This problem involves two similar triangles within a larger triangle. Since the two triangles have proportional sides, we can set up a proportion to solve for $x$.

In triangle $\triangle GHI$, we have:

• Side $GH = 32$
• Side $HI = 16$

In triangle $\triangle FJI$, we have:

• Side $FJ = 16x - 8$
• Side $JI = 6x$

Since the triangles are similar, we can set up the proportion: $\frac{GH}{FJ} = \frac{HI}{JI}$

Substitute the given values: $\frac{32}{16x - 8} = \frac{16}{6x}$

Cross-multiply to solve for $x$: $32 \cdot 6x = 16 \cdot (16x - 8)$ $192x = 256x - 128$

Now, simplify and solve for $x$: $192x - 256x = -128$ $-64x = -128$ $x = 2$

Therefore, the value of $x$ is $2$.

Would you like more details on any step, or do you have further questions?

Here are some related questions to deepen understanding:

1. What properties make triangles similar?
2. How does the proportionality of sides help in finding unknown lengths?
3. Why did we set up a proportion between $GH/FJ$ and $HI/JI$?
4. How would the problem change if one of the side lengths in $\triangle GHI$ were different?
5. Can we use this method for triangles that are not similar? Why or why not?

Tip: When working with similar triangles, ensure all corresponding sides are correctly paired in your proportions for accurate solutions.