Precalculus Examples

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

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Combine $- 1$ and $\frac{x + 1}{x + 1}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 1$ by $1$ .

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

Step 3.3

Subtract $1$ from $1$ .

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

Step 3.4

Add $- x$ and $0$ .

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

Step 4

Simplify by multiplying through.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Rewrite using the commutative property of multiplication.

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

Step 5

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

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Step 5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2

Multiply $x$ by $1$ .

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

Step 5.3

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

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

Step 5.4

Raise $x$ to the power of $1$ .

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

Step 5.5

Raise $x$ to the power of $1$ .

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

Step 5.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.7

Add $1$ and $1$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x$ .

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Step 6.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

Reorder $- 1^{2}$ and $x^{2}$ .

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

Step 7.3

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 $b = 1$ .

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

Step 8

Cancel the common factor of $x + 1$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Divide $x - 1$ by $1$ .

$x - 1$