Solve for x (x+2)/x+(x-5)/(2x)=(2x+4)/(3x)

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

Factor ${\displaystyle 2}$ out of ${\displaystyle 4}$.

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

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

Since ${\displaystyle x,2x,3x}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,2,3}$ then find LCM for the variable part ${\displaystyle x^1,x^1,x^1}$.

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.

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

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

The LCM of ${\displaystyle 1,2,3}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

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

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

The LCM for ${\displaystyle x,2x,3x}$ is the numeric part ${\displaystyle 6}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{x+2}{x}+\frac{x-5}{2x}=\frac{2(x+2)}{3x}}$ by ${\displaystyle 6x}$.

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 ${\displaystyle 5x=11}$ by ${\displaystyle 5}$ and simplify.