$$  $$(xt3) (x-2)  >0 over (xt2) (x-1) rational inequalities with graph direct to the answer 

To solve and graph the rational inequality $\frac{x^3 (x - 2)}{x^2 (x - 1)} > 0$, let's directly address the critical points and the solution set.

### Step 1: Simplify the Inequality
Simplify the expression:

\[
\frac{x^3 (x - 2)}{x^2 (x - 1)} = \frac{x^3}{x^2} \cdot \frac{x - 2}{x - 1} = x \cdot \frac{x - 2}{x - 1} = \frac{x(x - 2)}{x - 1}
\]

The inequality becomes:

\[
\frac{x(x - 2)}{x - 1} > 0
\]

### Step 2: Find Critical Points
The critical points occur where the numerator and denominator equal zero:

- **Numerator:** $x(x - 2) = 0 \Rightarrow x = 0, x = 2$
- **Denominator:** $x - 1 = 0 \Rightarrow x = 1$

### Step 3: Analyze the Sign Changes
To determine where the inequality is positive, evaluate the sign of $\frac{x(x - 2)}{x - 1}$ in the intervals determined by the critical points:

- **Intervals:** $x < 0$, $0 < x < 1$, $1 < x < 2$, and $x > 2$

### Step 4: Solution Set
- **For $x < 0$:** The expression is positive.
- **For $0 < x < 1$:** The expression is positive.
- **For $1 < x < 2$:** The expression is negative.
- **For $x > 2$:** The expression is positive.

### Final Answer:
The inequality holds true for:

\[
(-\infty, 0) \cup (0, 1) \cup (2, \infty)
\]

### Graphing the Solution
- **Open circles** at $x = 1$ because the expression is undefined.
- **Solid circles** at $x = 0$ and $x = 2$ where the numerator equals zero (inequality changes sign).

The regions $(-∞, 0)$, $(0, 1)$, and $(2, ∞)$ are shaded on the number line.

### Do you have any questions or want more details?

### Additional Questions:

1. What happens if the inequality is non-strict (≥ instead of >)?
2. How does the graph change if the expression is squared?
3. What is the importance of checking for undefined points?
4. How would a different denominator affect the solution?
5. What are some common mistakes when solving rational inequalities?
6. Can you solve a similar inequality with more complex factors?
7. How can you verify the solution using a graphing calculator?
8. What is the significance of critical points in determining the solution?

### Tip:
When solving rational inequalities, always consider the sign of the numerator and denominator separately to determine the correct intervals.

Learn how to solve and graph the rational inequality (x^3 (x - 2)) / (x^2 (x - 1)) > 0 step-by-step. Understand the critical points, sign analysis, and graphical representation of the solution. Suitable for high school students studying inequalities and rational expressions.

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