This is a geometry problem that involves similar triangles. When two triangles are similar, their corresponding sides are proportional.

In the diagram, you have two triangles, and based on the way the sides are labeled, we can apply the proportionality rule:

$\frac{5}{17} = \frac{2x + 3}{10x - 9}$

Now, let's solve for $x$:

Step 1: Set up the proportion

$\frac{5}{17} = \frac{2x + 3}{10x - 9}$

Step 2: Cross-multiply

$5(10x - 9) = 17(2x + 3)$

Step 3: Expand both sides

$50x - 45 = 34x + 51$

Step 4: Move all terms involving $x$ to one side and constant terms to the other side

$50x - 34x = 51 + 45$ $16x = 96$

Step 5: Solve for $x$

$x = \frac{96}{16} = 6$

Thus, the value of $x$ is $6$.

Let me know if you'd like further details or have any questions.

Here are 5 related questions you could explore:

1. How do we identify when triangles are similar?
2. What are other methods to solve proportions in similar triangles?
3. How do we apply the Pythagorean theorem to right triangles in problems like this?
4. What is the relationship between angles and sides in similar triangles?
5. Can you always solve for $x$ using just one proportion in a problem like this?

Tip: When working with similar triangles, ensure that you're matching the corresponding sides correctly before setting up the proportion!