Examples

\frac{3}{x + 1} + \frac{6}{x - 3}

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

Step 1

To write

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

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

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

Step 2

To write

$\frac{6}{x - 3}$ as a fraction with a common denominator, multiply by

$\frac{x + 1}{x + 1}$ .

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

Step 3

Write each expression with a common denominator of

$(x + 1) (x - 3)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.1

Multiply

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

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

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

Step 3.2

Multiply

$\frac{6}{x - 3}$ by

$\frac{x + 1}{x + 1}$ .

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

Step 3.3

Reorder the factors of

$(x - 3) (x + 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Factor

$3$ out of

$3 (x - 3) + 6 (x + 1)$ .

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Step 5.1.1

Factor

$3$ out of

$6 (x + 1)$ .

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

Step 5.1.2

Factor

$3$ out of

$3 (x - 3) + 3 (2 (x + 1))$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply

$2$ by

$1$ .

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

Step 5.4

Add

$x$ and

$2 x$ .

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

Step 5.5

Add

$- 3$ and

$2$ .

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

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