Simplify (-1/(x-2)+4/(x-6))/(-8/(x-6))

$\frac{- \frac{1}{x - 2} + \frac{4}{x - 6}}{- \frac{8}{x - 6}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{1}{x - 2} + \frac{4}{x - 6}) (- \frac{x - 6}{8})$

Step 2

To write $- \frac{1}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$(- \frac{1}{x - 2} \cdot \frac{x - 6}{x - 6} + \frac{4}{x - 6}) (- \frac{x - 6}{8})$

Step 3

To write $\frac{4}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$(- \frac{1}{x - 2} \cdot \frac{x - 6}{x - 6} + \frac{4}{x - 6} \cdot \frac{x - 2}{x - 2}) (- \frac{x - 6}{8})$

Step 4

Write each expression with a common denominator of $(x - 2) (x - 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x - 2}$ by $\frac{x - 6}{x - 6}$ .

$(- \frac{x - 6}{(x - 2) (x - 6)} + \frac{4}{x - 6} \cdot \frac{x - 2}{x - 2}) (- \frac{x - 6}{8})$

Multiply $\frac{4}{x - 6}$ by $\frac{x - 2}{x - 2}$ .

$(- \frac{x - 6}{(x - 2) (x - 6)} + \frac{4 (x - 2)}{(x - 6) (x - 2)}) (- \frac{x - 6}{8})$

Reorder the factors of $(x - 6) (x - 2)$ .

$(- \frac{x - 6}{(x - 2) (x - 6)} + \frac{4 (x - 2)}{(x - 2) (x - 6)}) (- \frac{x - 6}{8})$

Step 5

Simplify terms.

Combine the numerators over the common denominator.

$\frac{- (x - 6) + 4 (x - 2)}{(x - 2) (x - 6)} (- \frac{x - 6}{8})$

Rewrite using the commutative property of multiplication.

$- \frac{- (x - 6) + 4 (x - 2)}{(x - 2) (x - 6)} \cdot \frac{x - 6}{8}$

Cancel the common factor of $x - 6$ .

Move the leading negative in $- \frac{- (x - 6) + 4 (x - 2)}{(x - 2) (x - 6)}$ into the numerator.

$\frac{- (- (x - 6) + 4 (x - 2))}{(x - 2) (x - 6)} \cdot \frac{x - 6}{8}$

Factor $x - 6$ out of $(x - 2) (x - 6)$ .

$\frac{- (- (x - 6) + 4 (x - 2))}{(x - 6) (x - 2)} \cdot \frac{x - 6}{8}$

Cancel the common factor.

$\frac{- (- (x - 6) + 4 (x - 2))}{(x - 6) (x - 2)} \cdot \frac{x - 6}{8}$

Rewrite the expression.

$\frac{- (- (x - 6) + 4 (x - 2))}{x - 2} \cdot \frac{1}{8}$

Multiply $\frac{- (- (x - 6) + 4 (x - 2))}{x - 2}$ by $\frac{1}{8}$ .

$\frac{- (- (x - 6) + 4 (x - 2))}{(x - 2) \cdot 8}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{- (- x - - 6 + 4 (x - 2))}{(x - 2) \cdot 8}$

Multiply $- 1$ by $- 6$ .

$\frac{- (- x + 6 + 4 (x - 2))}{(x - 2) \cdot 8}$

Apply the distributive property.

$\frac{- (- x + 6 + 4 x + 4 \cdot - 2)}{(x - 2) \cdot 8}$

Multiply $4$ by $- 2$ .

$\frac{- (- x + 6 + 4 x - 8)}{(x - 2) \cdot 8}$

Add $- x$ and $4 x$ .

$\frac{- (3 x + 6 - 8)}{(x - 2) \cdot 8}$

Subtract $8$ from $6$ .

$\frac{- (3 x - 2)}{(x - 2) \cdot 8}$

Step 7

Simplify the expression.

Move $8$ to the left of $x - 2$ .

$\frac{- (3 x - 2)}{8 \cdot (x - 2)}$

Move the negative in front of the fraction.

$- \frac{3 x - 2}{8 (x - 2)}$

Simplify (-1/(x-2)+4/(x-6))/(-8/(x-6))

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