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

To solve the rational inequality \(\frac{x^3 (x - 2)}{x^2 (x - 1)} > 0\) and graph the solution, let's break down the process:

### Step 1: Simplify the Inequality
First, simplify the expression:

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

So the inequality becomes:

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

### Step 2: Determine Critical Points
Identify the points where the numerator and denominator are zero:

- **Numerator:** \(x(x - 2) = 0\) gives \(x = 0\) or \(x = 2\).
- **Denominator:** \(x - 1 = 0\) gives \(x = 1\).

These points are the critical points where the inequality can change its sign.

### Step 3: Test Intervals Around the Critical Points
The critical points split the number line into four intervals:

1. \(x < 0\)
2. \(0 < x < 1\)
3. \(1 < x < 2\)
4. \(x > 2\)

We need to test each interval to determine where the inequality holds true.

- **Interval 1:** \(x < 0\)  
  Choose \(x = -1\): \(\frac{-1(-1 - 2)}{-1 - 1} = \frac{-1(-3)}{-2} = \frac{3}{2} > 0\)

- **Interval 2:** \(0 < x < 1\)  
  Choose \(x = \frac{1}{2}\): \(\frac{\frac{1}{2}(\frac{1}{2} - 2)}{\frac{1}{2} - 1} = \frac{\frac{1}{2} \times \left(-\frac{3}{2}\right)}{-\frac{1}{2}} = \frac{-\frac{3}{4}}{-\frac{1}{2}} = \frac{3}{2} > 0\)

- **Interval 3:** \(1 < x < 2\)  
  Choose \(x = 1.5\): \(\frac{1.5(1.5 - 2)}{1.5 - 1} = \frac{1.5(-0.5)}{0.5} = \frac{-0.75}{0.5} = -1.5 < 0\)

- **Interval 4:** \(x > 2\)  
  Choose \(x = 3\): \(\frac{3(3 - 2)}{3 - 1} = \frac{3(1)}{2} = \frac{3}{2} > 0\)

### Step 4: Determine the Solution Set
The inequality is positive (i.e., holds true) for the intervals:

- \(x < 0\)
- \(0 < x < 1\)
- \(x > 2\)

The critical points \(x = 0\) and \(x = 2\) are included since they make the numerator zero (and the inequality allows for \(\geq 0\)) but \(x = 1\) is excluded since it makes the denominator zero.

### Final Answer:
\[
(-\infty, 0) \cup (0, 1) \cup (2, \infty)
\]

### Graph of the Solution Set:
The graph of the solution set would show open circles at \(x = 1\) and solid circles at \(x = 0\) and \(x = 2\). The regions where the inequality holds true would be highlighted:

1. The interval \((-∞, 0)\) is shaded.
2. The interval \((0, 1)\) is shaded.
3. The interval \((2, ∞)\) is shaded.

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

### Additional Questions:

1. What happens if the inequality is non-strict (≥ instead of >)?
2. How does the graph change if we modify the numerator?
3. How does the sign change affect the intervals?
4. Can you find the solution using a different method?
5. What are the implications of including or excluding critical points?
6. How does this inequality relate to polynomial inequalities?
7. What would be the solution if the denominator had a higher degree?
8. Can you describe the behavior of the function near the critical points?

### Tip:
Always check the endpoints and points where the expression is undefined when solving inequalities; these are critical for determining the correct intervals.

Learn how to solve the rational inequality (x^3(x - 2))/(x^2(x - 1)) > 0 with a detailed step-by-step solution and graph representation. This problem covers key concepts in inequalities, rational functions, critical points, and interval testing. Suitable for advanced high school students in Grades 11-12.

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