y+1/y-1 - y-1/y+1 = 8/3
I know that I have to multiply by the LCD of the denominaters, but what is the LCD of y-1, y+1, and 3?????
PLEASE HELP!!
~aShLeY
You want the denominator to be the least common multiple, LCM.
That LCM is 3(y-1)(y+1).
Once the denominators are the same on both sides, you can write this equation for the numerators only:
3(y+1)^2 - 3(y-1)^2 = 8(y+1)(y-1)
12y = 8 y^2 - 8
2y^2 -3y -2 = 0
(2y +1)(y -2) = 0
y = 2 or -1/2
3(y+1)(y-1) = 3 (y^2-1)
but you do not really need to know tha, use it in the original form.
for example the first term:
3(y+1)(y-1) [ (y+1)/(y-1) ]
is
3 (y+1)(y+1) = 3 (y+1)^2
the second term will give similarly
- 3 (y-1)^1
the comibation is then (again the difference of two squares)as in (A^2-B^2)=(A+B)(A-B)
3 [ (y+1)^2 -(y-1)^2 ]
which is
3 [ (y+1)+(y-1)] [ (y+1)-(y-1) ]
= 3 [ (2y)(2)] = 12 y
I will leave the right side for you to do. Check my arithmetic.
To find the least common denominator (LCD) for the denominators y-1, y+1, and 3, we need to factor each denominator completely and then identify the common factors.
Let's start by factoring the denominators:
1. Denominator y-1:
It is already in its factored form.
2. Denominator y+1:
It is already in its factored form.
3. Denominator 3:
Since 3 is a prime number, it cannot be factored any further.
Now, let's identify the common factors among the denominators:
1. Denominator y-1:
There are no common factors yet.
2. Denominator y+1:
There are no common factors yet.
3. Denominator 3:
There are no common factors yet.
As we can see, there are no common factors among y-1, y+1, and 3. Therefore, the LCD of these denominators is simply the product of all three denominators:
LCD = (y-1)(y+1)(3)
So, to solve the equation y+1/y-1 - y-1/y+1 = 8/3, you need to multiply both sides of the equation by the LCD:
(LCD) * (y+1/y-1 - y-1/y+1) = (LCD) * (8/3)
((y-1)(y+1)(3)) * (y+1/y-1) - ((y-1)(y+1)(3)) * (y-1/y+1) = (y-1)(y+1)(3) * (8/3)
Now, you can simplify the equation using the distributive property and solve for y.