How do you solve 1/2(x-3)+3/(2-x)=5x?

Dec 1, 2015

$x = 0 , \frac{15}{9}$

Explanation:

$\frac{x - 3}{2} + \frac{3}{2 - x} = 5 x$

Achieve a common denominator of $2 \left(2 - x\right)$.

$\frac{x - 3}{2} \left(\frac{2 - x}{2 - x}\right) + \left(\frac{3}{2 - x}\right) \left(\frac{2}{2}\right) = 5 x$

$\frac{\left(- {x}^{2} + 2 x + 3 x - 6\right) + 6}{4 - 2 x} = 5 x$

$- {x}^{2} + 5 x = 5 x \left(4 - 2 x\right)$

$- {x}^{2} + 5 x = - 10 {x}^{2} + 20 x$

$9 {x}^{2} - 15 x = 0$

$x \left(9 x - 15\right) = 0$

$\left\{\begin{matrix}x = 0 \\ 9 x - 15 = 0 \rightarrow x = \frac{15}{9}\end{matrix}\right.$

Check to make sure that neither answer will cause a denominator to be $0$.