Pre-Algebra Examples

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

Step 1

Rewrite the function as an equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

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

Step 2.5.1.2

Apply the distributive property.

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

Step 2.5.1.3

Apply the distributive property.

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

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $x$ by $x$ .

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

Step 2.5.2.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.5.2.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.5.2.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.5.2.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.5.2.2

Subtract $x$ from $- x$ .

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

Step 2.5.3

Subtract $2 x$ from $x$ .

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

Step 2.5.4

Add $1$ and $1$ .

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

Step 2.5.5

Reorder terms.

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