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

##### 1 Answer

The solution is

#### Explanation:

In an equality you have that the left side of the

In this case

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

we start multiplying left and right for

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

$$(x-3)\frac{x}{x-3}=2(x-3)-(x-3)\frac{2}{x-3}$$

we simplify the

$$x=2(x-3)-2$$

$$x=2x-6-2$$

$$x=2x-8$$

now we add left and right $-2x$

$$x-2x=2x-2x-8$$

$$-x=-8$$

and finally we multiply for

$$-1(-x)=-1(-8)$$

$$x=8.$$

In order to verify if our result is correct we substitute it in the initial equation and we must obtain an identity.

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

$$\frac{8}{8-3}=2-\frac{2}{8-3}$$

$$\frac{8}{5}=2-\frac{2}{5}$$

$$\frac{8}{5}=\frac{2\times 5-2}{5}$$

$$\frac{8}{5}=\frac{10-2}{5}$$

$$\frac{8}{5}=\frac{8}{5}$$

so we are sure that our solution is correct.