Algebra Examples

$\frac{\frac{x + 5}{x} - \frac{8}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$

Step 1

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

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{x + 5}{x} - \frac{8}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$ by $\frac{x (x - 1)}{x (x - 1)}$ .

$\frac{x (x - 1)}{x (x - 1)} \cdot \frac{\frac{x + 5}{x} - \frac{8}{x - 1}}{\frac{x + 1}{x} + \frac{x + 1}{x - 1}}$

Step 1.2

Combine.

$\frac{x (x - 1) (\frac{x + 5}{x} - \frac{8}{x - 1})}{x (x - 1) (\frac{x + 1}{x} + \frac{x + 1}{x - 1})}$

Step 2

Apply the distributive property.

$\frac{x (x - 1) \frac{x + 5}{x} + x (x - 1) (- \frac{8}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

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.

$\frac{x (x - 1) \frac{x + 5}{x} + x (x - 1) (- \frac{8}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.1.2

Rewrite the expression.

$\frac{(x - 1) (x + 5) + x (x - 1) (- \frac{8}{x - 1})}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.2

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 3.2.1

Move the leading negative in $- \frac{8}{x - 1}$ into the numerator.

$\frac{(x - 1) (x + 5) + x (x - 1) \frac{- 8}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.2.2

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

$\frac{(x - 1) (x + 5) + (x - 1) x \frac{- 8}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.2.3

Cancel the common factor.

$\frac{(x - 1) (x + 5) + (x - 1) x \frac{- 8}{x - 1}}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.2.4

Rewrite the expression.

$\frac{(x - 1) (x + 5) + x \cdot - 8}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.1

Cancel the common factor.

$\frac{(x - 1) (x + 5) + x \cdot - 8}{x (x - 1) \frac{x + 1}{x} + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.3.2

Rewrite the expression.

$\frac{(x - 1) (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + x (x - 1) \frac{x + 1}{x - 1}}$

Step 3.4

Cancel the common factor of $x - 1$ .

Tap for more steps...

Step 3.4.1

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

$\frac{(x - 1) (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + (x - 1) x \frac{x + 1}{x - 1}}$

Step 3.4.2

Cancel the common factor.

$\frac{(x - 1) (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + (x - 1) x \frac{x + 1}{x - 1}}$

Step 3.4.3

Rewrite the expression.

$\frac{(x - 1) (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Expand $(x - 1) (x + 5)$ using the FOIL Method.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$\frac{x (x + 5) - 1 (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 1 (x + 5) + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot 5 - 1 x - 1 \cdot 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.2

Simplify and combine like terms.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 5 - 1 x - 1 \cdot 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.2.1.2

Move $5$ to the left of $x$ .

$\frac{x^{2} + 5 \cdot x - 1 x - 1 \cdot 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.2.1.3

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

$\frac{x^{2} + 5 x - x - 1 \cdot 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.2.1.4

Multiply $- 1$ by $5$ .

$\frac{x^{2} + 5 x - x - 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.2.2

Subtract $x$ from $5 x$ .

$\frac{x^{2} + 4 x - 5 + x \cdot - 8}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.3

Move $- 8$ to the left of $x$ .

$\frac{x^{2} + 4 x - 5 - 8 \cdot x}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.4

Subtract $8 x$ from $4 x$ .

$\frac{x^{2} - 4 x - 5}{(x - 1) (x + 1) + x (x + 1)}$

Step 4.5

Factor $x^{2} - 4 x - 5$ using the AC method.

Tap for more steps...

Step 4.5.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 4.5.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 1)}{(x - 1) (x + 1) + x (x + 1)}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

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

$\frac{(x - 5) (x + 1)}{(x + 1) (x - 1) + x (x + 1)}$

Step 5.1.2

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

$\frac{(x - 5) (x + 1)}{(x + 1) (x - 1) + (x + 1) x}$

Step 5.1.3

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

$\frac{(x - 5) (x + 1)}{(x + 1) (x - 1 + x)}$

Step 5.2

Add $x$ and $x$ .

$\frac{(x - 5) (x + 1)}{(x + 1) (2 x - 1)}$

Step 6

Cancel the common factor of $x + 1$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{(x - 5) (x + 1)}{(x + 1) (2 x - 1)}$

Step 6.2

Rewrite the expression.

$\frac{x - 5}{2 x - 1}$