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

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

Step 1

Simplify terms.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Subtract $2$ from $1$ .

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

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

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

Step 2

Simplify the numerator.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Step 3

Simplify terms.

Factor $\frac{x + 1}{x}$ out of $\frac{\frac{x + 1}{x} (x - 1)}{x + 1}$ .

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

Cancel the common factor of $x + 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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