https://file.aiquickdraw.com/mathbot/file/1726349078725dvfy25ax.jpg

Let's solve the equation shown in the image:

\[
\frac{3}{x - 3} = \frac{3x}{x + 5}
\]

### Step 1: Cross multiply to eliminate the denominators
\[
3(x + 5) = 3x(x - 3)
\]

### Step 2: Expand both sides
On the left side:
\[
3(x + 5) = 3x + 15
\]

On the right side:
\[
3x(x - 3) = 3x^2 - 9x
\]

So, the equation becomes:
\[
3x + 15 = 3x^2 - 9x
\]

### Step 3: Move all terms to one side of the equation
\[
0 = 3x^2 - 9x - 3x - 15
\]

Simplify the terms:
\[
0 = 3x^2 - 12x - 15
\]

### Step 4: Divide the equation by 3 to simplify
\[
0 = x^2 - 4x - 5
\]

### Step 5: Factor the quadratic equation
\[
(x - 5)(x + 1) = 0
\]

### Step 6: Solve for \(x\)
Setting each factor equal to zero:
\[
x - 5 = 0 \quad \text{or} \quad x + 1 = 0
\]
\[
x = 5 \quad \text{or} \quad x = -1
\]

### Step 7: Verify the solution by checking the restrictions
The original denominators were \(x - 3\) and \(x + 5\), so we must check if the solutions make these expressions equal to zero:
- \(x - 3 = 0\) when \(x = 3\) (but \(x = 5\) and \(x = -1\) do not cause this issue).
- \(x + 5 = 0\) when \(x = -5\) (but this is not one of the solutions).

Thus, both solutions are valid.

The correct answer is:
\[
\boxed{d. \, x = -1 \, \text{or} \, 5}
\]

Would you like further details or have any other questions? Here are five related questions to expand on this topic:

1. How can you solve quadratic equations using the quadratic formula?
2. What are the methods to solve rational equations effectively?
3. What happens if there are restrictions on the values of \(x\) in rational equations?
4. Can you apply factoring techniques to all types of quadratic equations?
5. How does cross-multiplication help simplify rational equations?

**Tip:** Always check for restrictions in rational equations to ensure the solutions don’t make any denominator zero!

Learn how to solve the rational equation 3 / (x - 3) = 3x / (x + 5) with a detailed step-by-step solution. This problem involves rational equations, factoring, and quadratic equation solving techniques, suitable for students in Grades 10-12.

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