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

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x - 3) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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

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

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

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

Reorder the factors of $(x - 3) (x - 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 3$ .

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

Subtract $x$ from $2 x$ .

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

Add $- 2$ and $3$ .

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