Solve for x (x-2)/(x-3)+3=(2x)/x

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

Step 1

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{2 x}{x}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

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

Step 2

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 3, 1, 1$

Remove parentheses.

$x - 3, 1, 1$

The LCM of one and any expression is the expression.

$x - 3$

Step 3

Multiply each term in $\frac{x - 2}{x - 3} + 3 = 2$ by $x - 3$ to eliminate the fractions.

Multiply each term in $\frac{x - 2}{x - 3} + 3 = 2$ by $x - 3$ .

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

$x - 2 + 3 x + 3 \cdot - 3 = 2 (x - 3)$

Multiply $3$ by $- 3$ .

$x - 2 + 3 x - 9 = 2 (x - 3)$

Simplify by adding terms.

Add $x$ and $3 x$ .

$4 x - 2 - 9 = 2 (x - 3)$

Subtract $9$ from $- 2$ .

$4 x - 11 = 2 (x - 3)$

Simplify the right side.

Apply the distributive property.

$4 x - 11 = 2 x + 2 \cdot - 3$

Multiply $2$ by $- 3$ .

$4 x - 11 = 2 x - 6$

Step 4

Solve the equation.

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$4 x - 11 - 2 x = - 6$

Subtract $2 x$ from $4 x$ .

$2 x - 11 = - 6$

Move all terms not containing $x$ to the right side of the equation.

Add $11$ to both sides of the equation.

$2 x = - 6 + 11$

Add $- 6$ and $11$ .

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Divide $x$ by $1$ .

$x = \frac{5}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}$

Decimal Form:

$x = 2.5$

Mixed Number Form:

$x = 2 \frac{1}{2}$

Solve for x (x-2)/(x-3)+3=(2x)/x

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