Solve by Factoring x/(x-2)=2/(x-2)+3

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

Step 1

Move all the expressions to the left side of the equation.

Subtract $\frac{2}{x - 2}$ from both sides of the equation.

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

Subtract $3$ from both sides of the equation.

$\frac{x}{x - 2} - \frac{2}{x - 2} - 3 = 0$

Step 2

Simplify $\frac{x}{x - 2} - \frac{2}{x - 2} - 3$ .

Find the common denominator.

Write $- 3$ as a fraction with denominator $1$ .

$\frac{x}{x - 2} - \frac{2}{x - 2} + \frac{- 3}{1} = 0$

Multiply $\frac{- 3}{1}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x}{x - 2} - \frac{2}{x - 2} + \frac{- 3}{1} \cdot \frac{x - 2}{x - 2} = 0$

Multiply $\frac{- 3}{1}$ by $\frac{x - 2}{x - 2}$ .

$\frac{x}{x - 2} - \frac{2}{x - 2} + \frac{- 3 (x - 2)}{x - 2} = 0$

Combine the numerators over the common denominator.

$\frac{x - 2 - 3 (x - 2)}{x - 2} = 0$

Simplify each term.

Apply the distributive property.

$\frac{x - 2 - 3 x - 3 \cdot - 2}{x - 2} = 0$

Multiply $- 3$ by $- 2$ .

$\frac{x - 2 - 3 x + 6}{x - 2} = 0$

Subtract $3 x$ from $x$ .

$\frac{- 2 x - 2 + 6}{x - 2} = 0$

Add $- 2$ and $6$ .

$\frac{- 2 x + 4}{x - 2} = 0$

Factor $2$ out of $- 2 x + 4$ .

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) + 4}{x - 2} = 0$

Factor $2$ out of $4$ .

$\frac{2 (- x) + 2 (2)}{x - 2} = 0$

Factor $2$ out of $2 (- x) + 2 (2)$ .

$\frac{2 (- x + 2)}{x - 2} = 0$

Cancel the common factor of $- x + 2$ and $x - 2$ .

Factor $- 1$ out of $- x$ .

$\frac{2 (- (x) + 2)}{x - 2} = 0$

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{2 (- (x) - 1 (- 2))}{x - 2} = 0$

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{2 (- (x - 2))}{x - 2} = 0$

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{2 (- 1 (x - 2))}{x - 2} = 0$

Cancel the common factor.

$\frac{2 (- 1 (x - 2))}{x - 2} = 0$

Divide $2 \cdot (- 1)$ by $1$ .

$2 \cdot (- 1) = 0$

Multiply $2$ by $- 1$ .

$- 2 = 0$

Step 3

Since $- 2 \neq 0$ , there are no solutions.

No solution

Solve by Factoring x/(x-2)=2/(x-2)+3

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