Question
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
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
Answers
GPT-4o mini
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 \( x > 3 \)**.
\[
A = \text{length} \times \text{width}
\]
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 \( x > 3 \)**.