Solve by Factoring -2/(x-1)=(x-8)/(x+1)

$\frac{- 2}{x - 1} = \frac{x - 8}{x + 1}$

Step 1

Subtract $\frac{x - 8}{x + 1}$ from both sides of the equation.

$\frac{- 2}{x - 1} - \frac{x - 8}{x + 1} = 0$

Step 2

Simplify $\frac{- 2}{x - 1} - \frac{x - 8}{x + 1}$ .

Move the negative in front of the fraction.

$- \frac{2}{x - 1} - \frac{x - 8}{x + 1} = 0$

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

$- \frac{2}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{x - 8}{x + 1} = 0$

To write $- \frac{x - 8}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$- \frac{2}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{x - 8}{x + 1} \cdot \frac{x - 1}{x - 1} = 0$

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

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

$- \frac{2 (x + 1)}{(x - 1) (x + 1)} - \frac{x - 8}{x + 1} \cdot \frac{x - 1}{x - 1} = 0$

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

$- \frac{2 (x + 1)}{(x - 1) (x + 1)} - \frac{(x - 8) (x - 1)}{(x + 1) (x - 1)} = 0$

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

$- \frac{2 (x + 1)}{(x + 1) (x - 1)} - \frac{(x - 8) (x - 1)}{(x + 1) (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{- 2 (x + 1) - (x - 8) (x - 1)}{(x + 1) (x - 1)} = 0$

Simplify the numerator.

Factor $- 1$ out of $- 2 (x + 1) - (x - 8) (x - 1)$ .

Reorder $- 2 (x + 1)$ and $- (x - 8) (x - 1)$ .

$\frac{- (x - 8) (x - 1) - 2 (x + 1)}{(x + 1) (x - 1)} = 0$

Factor $- 1$ out of $- (x - 8) (x - 1)$ .

$\frac{- ((x - 8) (x - 1)) - 2 (x + 1)}{(x + 1) (x - 1)} = 0$

Factor $- 1$ out of $- 2 (x + 1)$ .

$\frac{- ((x - 8) (x - 1)) - (2 (x + 1))}{(x + 1) (x - 1)} = 0$

Factor $- 1$ out of $- ((x - 8) (x - 1)) - (2 (x + 1))$ .

$\frac{- ((x - 8) (x - 1) + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Expand $(x - 8) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$\frac{- (x (x - 1) - 8 (x - 1) + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot - 1 - 8 (x - 1) + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot - 1 - 8 x - 8 \cdot - 1 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{- (x^{2} + x \cdot - 1 - 8 x - 8 \cdot - 1 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Move $- 1$ to the left of $x$ .

$\frac{- (x^{2} - 1 \cdot x - 8 x - 8 \cdot - 1 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

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

$\frac{- (x^{2} - x - 8 x - 8 \cdot - 1 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Multiply $- 8$ by $- 1$ .

$\frac{- (x^{2} - x - 8 x + 8 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Subtract $8 x$ from $- x$ .

$\frac{- (x^{2} - 9 x + 8 + 2 (x + 1))}{(x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{- (x^{2} - 9 x + 8 + 2 x + 2 \cdot 1)}{(x + 1) (x - 1)} = 0$

Multiply $2$ by $1$ .

$\frac{- (x^{2} - 9 x + 8 + 2 x + 2)}{(x + 1) (x - 1)} = 0$

Add $- 9 x$ and $2 x$ .

$\frac{- (x^{2} - 7 x + 8 + 2)}{(x + 1) (x - 1)} = 0$

Add $8$ and $2$ .

$\frac{- (x^{2} - 7 x + 10)}{(x + 1) (x - 1)} = 0$

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Write the factored form using these integers.

$\frac{- ((x - 5) (x - 2))}{(x + 1) (x - 1)} = 0$

$\frac{- (x - 5) (x - 2)}{(x + 1) (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{(x - 5) (x - 2)}{(x + 1) (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 5) (x - 2) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 5 = 0$

$x - 2 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 5) (x - 2) = 0$ true.

$x = 5, 2$

Solve by Factoring -2/(x-1)=(x-8)/(x+1)

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