A rectangle has a width of x−1

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1
inches and a length of 3x
3
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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
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>
3
all values of x such that x is greater than 3 - no response given

all values of x such that x<−3
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<

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

3
or x is greater than 2 - incorrect

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

all values of x such that x<−2
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<

2
or x>3

1 answer

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 \( x > 3 \).