A 5 inch × 7 inch photograph is placed inside a picture frame. Both the length and width of the frame are 2x inches larger than the width and length of the photograph. Which expression represents the perimeter of the frame?

Let's solve this step by step:

### Given:
- The dimensions of the photograph are \(5\) inches by \(7\) inches.
- The frame adds \(2x\) inches to both the length and width of the photograph.

### Find:
The expression for the perimeter of the frame.

### Step 1: Determine the dimensions of the frame
- **Length of the frame**: The length of the photograph is \(7\) inches. Since the frame adds \(2x\) inches to this dimension, the length of the frame is:
  \[
  7 + 2x \text{ inches}
  \]
  
- **Width of the frame**: The width of the photograph is \(5\) inches. Since the frame adds \(2x\) inches to this dimension, the width of the frame is:
  \[
  5 + 2x \text{ inches}
  \]

### Step 2: Expression for the perimeter
The perimeter \(P\) of a rectangle is given by:
\[
P = 2 \times \text{(Length + Width)}
\]

Substitute the values for the length and width:
\[
P = 2 \times \left( (7 + 2x) + (5 + 2x) \right)
\]

### Step 3: Simplify the expression
Combine the like terms inside the parentheses:
\[
P = 2 \times \left( 7 + 5 + 2x + 2x \right)
\]
\[
P = 2 \times \left( 12 + 4x \right)
\]

Finally, distribute the 2:
\[
P = 24 + 8x
\]

### Answer:
The expression that represents the perimeter of the frame is \(24 + 8x\).

Would you like more details or have any further questions?

Here are five related questions to explore:

1. How do you find the area of the frame, given these dimensions?
2. What happens to the perimeter if \(x\) doubles?
3. How does the expression for the perimeter change if the frame adds \(3x\) inches instead of \(2x\) inches?
4. How would you represent the area of the frame in terms of \(x\)?
5. How do the length and width affect the perimeter of a rectangle?

**Tip:** When dealing with problems involving rectangles, it's useful to remember that both the length and width contribute equally to the perimeter calculation.

Learn how to calculate the perimeter of a frame around a photograph with dimensions 5 inches by 7 inches. This problem involves finding the perimeter of a rectangle, adding 2x inches to both length and width dimensions. Suitable for students in Grades 6-8.

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