Simplify 1/(x+1)-(5x-2)/(x^2-5x-6)

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

Factor $x^{2} - 5 x - 6$ using the AC method.

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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $- 1$ .

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

Multiply $- 1$ by $- 2$ .

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

Subtract $5 x$ from $x$ .

$\frac{- 4 x - 6 + 2}{(x + 1) (x - 6)}$

Add $- 6$ and $2$ .

$\frac{- 4 x - 4}{(x + 1) (x - 6)}$

Factor $4$ out of $- 4 x - 4$ .

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) - 4}{(x + 1) (x - 6)}$

Factor $4$ out of $- 4$ .

$\frac{4 (- x) + 4 (- 1)}{(x + 1) (x - 6)}$

Factor $4$ out of $4 (- x) + 4 (- 1)$ .

$\frac{4 (- x - 1)}{(x + 1) (x - 6)}$

Cancel the common factor of $- x - 1$ and $x + 1$ .

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

$\frac{4 (- (x) - 1)}{(x + 1) (x - 6)}$

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

$\frac{4 (- (x) - 1 (1))}{(x + 1) (x - 6)}$

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

$\frac{4 (- (x + 1))}{(x + 1) (x - 6)}$

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

$\frac{4 (- 1 (x + 1))}{(x + 1) (x - 6)}$

Cancel the common factor.

$\frac{4 (- 1 (x + 1))}{(x + 1) (x - 6)}$

Rewrite the expression.

$\frac{4 \cdot (- 1)}{x - 6}$

Multiply $4$ by $- 1$ .

$\frac{- 4}{x - 6}$

Move the negative in front of the fraction.

$- \frac{4}{x - 6}$

Simplify 1/(x+1)-(5x-2)/(x^2-5x-6)

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