Solve for x 1/(2x+2)+5/(x^2-1)=1/(x-1)

Factor ${\displaystyle 2}$ out of ${\displaystyle 2x+2}$.

Rewrite ${\displaystyle 1}$ as ${\displaystyle 1^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=1}$.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since ${\displaystyle 2}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 2}$.

${\displaystyle 2}$ is a prime number

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of ${\displaystyle 2,1,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle x+1}$ is ${\displaystyle x+1}$ itself.

${\displaystyle (x+1)}$ occurs ${\displaystyle 1}$ time.

The factor for ${\displaystyle x-1}$ is ${\displaystyle x-1}$ itself.

${\displaystyle (x-1)}$ occurs ${\displaystyle 1}$ time.

The LCM of ${\displaystyle x+1,x+1,x-1,x-1}$ is the result of multiplying all factors the greatest number of times they occur in either term.

The Least Common Multiple ${\displaystyle \mathrm{LCM}}$ of some numbers is the smallest number that the numbers are factors of.

Multiply each term in ${\displaystyle \frac{1}{2(x+1)}+\frac{5}{(x+1)(x-1)}=\frac{1}{x-1}}$ by ${\displaystyle 2(x+1)(x-1)}$.

Simplify the left side.

Simplify the right side.

Move all terms containing ${\displaystyle x}$ to the left side of the equation.

Move all terms not containing ${\displaystyle x}$ to the right side of the equation.

Divide each term in ${\displaystyle -x=-7}$ by ${\displaystyle -1}$ and simplify.