Divide. Fraction 1: The quantity x squared plus 2 times x plus 1 over the quantity x squared minus 8 times x plus 16. Fraction 2: the quantity x plus 1 over the quantity x squared minus 16. Find Fraction 1 over Fraction 2.

Question

A. The quantity x minus 1 times the quantity x plus 4, all over the quantity x plus 1
B. The quantity x plus 1 times the quantity x plus 4, all over the quantity x minus 4
C. The quantity x plus 1 times the quantity x plus 4, all over the quantity x minus 1
D. The quantity x minus 1 times the quantity x minus 4, all over the quantity x plus 4

Answer 1

To divide the two fractions, we need to multiply the first fraction by the reciprocal of the second fraction. The reciprocal of Fraction 2 is the quantity x squared minus 16 over the quantity x plus 1.

Therefore,

Fraction 1 ÷ Fraction 2 = (x^2 + 2x + 1)/(x^2 - 8x + 16) ÷ (x + 1)/(x^2 - 16)
= (x^2 + 2x + 1)/(x^2 - 8x + 16) * (x^2 - 16)/(x + 1)
= [(x + 1)(x + 1)]/[(x - 4)(x - 4)] * [(x - 4)(x + 4)]/(x + 1)
= (x + 1) * (x + 4) / (x - 4)

Therefore, the answer is (A) The quantity x minus 1 times the quantity x plus 4, all over the quantity x plus 1.

Answer 2

Simplify into one fraction.
start fraction 3 over x minus 3 end fraction minus start fraction 5 over x minus 2 end fraction
A. start fraction negative 2 x plus 9 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis
B. start fraction negative 2 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction
C. start fraction negative 2 left parenthesis x plus 9 right parenthesis over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction
D. start fraction 8 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis end fraction

Answer 3

To simplify the expression, we need to find a common denominator for the two fractions. The common denominator is (x - 3)(x - 2).

Therefore,

start fraction 3 over x minus 3 end fraction minus start fraction 5 over x minus 2 end fraction
= start fraction 3(x - 2)/[(x - 3)(x - 2)] - 5(x - 3)/[(x - 3)(x - 2)] end fraction
= start fraction (3x - 6 - 5x + 15)/[(x - 3)(x - 2)] end fraction
= start fraction (-2x + 9)/[(x - 3)(x - 2)] end fraction

Therefore, the answer is (A) start fraction negative 2 x plus 9 over left parenthesis x minus 3 right parenthesis left parenthesis x minus 2 right parenthesis.

Answer 4

Solve.

The quotient 1 over the quantity 3 times x minus 6 minus the quotient 5 over the quantity x minus 2 equals 12
A. x equals 34 over 9
B. x equals negative 29 over 18
C. x equals negative 34 over 9
D. x equals 29 over 18

Answer 5

To solve the equation, we need to simplify the left side first.

Start with the first quotient:

start fraction 1 over 3x - 6 end fraction

We can factor out a 3 from the denominator:

start fraction 1 over 3(x - 2) end fraction

Now for the second quotient:

start fraction 5 over x - 2 end fraction

The LCD for this expression and 3(x - 2) is (x - 2)(3x - 6) = 3(x - 2)^2. We can rewrite the second quotient with this denominator:

start fraction 5(3x - 6) over 3(x - 2)^2 end fraction

Simplifying this expression:

start fraction 15x - 30 over 3(x - 2)^2 end fraction

Combining the two quotients:

start fraction 1 over 3(x - 2) end fraction - start fraction 15x - 30 over 3(x - 2)^2 end fraction

LCD for these two fractions is 3(x - 2)^2. Rewriting the expression with this denominator and simplifying:

start fraction (x - 2) - 5(3x - 6) end fraction / 3(x - 2)^2 = start fraction (-14x + 32)/3(x - 2)^2 end fraction

Now we can solve for x:

start fraction (-14x + 32)/3(x - 2)^2 = 12 end fraction

Multiplying both sides by 3(x - 2)^2:

-14x + 32 = 36(x - 2)^2

Expanding the right side:

-14x + 32 = 36(x^2 - 4x + 4)

Simplifying:

-14x + 32 = 36x^2 - 144x + 144

Rearranging and simplifying:

36x^2 - 130x + 112 = 0

We can solve this quadratic equation by factoring:

(9x - 28)(4x - 4) = 0

Solving for x, we get:

x = 28/9 or x = 1

However, we must check these solutions in the original equation to see if any of them produce denominators of zero.

For x = 1, the first quotient has a denominator of zero, which is not allowed.

For x = 28/9, the expression is valid:

start fraction 1 over 3(28/9) - 6 end fraction - start fraction 5 over 28/9 - 2 end fraction = 12

Simplifying and solving:

start fraction -1 over 25/3 end fraction = 12

-3/25 = 12

This is false, so there are no solutions to the equation.

Therefore, the answer is none of the above.