Question

1. use the figure on the right to find the following.

a) find the area of the larger rectangle in simplest form.
(3x-7)5x 15x²-35x units²
b) find the area of the inside rectangle.
(x-8)(x-3) x²-3x-8x+24 x²-11x+24 units²
c) find the area of the shaded region.

Explanation:

Part (a)

Step1: Recall area of rectangle formula

The area of a rectangle is given by \( A = \text{length} \times \text{width} \). For the larger rectangle, the length is \( 3x - 7 \) and the width is \( 5x \).

Step2: Multiply the length and width

Using the distributive property (also known as the distributive law of multiplication over addition), we multiply \( 5x \) with each term in \( 3x - 7 \):
\[

$$\begin{align*} (3x - 7)(5x) &= 3x \times 5x - 7 \times 5x \\ &= 15x^2 - 35x \end{align*}$$

\]

Step1: Recall area of rectangle formula

The area of a rectangle is \( A = \text{length} \times \text{width} \). For the inside rectangle, the length is \( x - 3 \) and the width is \( x - 8 \).

Step2: Multiply the length and width

Using the FOIL method (First, Outer, Inner, Last) to multiply \( (x - 3)(x - 8) \):
\[

$$\begin{align*} (x - 3)(x - 8) &= x \times x + x \times (-8) + (-3) \times x + (-3) \times (-8) \\ &= x^2 - 8x - 3x + 24 \\ &= x^2 - 11x + 24 \end{align*}$$

\]

Step1: Understand the shaded region area

The area of the shaded region is the area of the larger rectangle minus the area of the inside rectangle.

Step2: Subtract the two areas

We have the area of the larger rectangle as \( 15x^2 - 35x \) and the area of the inside rectangle as \( x^2 - 11x + 24 \). So we subtract:
\[

$$\begin{align*} (15x^2 - 35x) - (x^2 - 11x + 24) &= 15x^2 - 35x - x^2 + 11x - 24 \\ &= (15x^2 - x^2) + (-35x + 11x) - 24 \\ &= 14x^2 - 24x - 24 \end{align*}$$

\]

Answer:

The area of the larger rectangle is \( 15x^2 - 35x \) square units.

Part (b)