Solve the Rational Equation for x 2/(3x)+3/4=1/x

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

Since ${\displaystyle 3x,x,4}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 3,1,4}$ then find LCM for the variable part ${\displaystyle x^1,x^1}$.

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

Since ${\displaystyle 3}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 3}$.

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

${\displaystyle 4}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 2}$.

The LCM of ${\displaystyle 3,1,4}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

Multiply ${\displaystyle 2\cdot 2\cdot 3}$.

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

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

The LCM for ${\displaystyle 3x,x,4}$ is the numeric part ${\displaystyle 12}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{2}{3x}=\frac{1}{x}-\frac{3}{4}}$ by ${\displaystyle 12x}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle 12-9x=8}$.

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

Divide each term in ${\displaystyle -9x=-4}$ by ${\displaystyle -9}$ and simplify.