Algebra Examples

$\frac{4}{x + 1} + \frac{x + 2}{x^{2} - 1} + \frac{3}{x - 1}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{4}{x + 1} + \frac{x + 2}{x^{2} - 1^{2}} + \frac{3}{x - 1}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{4}{x + 1} + \frac{x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 2

To write $\frac{4}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{4}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{4}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{4 (x - 1)}{(x + 1) (x - 1)} + \frac{x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 (x - 1) + x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{4 x + 4 \cdot - 1 + x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 4.2

Multiply $4$ by $- 1$ .

$\frac{4 x - 4 + x + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 4.3

Add $4 x$ and $x$ .

$\frac{5 x - 4 + 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 4.4

Add $- 4$ and $2$ .

$\frac{5 x - 2}{(x + 1) (x - 1)} + \frac{3}{x - 1}$

Step 5

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

$\frac{5 x - 2}{(x + 1) (x - 1)} + \frac{3}{x - 1} \cdot \frac{x + 1}{x + 1}$

Step 6

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

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Step 6.1

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

$\frac{5 x - 2}{(x + 1) (x - 1)} + \frac{3 (x + 1)}{(x - 1) (x + 1)}$

Step 6.2

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

$\frac{5 x - 2}{(x + 1) (x - 1)} + \frac{3 (x + 1)}{(x + 1) (x - 1)}$

Step 7

Combine the numerators over the common denominator.

$\frac{5 x - 2 + 3 (x + 1)}{(x + 1) (x - 1)}$

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

$\frac{5 x - 2 + 3 x + 3 \cdot 1}{(x + 1) (x - 1)}$

Step 8.2

Multiply $3$ by $1$ .

$\frac{5 x - 2 + 3 x + 3}{(x + 1) (x - 1)}$

Step 8.3

Add $5 x$ and $3 x$ .

$\frac{8 x - 2 + 3}{(x + 1) (x - 1)}$

Step 8.4

Add $- 2$ and $3$ .

$\frac{8 x + 1}{(x + 1) (x - 1)}$