solve the inequality: 3x^3 > 6x^2 in interval notation

To solve the inequality \( 3x^3 > 6x^2 \), let's proceed step-by-step:

### Step 1: Simplify the inequality
\[
3x^3 > 6x^2
\]
Divide both sides by 3:
\[
x^3 > 2x^2
\]
Next, subtract \( 2x^2 \) from both sides:
\[
x^3 - 2x^2 > 0
\]
Factor out the common term \( x^2 \):
\[
x^2(x - 2) > 0
\]

### Step 2: Identify critical points
The critical points are obtained by setting \( x^2(x - 2) = 0 \):
\[
x^2 = 0 \quad \Rightarrow \quad x = 0
\]
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
Thus, the critical points are \( x = 0 \) and \( x = 2 \).

### Step 3: Analyze the sign changes
We will test the sign of \( x^2(x - 2) \) in the intervals determined by the critical points: \( (-\infty, 0) \), \( (0, 2) \), and \( (2, \infty) \).

1. **For \( x \in (-\infty, 0) \):** 
   - \( x^2 > 0 \) (since the square of any real number is non-negative)
   - \( x - 2 < 0 \) (since \( x < 2 \))
   - Thus, \( x^2(x - 2) < 0 \).

2. **For \( x \in (0, 2) \):**
   - \( x^2 > 0 \)
   - \( x - 2 < 0 \)
   - Thus, \( x^2(x - 2) < 0 \).

3. **For \( x \in (2, \infty) \):**
   - \( x^2 > 0 \)
   - \( x - 2 > 0 \)
   - Thus, \( x^2(x - 2) > 0 \).

### Step 4: Determine the solution set
The inequality \( x^2(x - 2) > 0 \) holds in the interval \( (2, \infty) \). At \( x = 0 \) and \( x = 2 \), the expression equals zero, which does not satisfy the strict inequality.

### Final answer in interval notation:
\[
(2, \infty)
\]

Would you like more details or have any questions about this solution?

### Related Questions:
1. How do you solve \( x^3 < 2x^2 \) in interval notation?
2. What are the steps to solve a rational inequality like \( \frac{x^2 - 4}{x + 1} > 0 \)?
3. How do you graphically represent the solution to the inequality \( 3x^3 > 6x^2 \)?
4. What changes if the inequality was \( 3x^3 \geq 6x^2 \)?
5. How do you test intervals when solving inequalities?

### Tip:
Always factor out common terms first when solving polynomial inequalities; it simplifies finding critical points and analyzing the intervals.

Learn how to solve the polynomial inequality 3x^3 > 6x^2 with a detailed step-by-step guide. This solution involves factorizing the expression, identifying critical points, and analyzing intervals. Suitable for high school students in Grades 9-12.

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