To solve the inequality \(x^2 - 3x > 0\), follow these steps:

1. **Find the roots of the corresponding equation**:
   \[
   x^2 - 3x = 0
   \]
   Factor the quadratic equation:
   \[
   x(x - 3) = 0
   \]
   So, the roots are:
   \[
   x = 0 \quad \text{and} \quad x = 3
   \]

2. **Determine the intervals** defined by the roots. These roots divide the real number line into three intervals:
   \[
   (-\infty, 0), \quad (0, 3), \quad (3, \infty)
   \]

3. **Test a point in each interval** to determine where the inequality \(x^2 - 3x > 0\) holds. 

   - For the interval \((-∞, 0)\), choose \(x = -1\):
     \[
     (-1)^2 - 3(-1) = 1 + 3 = 4 \quad (\text{positive})
     \]

   - For the interval \((0, 3)\), choose \(x = 1\):
     \[
     1^2 - 3(1) = 1 - 3 = -2 \quad (\text{negative})
     \]

   - For the interval \((3, \infty)\), choose \(x = 4\):
     \[
     4^2 - 3(4) = 16 - 12 = 4 \quad (\text{positive})
     \]

4. **Combine the results**:
   The inequality \(x^2 - 3x > 0\) is satisfied in the intervals \((-∞, 0)\) and \((3, ∞)\).

5. **Write the solution**:
   \[
   x \in (-\infty, 0) \cup (3, \infty)
   \]

Would you like more details or have any questions?

Here are some related questions you might want to explore:
1. How do you solve a quadratic inequality?
2. What are the methods to solve polynomial inequalities?
3. Can you explain the factorization of quadratic equations?
4. How do you determine the intervals for testing inequalities?
5. What is the importance of roots in solving inequalities?

**Tip:** When solving inequalities, always consider the critical points (roots) and test the intervals between these points to determine where the inequality holds.

Learn how to solve the quadratic inequality x^2 - 3x > 0 with a detailed step-by-step solution. This problem involves finding roots, testing intervals, and determining where the inequality holds. Suitable for students in Grades 7-9 studying quadratic equations and inequalities.

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