Simplify ((x+2)/x-2/(x-1))/((x+1)/x+(x+1)/(x-1))

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Move the leading negative in $- \frac{2}{x - 1}$ into the numerator.

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Expand $(x - 1) (x + 2)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $2$ to the left of $x$ .

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

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

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

Multiply $- 1$ by $2$ .

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

Subtract $x$ from $2 x$ .

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

Move $- 2$ to the left of $x$ .

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

Subtract $2 x$ from $x$ .

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

Factor $x^{2} - x - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Write the factored form using these integers.

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

Step 5

Simplify the denominator.

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

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

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

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

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

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

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

Add $x$ and $x$ .

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

Step 6

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x+2)/x-2/(x-1))/((x+1)/x+(x+1)/(x-1))

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