The problem shown in the image is asking to find the perimeter of a rectangle, with sides labeled by expressions involving the variable $w$. The lengths of the sides are:

• Two of the sides are labeled $7w - 6$
• The other two sides are labeled $2w$

Perimeter of a Rectangle Formula:

The perimeter $P$ of a rectangle is given by:

$P = 2 \times (\text{Length} + \text{Width})$

In this case, the lengths are $7w - 6$ and the widths are $2w$. Applying this to the formula:

$P = 2 \times \left( (7w - 6) + 2w \right)$

Simplification:

First, simplify the expression inside the parentheses:

$P = 2 \times (7w - 6 + 2w) = 2 \times (9w - 6)$

Now distribute the $2$:

$P = 2 \times 9w - 2 \times 6 = 18w - 12$

Final Answer:

The simplified expression for the perimeter is:

$P = 18w - 12$

Would you like more details or have any questions about this solution?

Related Questions:

1. What is the perimeter of the rectangle if $w = 2$?
2. How do you find the perimeter if the lengths were labeled $8w - 3$ and the widths were $w + 5$?
3. How does the perimeter change if the expressions for the sides were different, such as $5w + 4$ and $3w - 2$?
4. Can you write a general formula for the perimeter if the length is $aw + b$ and the width is $cw + d$?
5. What happens to the perimeter when $w$ becomes a negative number?

Tip:

Always remember to simplify the expression inside the parentheses before multiplying by 2 when calculating the perimeter of rectangles using algebraic expressions.