To solve for $x$ in the given pair of similar right triangles, we can use the property that corresponding sides of similar triangles are proportional.

The sides of the larger triangle are $2x - 4$ and 39, while the sides of the smaller triangle are $x + 6$ and 24. Since the triangles are similar, the ratios of corresponding sides are equal:

$\frac{2x - 4}{39} = \frac{x + 6}{24}$

Now, we'll cross-multiply and solve for $x$:

$(2x - 4) \times 24 = 39 \times (x + 6)$

$48x - 96 = 39x + 234$

$48x - 39x = 234 + 96$

$9x = 330$

$x = \frac{330}{9}$

$x \approx 36.67$

So, $x \approx 36.67$.

Would you like more details or have any questions?

Here are 5 questions you might want to ask:

1. How can I check if my solution for similar triangles is correct?
2. What are some common properties of similar triangles?
3. Can you explain the concept of proportionality in geometry?
4. How do similar triangles relate to the concept of scale factors?
5. What are some real-world applications of similar triangles?

Tip: When dealing with similar triangles, always set up the proportions carefully to ensure you're comparing corresponding sides correctly.