Simplify (1/(2x)+2/(3x))/((x-1)/(x-3))

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{1}{2 x} + \frac{2}{3 x}) \frac{x - 3}{x - 1}$

Step 2

To write $\frac{1}{2 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{1}{2 x} \cdot \frac{3}{3} + \frac{2}{3 x}) \frac{x - 3}{x - 1}$

Step 3

To write $\frac{2}{3 x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(\frac{1}{2 x} \cdot \frac{3}{3} + \frac{2}{3 x} \cdot \frac{2}{2}) \frac{x - 3}{x - 1}$

Step 4

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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

$(\frac{3}{2 x \cdot 3} + \frac{2}{3 x} \cdot \frac{2}{2}) \frac{x - 3}{x - 1}$

Multiply $3$ by $2$ .

$(\frac{3}{6 x} + \frac{2}{3 x} \cdot \frac{2}{2}) \frac{x - 3}{x - 1}$

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

$(\frac{3}{6 x} + \frac{2 \cdot 2}{3 x \cdot 2}) \frac{x - 3}{x - 1}$

Multiply $2$ by $3$ .

$(\frac{3}{6 x} + \frac{2 \cdot 2}{6 x}) \frac{x - 3}{x - 1}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 + 2 \cdot 2}{6 x} \cdot \frac{x - 3}{x - 1}$

Step 6

Simplify the numerator.

Multiply $2$ by $2$ .

$\frac{3 + 4}{6 x} \cdot \frac{x - 3}{x - 1}$

Add $3$ and $4$ .

$\frac{7}{6 x} \cdot \frac{x - 3}{x - 1}$

Step 7

Multiply $\frac{7}{6 x}$ by $\frac{x - 3}{x - 1}$ .

$\frac{7 (x - 3)}{6 x (x - 1)}$