Pre-Algebra Examples

\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1}

$\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1^{2}}

$\frac{7 x}{x + 1} + \frac{7}{x^{2} - 1^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

\frac{7 x}{x + 1} + \frac{7}{(x + 1) (x - 1)}

$\frac{7 x}{x + 1} + \frac{7}{(x + 1) (x - 1)}$

\frac{7 x}{x + 1} + \frac{7}{(x + 1) (x - 1)}

$\frac{7 x}{x + 1} + \frac{7}{(x + 1) (x - 1)}$

Step 2

To write

\frac{7 x}{x + 1}

$\frac{7 x}{x + 1}$ as a fraction with a common denominator, multiply by

\frac{x - 1}{x - 1}

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

\frac{7 x}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{7}{(x + 1) (x - 1)}

$\frac{7 x}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{7}{(x + 1) (x - 1)}$

Step 3

Simplify terms.

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Step 3.1

Multiply

\frac{7 x}{x + 1}

$\frac{7 x}{x + 1}$ by

\frac{x - 1}{x - 1}

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

\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{7}{(x + 1) (x - 1)}

$\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{7}{(x + 1) (x - 1)}$

Step 3.2

Combine the numerators over the common denominator.

\frac{7 x (x - 1) + 7}{(x + 1) (x - 1)}

$\frac{7 x (x - 1) + 7}{(x + 1) (x - 1)}$

\frac{7 x (x - 1) + 7}{(x + 1) (x - 1)}

$\frac{7 x (x - 1) + 7}{(x + 1) (x - 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor

7

$7$ out of

7 x (x - 1) + 7

$7 x (x - 1) + 7$ .

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Step 4.1.1

Factor

7

$7$ out of

7 x (x - 1)

$7 x (x - 1)$ .

\frac{7 (x (x - 1)) + 7}{(x + 1) (x - 1)}

$\frac{7 (x (x - 1)) + 7}{(x + 1) (x - 1)}$

Step 4.1.2

Factor

7

$7$ out of

7

$7$ .

\frac{7 (x (x - 1)) + 7 (1)}{(x + 1) (x - 1)}

$\frac{7 (x (x - 1)) + 7 (1)}{(x + 1) (x - 1)}$

Step 4.1.3

Factor

7

$7$ out of

7 (x (x - 1)) + 7 (1)

$7 (x (x - 1)) + 7 (1)$ .

\frac{7 (x (x - 1) + 1)}{(x + 1) (x - 1)}

$\frac{7 (x (x - 1) + 1)}{(x + 1) (x - 1)}$

\frac{7 (x (x - 1) + 1)}{(x + 1) (x - 1)}

$\frac{7 (x (x - 1) + 1)}{(x + 1) (x - 1)}$

Step 4.2

Apply the distributive property.

\frac{7 (x \cdot x + x \cdot - 1 + 1)}{(x + 1) (x - 1)}

$\frac{7 (x \cdot x + x \cdot - 1 + 1)}{(x + 1) (x - 1)}$

Step 4.3

Multiply

x

$x$ by

x

$x$ .

\frac{7 (x^{2} + x \cdot - 1 + 1)}{(x + 1) (x - 1)}

$\frac{7 (x^{2} + x \cdot - 1 + 1)}{(x + 1) (x - 1)}$

Step 4.4

Move

- 1

$- 1$ to the left of

x

$x$ .

\frac{7 (x^{2} - 1 \cdot x + 1)}{(x + 1) (x - 1)}

$\frac{7 (x^{2} - 1 \cdot x + 1)}{(x + 1) (x - 1)}$

Step 4.5

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

\frac{7 (x^{2} - x + 1)}{(x + 1) (x - 1)}

$\frac{7 (x^{2} - x + 1)}{(x + 1) (x - 1)}$

\frac{7 (x^{2} - x + 1)}{(x + 1) (x - 1)}

$\frac{7 (x^{2} - x + 1)}{(x + 1) (x - 1)}$