Simplify (1/x-3/(x+1))/(2/(x+1)-3/x)

$\frac{\frac{1}{x} - \frac{3}{x + 1}}{\frac{2}{x + 1} - \frac{3}{x}}$

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

Multiply $\frac{\frac{1}{x} - \frac{3}{x + 1}}{\frac{2}{x + 1} - \frac{3}{x}}$ by $\frac{x (x + 1)}{x (x + 1)}$ .

$\frac{x (x + 1)}{x (x + 1)} \cdot \frac{\frac{1}{x} - \frac{3}{x + 1}}{\frac{2}{x + 1} - \frac{3}{x}}$

Combine.

$\frac{x (x + 1) (\frac{1}{x} - \frac{3}{x + 1})}{x (x + 1) (\frac{2}{x + 1} - \frac{3}{x})}$

Apply the distributive property.

$\frac{x (x + 1) \frac{1}{x} + x (x + 1) (- \frac{3}{x + 1})}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Simplify by cancelling.

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{x (x + 1) \frac{1}{x} + x (x + 1) (- \frac{3}{x + 1})}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Rewrite the expression.

$\frac{x + 1 + x (x + 1) (- \frac{3}{x + 1})}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Cancel the common factor of $x + 1$ .

Move the leading negative in $- \frac{3}{x + 1}$ into the numerator.

$\frac{x + 1 + x (x + 1) \frac{- 3}{x + 1}}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

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

$\frac{x + 1 + (x + 1) x \frac{- 3}{x + 1}}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Cancel the common factor.

$\frac{x + 1 + (x + 1) x \frac{- 3}{x + 1}}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Rewrite the expression.

$\frac{x + 1 + x \cdot - 3}{x (x + 1) \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Cancel the common factor of $x + 1$ .

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

$\frac{x + 1 + x \cdot - 3}{(x + 1) x \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Cancel the common factor.

$\frac{x + 1 + x \cdot - 3}{(x + 1) x \frac{2}{x + 1} + x (x + 1) (- \frac{3}{x})}$

Rewrite the expression.

$\frac{x + 1 + x \cdot - 3}{x \cdot 2 + x (x + 1) (- \frac{3}{x})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{3}{x}$ into the numerator.

$\frac{x + 1 + x \cdot - 3}{x \cdot 2 + x (x + 1) \frac{- 3}{x}}$

Cancel the common factor.

$\frac{x + 1 + x \cdot - 3}{x \cdot 2 + x (x + 1) \frac{- 3}{x}}$

Rewrite the expression.

$\frac{x + 1 + x \cdot - 3}{x \cdot 2 + (x + 1) \cdot - 3}$

Simplify the numerator.

Move $- 3$ to the left of $x$ .

$\frac{x + 1 - 3 \cdot x}{x \cdot 2 + (x + 1) \cdot - 3}$

Subtract $3 x$ from $x$ .

$\frac{- 2 x + 1}{x \cdot 2 + (x + 1) \cdot - 3}$

Simplify the denominator.

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

$\frac{- 2 x + 1}{2 \cdot x + (x + 1) \cdot - 3}$

Apply the distributive property.

$\frac{- 2 x + 1}{2 x + x \cdot - 3 + 1 \cdot - 3}$

Move $- 3$ to the left of $x$ .

$\frac{- 2 x + 1}{2 x - 3 \cdot x + 1 \cdot - 3}$

Multiply $- 3$ by $1$ .

$\frac{- 2 x + 1}{2 x - 3 \cdot x - 3}$

Subtract $3 x$ from $2 x$ .

$\frac{- 2 x + 1}{- x - 3}$

Simplify terms.

Factor $- 1$ out of $- 2 x$ .

$\frac{- (2 x) + 1}{- x - 3}$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (2 x) - 1 (- 1)}{- x - 3}$

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

$\frac{- (2 x - 1)}{- x - 3}$

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$\frac{- 1 (2 x - 1)}{- x - 3}$

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

$\frac{- 1 (2 x - 1)}{- (x) - 3}$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{- 1 (2 x - 1)}{- (x) - 1 (3)}$

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{- 1 (2 x - 1)}{- (x + 3)}$

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{- 1 (2 x - 1)}{- 1 (x + 3)}$

Cancel the common factor.

$\frac{- 1 (2 x - 1)}{- 1 (x + 3)}$

Rewrite the expression.

$\frac{2 x - 1}{x + 3}$

Simplify (1/x-3/(x+1))/(2/(x+1)-3/x)

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