Divide and simplify

(3y-6)/14÷(y-2)/4y

I think I am supposed to multiply both sides by 4y. Help.

(3y - 6) / 14 + (y - 2) / 4y ,

Find a common denominator:
(6y^2 - 12y + 7y - 14) / 28y,
Combine like-terms:
(6y^2 - 5y - 14) / 28y,

(3y - 6) / 14 + (y - 2) / 4y ,

Find a common denominator:
(6y^2 - 12y + 7y - 14) / 28y,
Combine like-terms:
(6y^2 - 5y - 14) / 28y,
Use A*C Method to factor numerator:
A*C = 6 * -14 = -84 = 7 * -12,
6y^2 + (7y - 12y) - 14,
Group the 4 terms into 2 factorable pairs:

(6y^2 - 12y) + (7y - 14),
6y(y - 2) + 7(y - 2),
(y - 2) (6y + 7) / 28y.

To divide a fraction by another fraction, you can multiply the first fraction by the reciprocal of the second fraction. In this case, you have a fraction divided by another fraction:

(3y-6)/14 ÷ (y-2)/4y

To simplify this expression, you can multiply the first fraction by the reciprocal of the second fraction:

(3y-6)/14 × (4y)/(y-2)

Now, for the second part of your question, you mentioned that you want to multiply both sides by 4y. However, there are no sides in this problem. It seems you mistakenly applied the concept of multiplying both sides of an equation to this fraction division problem.

To solve this fraction division problem correctly, you need to multiply the fractions together, not multiply both sides by a value.

So, let's multiply the fractions together:

(3y-6)/14 × (4y)/(y-2)

To do this, multiply the numerators (top numbers) together, and multiply the denominators (bottom numbers) together:

(3y-6)(4y)/(14)(y-2)

Simplifying the numerator, you have:

(12y^2 - 24y)/(14)(y-2)

Now, you can simplify the denominator by multiplying 14 by (y-2):

(12y^2 - 24y)/(14y-28)

And that's your simplified result.