Simplify (x-7)/(x+1)-5/(x+6)

$\frac{x - 7}{x + 1} - \frac{5}{x + 6}$

Step 1

To write $\frac{x - 7}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{x - 7}{x + 1} \cdot \frac{x + 6}{x + 6} - \frac{5}{x + 6}$

Step 2

To write $- \frac{5}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{x - 7}{x + 1} \cdot \frac{x + 6}{x + 6} - \frac{5}{x + 6} \cdot \frac{x + 1}{x + 1}$

Step 3

Write each expression with a common denominator of $(x + 1) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x - 7}{x + 1}$ by $\frac{x + 6}{x + 6}$ .

$\frac{(x - 7) (x + 6)}{(x + 1) (x + 6)} - \frac{5}{x + 6} \cdot \frac{x + 1}{x + 1}$

Multiply $\frac{5}{x + 6}$ by $\frac{x + 1}{x + 1}$ .

$\frac{(x - 7) (x + 6)}{(x + 1) (x + 6)} - \frac{5 (x + 1)}{(x + 6) (x + 1)}$

Reorder the factors of $(x + 6) (x + 1)$ .

$\frac{(x - 7) (x + 6)}{(x + 1) (x + 6)} - \frac{5 (x + 1)}{(x + 1) (x + 6)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x - 7) (x + 6) - 5 (x + 1)}{(x + 1) (x + 6)}$

Step 5

Simplify the numerator.

Expand $(x - 7) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$\frac{x (x + 6) - 7 (x + 6) - 5 (x + 1)}{(x + 1) (x + 6)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 - 7 (x + 6) - 5 (x + 1)}{(x + 1) (x + 6)}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot 6 - 7 x - 7 \cdot 6 - 5 (x + 1)}{(x + 1) (x + 6)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 6 - 7 x - 7 \cdot 6 - 5 (x + 1)}{(x + 1) (x + 6)}$

Move $6$ to the left of $x$ .

$\frac{x^{2} + 6 \cdot x - 7 x - 7 \cdot 6 - 5 (x + 1)}{(x + 1) (x + 6)}$

Multiply $- 7$ by $6$ .

$\frac{x^{2} + 6 x - 7 x - 42 - 5 (x + 1)}{(x + 1) (x + 6)}$

Subtract $7 x$ from $6 x$ .

$\frac{x^{2} - x - 42 - 5 (x + 1)}{(x + 1) (x + 6)}$

Apply the distributive property.

$\frac{x^{2} - x - 42 - 5 x - 5 \cdot 1}{(x + 1) (x + 6)}$

Multiply $- 5$ by $1$ .

$\frac{x^{2} - x - 42 - 5 x - 5}{(x + 1) (x + 6)}$

Subtract $5 x$ from $- x$ .

$\frac{x^{2} - 6 x - 42 - 5}{(x + 1) (x + 6)}$

Subtract $5$ from $- 42$ .

$\frac{x^{2} - 6 x - 47}{(x + 1) (x + 6)}$