Question

A rectangle has a width of x−1
𝑥
−
1
inches and a length of 3x
3
𝑥
inches. Find the values of x that will produce a rectangle that has an area greater than 18 square inches.(1 point)
Responses

all values of x such that x>3
𝑥
>
3
all values of x such that x is greater than 3 - no response given

all values of x such that x<−3
𝑥
<
−
3
or x>2
𝑥
>
2
all values of x such that x<−3
𝑥
<
−
3
or x is greater than 2 - incorrect

all values of x such that x>2
𝑥
>
2
all values of x such that x is greater than 2 - no response given

all values of x such that x<−2
𝑥
<
−
2
or x>3

Answer 1

To find the values of \( x \) that will produce a rectangle with an area greater than 18 square inches, we can start by calculating the area \( A \) of the rectangle using the given dimensions:

- Width: \( x - 1 \)
- Length: \( 3x \)

The area \( A \) is given by:

\[
A = \text{Width} \times \text{Length} = (x - 1)(3x)
\]

Expanding this expression:

\[
A = 3x(x - 1) = 3x^2 - 3x
\]

We want to find when this area is greater than 18 square inches:

\[
3x^2 - 3x > 18
\]

To solve this inequality, we first rearrange it:

\[
3x^2 - 3x - 18 > 0
\]

Next, we can simplify it by dividing everything by 3:

\[
x^2 - x - 6 > 0
\]

Now, we can factor the quadratic expression:

\[
(x - 3)(x + 2) > 0
\]

Now we will determine the critical points by setting the factors equal to zero:

1. \( x - 3 = 0 \Rightarrow x = 3 \)
2. \( x + 2 = 0 \Rightarrow x = -2 \)

These critical points divide the number line into intervals. We will test the intervals to see where the product is positive:

1. Interval \( (-\infty, -2) \):
- Choose \( x = -3 \): \(( -3 - 3)( -3 + 2) = (-6)(-1) > 0\) (True)

2. Interval \( (-2, 3) \):
- Choose \( x = 0 \): \((0 - 3)(0 + 2) = (-3)(2) < 0\) (False)

3. Interval \( (3, \infty) \):
- Choose \( x = 4 \): \((4 - 3)(4 + 2) = (1)(6) > 0\) (True)

So the solution to the inequality \( (x - 3)(x + 2) > 0 \) is:

\[
x < -2 \quad \text{or} \quad x > 3
\]

Thus, the values of \( x \) that will produce a rectangle with an area greater than 18 square inches are all values of \( x \) such that:

\[
x < -2 \quad \text{or} \quad x > 3
\]

Given the provided choices, the correct response would be: **all values of x such that \( x < -2 \) or \( x > 3 \)**.