Pre-Algebra Examples

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

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

Step 1

To write

\frac{x}{x - 1}

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

\frac{x + 1}{x + 1}

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

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

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

Step 2

To write

\frac{x}{x + 1}

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

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

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

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

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

Step 3

Write each expression with a common denominator of

(x - 1) (x + 1)

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

1

$1$ .

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Step 3.1

Multiply

\frac{x}{x - 1}

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

\frac{x + 1}{x + 1}

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

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

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

Step 3.2

Multiply

\frac{x}{x + 1}

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

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

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

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

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

Step 3.3

Reorder the factors of

(x - 1) (x + 1)

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Factor

x

$x$ out of

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

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

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

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

Step 5.2

Add

x

$x$ and

x

$x$ .

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

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

Step 5.3

Subtract

1

$1$ from

1

$1$ .

\frac{x (2 x + 0)}{(x + 1) (x - 1)}

$\frac{x (2 x + 0)}{(x + 1) (x - 1)}$

Step 5.4

Add

2 x

$2 x$ and

0

$0$ .

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

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

Step 5.5

Combine exponents.

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Step 5.5.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 5.5.2

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 5.5.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 5.5.4

Add

1

$1$ and

1

$1$ .

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

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

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

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

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

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