https://file.aiquickdraw.com/mathbot/file/1728497625038buvth626.jpg

The image presents four puzzles asking to find the perimeters of different rectangles, each with algebraic expressions for the side lengths. Here's how to approach each:

### 1. First Rectangle:
- **Length**: \( 3x + 2 \)
- **Width**: \( 2x \)
  
  **Perimeter formula**: 
  \[
  P = 2 \times (\text{Length} + \text{Width}) = 2 \times ((3x + 2) + 2x)
  \]
  Simplifying:
  \[
  P = 2 \times (5x + 2) = 10x + 4
  \]
  **Perimeter**: \( 10x + 4 \)

### 2. Second Rectangle:
- **Length**: \( x \)
- **Width**: \( 4x + 1 \)
  
  **Perimeter formula**:
  \[
  P = 2 \times (x + (4x + 1))
  \]
  Simplifying:
  \[
  P = 2 \times (5x + 1) = 10x + 2
  \]
  **Perimeter**: \( 10x + 2 \)

### 3. Third Rectangle:
- **Length**: \( x + 5 \)
- **Width**: \( 7 \)
  
  **Perimeter formula**:
  \[
  P = 2 \times ((x + 5) + 7)
  \]
  Simplifying:
  \[
  P = 2 \times (x + 12) = 2x + 24
  \]
  **Perimeter**: \( 2x + 24 \)

### 4. Fourth Rectangle:
- **Length**: \( 4x - 3 \)
- **Width**: \( 5 \)
  
  **Perimeter formula**:
  \[
  P = 2 \times ((4x - 3) + 5)
  \]
  Simplifying:
  \[
  P = 2 \times (4x + 2) = 8x + 4
  \]
  **Perimeter**: \( 8x + 4 \)

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Would you like any details on the steps or further explanations? Here are five questions to expand your learning:

1. How do you simplify expressions within parentheses when solving for the perimeter?
2. Can you explain why multiplying by 2 is necessary for the perimeter formula?
3. How would you solve these perimeter expressions if you were given a specific value for \( x \)?
4. What happens to the perimeter if you double the length and width expressions?
5. Can you calculate the area for any of these rectangles?

**Tip**: Always check if the algebraic expressions in a problem can be simplified before substituting values.

Can you find the perimeter of the rectangles with algebraic side lengths?

Learn how to solve perimeter problems for rectangles with algebraic side lengths. These problems involve applying algebra and geometry concepts to find perimeters using expressions like 3x+2 and 4x-3. Suitable for students in Grades 8-10.

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