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Let's analyze the given rational inequality:

\[
\frac{4x}{3 - x} \geq 4x
\]

### Step 1: Move all terms to one side
We subtract \(4x\) from both sides to get:

\[
\frac{4x}{3 - x} - 4x \geq 0
\]

### Step 2: Get a common denominator
To simplify the expression, let's combine terms by finding a common denominator:

\[
\frac{4x}{3 - x} - \frac{4x(3 - x)}{3 - x} \geq 0
\]

This simplifies to:

\[
\frac{4x - 4x(3 - x)}{3 - x} \geq 0
\]

Now distribute the \(4x\) in the second term:

\[
\frac{4x - (12x + 4x^2)}{3 - x} \geq 0
\]

Simplify the numerator:

\[
\frac{4x - 12x + 4x^2}{3 - x} = \frac{-8x + 4x^2}{3 - x}
\]

### Step 3: Factor the numerator
The numerator can be factored as:

\[
\frac{4x(x - 2)}{3 - x} \geq 0
\]

### Step 4: Solve the inequality
We need to solve:

\[
\frac{4x(x - 2)}{3 - x} \geq 0
\]

This inequality holds when the expression is zero or positive. To find the critical points, solve for when the numerator and denominator are zero:

- Numerator \(4x(x - 2) = 0\) gives \(x = 0\) and \(x = 2\).
- Denominator \(3 - x = 0\) gives \(x = 3\) (which makes the expression undefined).

### Step 5: Test intervals
Now, test intervals between the critical points: \( (-\infty, 0), (0, 2), (2, 3), (3, \infty) \).

- For \(x < 0\), the expression is positive.
- For \(0 < x < 2\), the expression is negative.
- For \(2 < x < 3\), the expression is positive.
- For \(x > 3\), the expression is negative.

### Step 6: Conclusion
The inequality is satisfied when \(x \in (-\infty, 0] \cup (2, 3)\).

So, the correct interval solution is:

\[
(-\infty, 0] \cup (2, 3)
\]

If you'd like more details or have any questions, feel free to ask! Here are five related questions that can expand your understanding:

1. How do you determine where a rational function is undefined?
2. What are the steps to solving inequalities with rational expressions?
3. How can you test the sign of an expression in different intervals?
4. What does interval notation represent, and how do you use it for inequalities?
5. Why do we exclude certain values in rational inequalities?

**Tip**: Always check for undefined values in the denominator when solving rational inequalities.

Solve the rational inequality: 4x / (3 - x) ≥ 4x and provide the answer in interval notation.

Learn how to solve the rational inequality 4x/(3 - x) ≥ 4x by simplifying it, finding critical points, and testing intervals. This problem focuses on rational expressions and interval notation, suitable for high school students.

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