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

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Factor $3$ out of $3 (x + 3) - 3 (x - 1)$ .

Factor $3$ out of $- 3 (x - 1)$ .

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

Factor $3$ out of $3 (x + 3) + 3 (- (x - 1))$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $x$ .

$\frac{3 (0 + 3 + 1)}{(x + 3) (x - 1)}$

Add $0$ and $3$ .

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

Add $3$ and $1$ .

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

Step 6

Multiply $3$ by $4$ .

$\frac{12}{(x + 3) (x - 1)}$