Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

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

Step 4.2

Subtract $3 x$ from $x$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

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

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

Step 5.4

Multiply $- 3$ by $1$ .

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

Step 5.5

Subtract $3 x$ from $2 x$ .

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

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

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

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

Cancel the common factor.

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

Step 6.10

Rewrite the expression.

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