Question

1. a rectangle has a side length of -7x + 9 and a width of 3(x - 2). part a: calculate the perimeter of the rectangle in simplest form.

Explanation:

Step1: Expand the width expression

$$\begin{align*} 3(x - 2)&=3\times x-3\times2\\ &=3x - 6 \end{align*}$$

Step2: Recall the perimeter formula for a rectangle

The perimeter formula for a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Here, \(l=-7x + 9\) and \(w = 3x-6\).

Step3: Substitute the length and width into the perimeter formula

$$\begin{align*} P&=2((-7x + 9)+(3x - 6))\\ &=2(-7x+9 + 3x-6) \end{align*}$$

Step4: Combine like - terms inside the parentheses

$$\begin{align*} -7x+3x+9 - 6&=(-7 + 3)x+(9 - 6)\\ &=-4x + 3 \end{align*}$$

Step5: Distribute the 2

$$\begin{align*} P&=2(-4x + 3)\\ &=2\times(-4x)+2\times3\\ &=-8x+6 \end{align*}$$

Answer:

\(-8x + 6\)