Pre-Algebra Examples

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

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

Step 1

To write

\frac{1}{1 + x}

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

\frac{x}{x}

$\frac{x}{x}$ .

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

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

Step 2

To write

\frac{1 - x}{x}

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

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

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

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

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

Step 3

Write each expression with a common denominator of

(1 + x) x

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

1

$1$ .

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Step 3.1

Multiply

\frac{1}{1 + x}

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

\frac{x}{x}

$\frac{x}{x}$ .

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

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

Step 3.2

Multiply

\frac{1 - x}{x}

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

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

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

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

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

Step 3.3

Reorder the factors of

(1 + x) x

$(1 + x) x$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Expand

(1 - x) (1 + x)

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

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Step 5.1.1

Apply the distributive property.

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

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

Step 5.1.2

Apply the distributive property.

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

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

Step 5.1.3

Apply the distributive property.

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

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

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

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Multiply

1

$1$ by

1

$1$ .

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

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

Step 5.2.1.2

Multiply

x

$x$ by

1

$1$ .

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

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

Step 5.2.1.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 5.2.1.4

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 5.2.1.4.1

Move

x

$x$ .

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

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

Step 5.2.1.4.2

Multiply

x

$x$ by

x

$x$ .

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

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

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

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

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

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

Step 5.2.2

Subtract

x

$x$ from

x

$x$ .

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

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

Step 5.2.3

Add

1

$1$ and

0

$0$ .

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

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

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

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

Step 5.3

Reorder terms.

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

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

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor

- 1

$- 1$ out of

- x^{2}

$- x^{2}$ .

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

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

Step 6.2

Factor

- 1

$- 1$ out of

x

$x$ .

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

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

Step 6.3

Factor

- 1

$- 1$ out of

- (x^{2}) - 1 (- x)

$- (x^{2}) - 1 (- x)$ .

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

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

Step 6.4

Rewrite

1

$1$ as

- 1 (- 1)

$- 1 (- 1)$ .

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

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

Step 6.5

Factor

- 1

$- 1$ out of

- (x^{2} - x) - 1 (- 1)

$- (x^{2} - x) - 1 (- 1)$ .

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

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

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite

- (x^{2} - x - 1)

$- (x^{2} - x - 1)$ as

- 1 (x^{2} - x - 1)

$- 1 (x^{2} - x - 1)$ .

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

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

Step 6.6.2

Move the negative in front of the fraction.

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

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

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

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

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

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