Given that the length of the rectangle is 3 units shorter than one-third of the width $x$, we can start by expressing the length in terms of $x$.

Step 1: Express the length in terms of $x$.

• The width is $x$.
• The length is $\frac{1}{3}x - 3$.

Step 2: Write the expression for the perimeter of the rectangle.

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

Substituting the expressions for the length and width: $P = 2 \times \left(\frac{1}{3}x - 3 + x\right)$

Step 3: Simplify the expression inside the parentheses.

Combine like terms: $P = 2 \times \left(\frac{1}{3}x + \frac{3}{3}x - 3\right) = 2 \times \left(\frac{4}{3}x - 3\right)$

Step 4: Distribute the 2.

$P = \frac{8}{3}x - 6$

So, the correct expression for the perimeter of the rectangle is: $\boxed{\frac{8}{3}x - 6}$

Would you like more details on this? If so, I can break down any specific step further.

Here are some related questions:

1. How do you determine the perimeter of different shapes?
2. What is the formula for the area of a rectangle?
3. How do you solve linear equations involving fractions?
4. Can you explain how to combine like terms in an algebraic expression?
5. What are other ways to express the perimeter of this rectangle?

Tip: Always simplify expressions inside parentheses first before multiplying or distributing terms.