The equation in the image is:

$\frac{2}{x - 3} = \frac{5}{x + 3}$

We are tasked with solving for $x$ and simplifying the result fully. Here's how to solve this step by step:

Step 1: Cross-multiply

To eliminate the fractions, cross-multiply the two expressions:

$2(x + 3) = 5(x - 3)$

Step 2: Expand both sides

Expand both sides of the equation:

$2x + 6 = 5x - 15$

Step 3: Move the terms with $x$ to one side

Now, move the $x$-terms to one side by subtracting $2x$ from both sides:

$6 = 3x - 15$

Step 4: Move constant terms to the other side

Add 15 to both sides:

$6 + 15 = 3x$

$21 = 3x$

Step 5: Solve for $x$

Now, divide by 3 to isolate $x$:

$x = \frac{21}{3} = 7$

Step 6: Check for any restrictions

The denominators $x - 3$ and $x + 3$ cannot be zero, which means $x \neq 3$ and $x \neq -3$. Since our solution $x = 7$ does not violate these conditions, it is valid.

Final answer:

$x = 7$

Let me know if you need further details or have any questions!

Here are 5 questions related to this topic:

1. What would happen if $x = 3$ or $x = -3$?
2. Can you cross-multiply with any fraction?
3. How would the equation change if the right-hand side had a different constant in the numerator?
4. How would you solve the equation if both fractions had different variables in the denominators?
5. What are the general steps to solve rational equations?

Tip: Always check for restrictions on the variable when solving equations involving fractions to avoid invalid solutions.