation:
\[
2x(x - 1) = 0
\]
This equation will be true if either \( 2x = 0 \) or \( x - 1 = 0 \).

- \( 2x = 0 \) gives \( x = 0 \).
- \( x - 1 = 0 \) gives \( x = 1 \).

So, the roots are \( x = 0 \) and \( x = 1 \).

### Step 2: Determine the sign of the expression in each interval
The roots divide the number line into three intervals:
- \( x < 0 \)
- \( 0 < x < 1 \)
- \( x > 1 \)

Now, let's test the sign of \( 2x(x - 1) \) in each interval:

1. **Interval \( x < 0 \):**
   - Choose a test point, say \( x = -1 \).
   - \( 2(-1)((-1) - 1) = 2(-1)(-2) = 4 \) (positive).

2. **Interval \( 0 < x < 1 \):**
   - Choose a test point, say \( x = \frac{1}{2} \).
   - \( 2\left(\frac{1}{2}\right)\left(\frac{1}{2} - 1\right) = 2\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right) = -\frac{1}{2} \) (negative).

3. **Interval \( x > 1 \):**
   - Choose a test point, say \( x = 2 \).
   - \( 2(2)(2 - 1) = 2(2)(1) = 4 \) (positive).

### Step 3: Find the solution set
We want the expression \( 2x(x - 1) \) to be **less than 0**. This occurs in the interval where the expression is negative, which is \( 0 < x < 1 \).

### Final Answer:
\[
\boxed{0 < x < 1}
\]

This is the solution to the inequality \( 2x(x - 1) < 0 \).

Would you like further details, or do you have any questions?

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Here are five related questions for further exploration:

1. How would the solution change if the inequality were \( 2x(x - 1) > 0 \)?
2. What are the solutions if the inequality is \( 2x(x - 1) \leq 0 \)?
3. How can this inequality be solved graphically?
4. What happens if we consider the inequality \( 2x(x - 1) \geq 0 \)?
5. How do you solve a similar inequality with three factors, such as \( x(x - 2)(x + 3) < 0 \)?

**Tip:** When solving inequalities, it's important to check the sign of the expression in each interval created by the roots to accurately determine the solution set.

Learn how to solve the inequality 2x(x - 1) < 0 step-by-step with detailed explanations and interval testing. This problem involves understanding quadratic inequalities and testing intervals. Suitable for students in Grades 10-12.

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