Algebra Examples

$\frac{\frac{x - 1}{x} - \frac{- 4}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$

Step 1

Multiply the numerator and denominator of the fraction by $x (x - 1)$ .

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Step 1.1

Multiply $\frac{\frac{x - 1}{x} - \frac{- 4}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$ by $\frac{x (x - 1)}{x (x - 1)}$ .

$\frac{x (x - 1)}{x (x - 1)} \cdot \frac{\frac{x - 1}{x} - \frac{- 4}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$

Step 1.2

Combine.

$\frac{x (x - 1) (\frac{x - 1}{x} - \frac{- 4}{x - 1})}{x (x - 1) (\frac{x + 1}{x} + \frac{x + 1}{x - 1})}$

Step 2

Apply the distributive property.

$\frac{x (x - 1) \frac{x - 1}{x} + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

$\frac{x (x - 1) \frac{x - 1}{x} + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.1.2

Rewrite the expression.

$\frac{(x - 1) (x - 1) + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.2

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

$\frac{{(x - 1)}^{1} (x - 1) + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.3

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

$\frac{{(x - 1)}^{1} {(x - 1)}^{1} + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.4

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

$\frac{{(x - 1)}^{1 + 1} + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.5

Add $1$ and $1$ .

$\frac{{(x - 1)}^{2} + x (x - 1) (- \frac{- 4}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.6

Cancel the common factor of $x - 1$ .

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Step 3.6.1

Move the leading negative in $- \frac{- 4}{x - 1}$ into the numerator.

$\frac{{(x - 1)}^{2} + x (x - 1) \frac{4}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.6.2

Factor $x - 1$ out of $x (x - 1)$ .

$\frac{{(x - 1)}^{2} + (x - 1) x \frac{4}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.6.3

Cancel the common factor.

$\frac{{(x - 1)}^{2} + (x - 1) x \frac{4}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.6.4

Rewrite the expression.

$\frac{{(x - 1)}^{2} + x \cdot 4}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.7

Cancel the common factor of $x$ .

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Step 3.7.1

Cancel the common factor.

$\frac{{(x - 1)}^{2} + x \cdot 4}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.7.2

Rewrite the expression.

$\frac{{(x - 1)}^{2} + x \cdot 4}{(x - 1) (x + 1) + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.8

Cancel the common factor of $x - 1$ .

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Step 3.8.1

Factor $x - 1$ out of $x (x - 1)$ .

$\frac{{(x - 1)}^{2} + x \cdot 4}{(x - 1) (x + 1) + (x - 1) x \frac{x + 1}{x - 1}}$

Step 3.8.2

Cancel the common factor.

$\frac{{(x - 1)}^{2} + x \cdot 4}{(x - 1) (x + 1) + (x - 1) x \frac{x + 1}{x - 1}}$

Step 3.8.3

Rewrite the expression.

$\frac{{(x - 1)}^{2} + x \cdot 4}{(x - 1) (x + 1) + x (x + 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

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

Step 4.2

Factor using the perfect square rule.

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Step 4.2.1

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

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

Step 4.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 u = 2 \cdot u \cdot 1$

Step 4.2.3

Rewrite the polynomial.

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

Step 4.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

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

Step 4.3

Replace all occurrences of $u$ with $x$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x + 1$ out of $(x - 1) (x + 1) + x (x + 1)$ .

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Step 5.1.1

Factor $x + 1$ out of $(x - 1) (x + 1)$ .

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

Step 5.1.2

Factor $x + 1$ out of $x (x + 1)$ .

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

Step 5.1.3

Factor $x + 1$ out of $(x + 1) (x - 1) + (x + 1) x$ .

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

Step 5.2

Add $x$ and $x$ .

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

Step 6

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 6.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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