# How do you solve for x: (4x)/(x-3)= 2+12/(x-3)?

Apr 30, 2018

The steps are as follows: Set the denominators equal, add the two terms on the right, cross multiply, then solve for x using factoring.

#### Explanation:

We begin by setting the denominator of the 2 equal to the rest of the terms in the equation. To do this, we can assume 2 to be equal to $\frac{2}{1}$ . To get the common denominator, we must multiply both the numerator and denominator in $\frac{2}{1}$ by $x - 3$ .

Once this has been completed, it should come out to $\frac{2 x - 6}{x - 3}$. This number can be added to 12 (in $\frac{12}{x - 3}$) to get $2 x + 6$. This addition step is important as it allows us to cross multiply the two remaining terms, getting rid of the pesky fractions.

To cross multiply, set the two terms you have set up your equation like this. $\frac{2 x + 6}{x - 3} = \frac{4 x}{x - 3}$ Now to cross multiply just multiply the $x - 3$s with their respective opposite term to come out with a final equation of $2 {x}^{2} - 18 = 4 {x}^{2} - 12 x$.

Moving all of the terms to one side of the equation and dividing by common multipliers results in the quadratic ${x}^{2} - 6 x - 9$. Simply factoring this quadratic will result in the solutions $\left(x - 3\right) \left(x - 3\right)$. Flip the sign to get $x = 3$ and that's your answer. After checking, however, you should be able to see that your answer does not work in the original equation, making the answer in fact, no solution.

I hope this helped, if you have any further questions don't hesitate to ask.