Algebra Examples

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

Step 1

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

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Step 1.1

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

$- 6, 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify each term.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $5$ by $- 1$ .

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

Step 5.1.3

Multiply $- 1$ by $- 2$ .

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

Step 5.1.4

Subtract $5 x$ from $x$ .

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

Step 5.1.5

Add $- 6$ and $2$ .

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

Step 5.1.6

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

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Step 5.1.6.1

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

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

Step 5.1.6.2

Factor $4$ out of $- 4$ .

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

Step 5.1.6.3

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

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

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

Step 5.2.5

Cancel the common factor.

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

Step 5.2.6

Rewrite the expression.

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

Step 5.3

Multiply $4$ by $- 1$ .

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

Step 5.4

Move the negative in front of the fraction.

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

Step 6

Simplify terms.

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Step 6.1

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 6.2

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

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

Step 6.3

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Move a negative from the denominator of $\frac{x}{- 1 (- 6 + x)}$ to the numerator.

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

Step 6.4.2

Reorder terms.

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

Step 6.5

Combine the numerators over the common denominator.

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

Step 6.6

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 6.7

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

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

Step 6.8

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

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

Step 6.9

Move the negative in front of the fraction.

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