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

Jul 10, 2016

The soln. set is $\left\{0 , \frac{5}{3}\right\} .$

#### Explanation:

$\frac{1}{2} \cdot \left(x - 3\right) + \frac{3}{2 - x} = 5 x .$

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

$\therefore \left(x - 3\right) \left(2 - x\right) + 6 = 10 x \left(2 - x\right) .$

$\therefore \left(x - 3\right) \left(2 - x\right) - 10 x \left(2 - x\right) + 6 = 0.$

$\therefore \left(2 - x\right) \left\{x - 3 - 10 x\right\} + 6 = 0.$

$\therefore \left(2 - x\right) \left(- 9 x - 3\right) + 6 = 0.$

$\therefore \left(x - 2\right) \left(9 x + 3\right) + 6 = 0.$

$\therefore 9 {x}^{2} - 18 x + 3 x - 6 + 6 = 0.$

$\therefore 9 {x}^{2} - 15 x = 0.$

$\therefore 3 x \left(3 x - 5\right) = 0.$

$\therefore x = 0 , \mathmr{and} , x = \frac{5}{3.}$

We can easily verify that these roots satisfy the given eqn. Therefore,

The soln. set is $\left\{0 , \frac{5}{3}\right\} .$