A rectangle has a width of

x

1
inches and a length of
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
>
2
all values of x such that x greater than 2

all values of x such that
x
>
3
all values of x such that x greater than 3

all values of x such that
x
<

2
or
x
>
3
all values of x such that x less than negative 2 or x greater than 3

all values of x such that
x
<

3
or
x
>
2

1 answer

To find the values of \( x \) that will produce a rectangle with an area greater than 18 square inches, we start with the formula for the area \( A \) of a rectangle, which is given by:

\[ A = \text{length} \times \text{width} \]

For this rectangle, the width is \( x - 1 \) inches and the length is \( 3x \) inches. Thus, we have:

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

We want the area to be greater than 18 square inches:

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

To solve this inequality, first, we can move all terms to one side:

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

Next, divide each term by 3 to simplify:

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

Now we can factor the quadratic:

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

Next, we find the critical points by setting the factors equal to zero:

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

Now, we need to test the intervals determined by these critical points: \( (-\infty, -2) \), \( (-2, 3) \), and \( (3, \infty) \).

  1. Testing \( x < -2 \) (e.g., \( x = -3 \)): \[ (-3 - 3)(-3 + 2) = (-6)(-1) = 6 > 0 \quad \text{(True)} \]

  2. Testing \( -2 < x < 3 \) (e.g., \( x = 0 \)): \[ (0 - 3)(0 + 2) = (-3)(2) = -6 < 0 \quad \text{(False)} \]

  3. Testing \( x > 3 \) (e.g., \( x = 4 \)): \[ (4 - 3)(4 + 2) = (1)(6) = 6 > 0 \quad \text{(True)} \]

Combining these results, 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 \( x < -2 \) or \( x > 3 \).

Therefore, the correct response is:

all values of x such that \( x < -2 \) or \( x > 3 \).