The problem asks for the equation to solve for $x$ when the perimeter of the rectangle is 384 inches.

Step 1: Identify the sides of the rectangle.

• The length of the rectangle is given as $5(3x - 9)$.
• The width of the rectangle is given as $2(8x - 4)$.

Step 2: Formula for the perimeter of a rectangle.

The perimeter $P$ of a rectangle is calculated using the formula: $P = 2 \times (\text{Length} + \text{Width})$

Step 3: Substitute the given expressions into the formula.

$P = 2 \times \left[5(3x - 9) + 2(8x - 4)\right]$

Step 4: Set the perimeter equal to 384 inches.

$2 \times \left[5(3x - 9) + 2(8x - 4)\right] = 384$

Step 5: Solve the equation.

1. Distribute the constants inside the parentheses: $5(3x - 9) = 15x - 45$ $2(8x - 4) = 16x - 8$
2. Combine like terms: $2 \times (15x - 45 + 16x - 8) = 384$ $2 \times (31x - 53) = 384$
3. Divide both sides by 2: $31x - 53 = 192$
4. Solve for $x$: $31x = 192 + 53$ $31x = 245$ $x = \frac{245}{31} \approx 7.9$

None of the multiple-choice answers seem to match this result exactly. It is possible that the correct answer was not simplified properly or involves an approximation. However, based on standard calculation, $x \approx 7.9$.

Would you like more details on how to interpret the problem further, or do you have any other questions?

Here are some related questions you might find helpful:

1. How do you calculate the perimeter of other geometric shapes?
2. What are common mistakes to avoid when solving perimeter problems?
3. How do you solve quadratic equations derived from geometry problems?
4. Can you explain the concept of distributing and combining like terms?
5. How do you handle decimal approximations in perimeter calculations?

Tip: Always double-check your work by substituting your solution back into the original equation to see if it satisfies the given condition.