Solve for x 9/(x-1)=18/(x^2-1)

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.

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

The LCM of ${\displaystyle 1,1}$ 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.

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

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

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

Multiply each term in ${\displaystyle \frac{9}{x-1}=\frac{18}{(x+1)(x-1)}}$ by ${\displaystyle (x-1)(x+1)}$.

Simplify the left side.

Simplify the right side.

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

Divide each term in ${\displaystyle 9x=9}$ by ${\displaystyle 9}$ and simplify.