The problem presents a square with side lengths labeled as expressions. We are tasked to find the value of $x$.

Given information:

• One side of the square (e.g., $AB$) is labeled $3x + 4$.
• The opposite side (e.g., $BC$) is labeled $9x - 8$.

Approach:

Since the figure is a square, all sides must be equal. Therefore, we can set the two expressions for the sides equal to each other and solve for $x$.

$3x + 4 = 9x - 8$

Step 1: Solve for $x$

1. Subtract $3x$ from both sides: $4 = 6x - 8$
2. Add 8 to both sides: $12 = 6x$
3. Divide by 6: $x = 2$

Thus, the value of $x$ is $\boxed{2}$.

Would you like more details or have any further questions?

Here are some related questions to consider:

1. What would happen if the figure was not a square but a rectangle?
2. How can we check that the solution for $x = 2$ is correct?
3. What is the length of each side of the square when $x = 2$?
4. What if a different pair of sides were given? Would the process change?
5. How do you solve for $x$ if more complex expressions were given for the sides?

Tip: Whenever solving for $x$ in geometry problems involving squares or rectangles, remember that equality of sides is key to setting up equations.