Multiply. The quotient of the quantity x minus 5 over the quantity 4 times x plus 8 times the quantity 12 times x squared plus 32 times x plus 16
A. The quantity 3 times x plus 2 over 4 times the quantity x minus 5
B. The quantity x minus 5 times the quantity 3 times x plus 2, all over 4
C. The quantity x minus 5 times the quantity 3 times x plus 2
D. The quantity x minus 5 times the quantity 12 times x plus 8
The first step is to simplify the expression inside the quotient:
x - 5 / (4x + 8(12x^2 + 32x + 16))
x - 5 / (4x + 96x^2 + 256x + 128) (distributing 8 to the terms inside the parentheses)
x - 5 / (4(x + 24x^2 + 64x + 32)) (factoring out 4)
Now we can multiply by the numerator of the answer choices to see which one gives us the original expression:
A. (3x + 2) / 4(x - 5)
[(3x + 2) / 4(x - 5)] * [4(x + 24x^2 + 64x + 32) / (x - 5)] = (3x + 2)(x + 24x^2 + 64x + 32) / 1, which is not the original expression.
B. (x - 5)(3x + 2) / 4
[(x - 5)(3x + 2) / 4] * [4(x + 24x^2 + 64x + 32) / (x - 5)] = (x - 5)(3x + 2)(x + 24x^2 + 64x + 32) / 1, which is the original expression. Therefore, the answer is B.
C. (x - 5)(3x + 2)
[(x - 5)(3x + 2)] * [4(x + 24x^2 + 64x + 32) / (x - 5)] = (x - 5)(3x + 2)(4x + 96x^2 + 256x + 128) / (x - 5), which simplifies to (x - 5)(3x + 2)(4x + 96x^2 + 256x + 128) / 1, not the original expression.
D. (x - 5)(12x + 8)
[(x - 5)(12x + 8)] * [4(x + 24x^2 + 64x + 32) / (x - 5)] = (x - 5)(12x + 8)(4x + 96x^2 + 256x + 128) / (x - 5), which simplifies to (x - 5)(12x + 8)(4x + 96x^2 + 256x + 128) / 1, not the original expression.
Therefore, the answer is B.
The time t required to drive a certain distance varies inversely with the speed r. If it takes 4 hours to drive the distance at 35 miles per hour, how long will it take to drive the same distance at 45 miles per hour?
A. about 3.11 hours
B. 140 hours
C. about 5.14 hours
D. 393.75 hours
We know that the time is inversely proportional to the speed, which means that their product is constant:
t * r = k
We can solve for k using the information given:
4 hours * 35 miles per hour = k
k = 140
Now we can use k to find the time required at 45 miles per hour:
t * 45 miles per hour = 140
t = 140 / 45
t ≈ 3.11 hours
Therefore, the answer is A.
Bob and James are finishing the roof of a house. Working alone, Bob can shingle the roof in 10 hours. James can shingle the same roof in 16 hours. How long will it take them working together to shingle the roof? Round your answer to the nearest hundredth if necessary.
A. 13 hours
B. 26.67 hours
C. 6.15 hours
D. 0.04 hours
Let x be the number of hours it takes them working together to shingle the roof. Then we can set up the equation:
1/10 + 1/16 = 1/x
We're adding the rates at which Bob and James can shingle the roof since they're working together. Now we can solve for x:
8/80 + 5/80 = 1/x
13/80 = 1/x
x = 80/13 ≈ 6.15 hours
Therefore, the answer is C.
Simplify. The quantity x squared plus 3 times x minus 4 over the quantity x plus 4
A. 1 over the quantity x minus 4, where x does not equal 4
B. x minus 4
C. x minus 1, where x does not equal 1
D. x minus 1, where x does not equal negative 4
We can factor the numerator using the AC method:
x^2 + 3x - 4 = x^2 + 4x - x - 4 = x(x + 4) - 1(x + 4) = (x - 1)(x + 4)
Now we can rewrite the expression:
(x^2 + 3x - 4) / (x + 4) = [(x - 1)(x + 4)] / (x + 4)
The (x + 4) terms cancel out, leaving us with:
(x - 1) / 1
Therefore, the answer is simply:
x - 1, where x is not equal to -4.
What are the excluded values of the function?
y equals 3 over the quantity 4 times x plus 64
A. x = 0
B. x equals negative 64
C. x equals negative 16
D. x equals negative 8
The given function is:
y = 3 / (4x + 64)
The denominator cannot be equal to zero, so we need to find the value(s) of x that make it zero. Setting the denominator equal to zero, we get:
4x + 64 = 0
Solving for x, we get:
x = -16
Therefore, the excluded value of the function is x = -16.
So the answer is C.
Solve.
start fraction 8 over x end fraction plus start fraction 15 over x squared end fraction equals negative 1
A. x equals 5 comma negative 3
B. x equals negative 5 comma negative 3
C. x equals negative 5 comma 3
D. x = 5, 3
To solve the equation, we first need to combine the fractions on the left-hand side:
8/x + 15/x^2 = -1
Multiplying both sides by x^2 to clear the denominators, we get:
8x + 15 = -x^2
Moving all the terms to one side, we get a quadratic equation:
x^2 + 8x + 15 = 0
We can factor this equation to get:
(x + 3)(x + 5) = 0
Setting each factor equal to zero, we get:
x + 3 = 0 or x + 5 = 0
Solving for x, we get:
x = -3 or x = -5
Therefore, the answer is B. x equals negative 5 comma negative 3.
Simplify into one fraction. Fraction 1: negative 5 times x over the quantity x plus 3; Fraction 2: 7 over the quantity x plus 3. Find Fraction 1 minus Fraction 2.
A. The quantity negative 5 times x minus 7 over the quantity x plus 3
B. The quantity negative 5 times x plus 7 over the quantity x plus 3
C. The quantity negative 5 times x minus 7 over the quantity x plus 3 squared
D. The quantity negative 5 times x plus 7 over the quantity x plus 3 squared
First, we need to find a common denominator. The denominator for Fraction 1 is x + 3, so we just need to multiply Fraction 2 by -5x on both the numerator and denominator to get a common denominator:
Fraction 1: -5x / (x + 3)
Fraction 2: (7(-5x)) / (-5x)(x + 3) = -35x / (x + 3)
Now we can subtract them:
Fraction 1 - Fraction 2 = (-5x / (x + 3)) - (-35x / (x + 3))
Simplifying the right-hand side, we get:
(-5x + 35x) / (x + 3) = 30x / (x + 3)
Therefore, the answer is D. The quantity negative 5 times x plus 7 over the quantity x plus 3 squared.