Solve by Factoring 6/(x-1)=9/(x+1)

$\frac{6}{x - 1} = \frac{9}{x + 1}$

Step 1

Subtract $\frac{9}{x + 1}$ from both sides of the equation.

$\frac{6}{x - 1} - \frac{9}{x + 1} = 0$

Step 2

Simplify $\frac{6}{x - 1} - \frac{9}{x + 1}$ .

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

$\frac{6}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{9}{x + 1} = 0$

To write $- \frac{9}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{6}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{9}{x + 1} \cdot \frac{x - 1}{x - 1} = 0$

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

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

$\frac{6 (x + 1)}{(x - 1) (x + 1)} - \frac{9}{x + 1} \cdot \frac{x - 1}{x - 1} = 0$

Multiply $\frac{9}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{6 (x + 1)}{(x - 1) (x + 1)} - \frac{9 (x - 1)}{(x + 1) (x - 1)} = 0$

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

$\frac{6 (x + 1)}{(x + 1) (x - 1)} - \frac{9 (x - 1)}{(x + 1) (x - 1)} = 0$

Combine the numerators over the common denominator.

$\frac{6 (x + 1) - 9 (x - 1)}{(x + 1) (x - 1)} = 0$

Simplify the numerator.

Factor $3$ out of $6 (x + 1) - 9 (x - 1)$ .

Factor $3$ out of $6 (x + 1)$ .

$\frac{3 (2 (x + 1)) - 9 (x - 1)}{(x + 1) (x - 1)} = 0$

Factor $3$ out of $- 9 (x - 1)$ .

$\frac{3 (2 (x + 1)) + 3 (- 3 (x - 1))}{(x + 1) (x - 1)} = 0$

Factor $3$ out of $3 (2 (x + 1)) + 3 (- 3 (x - 1))$ .

$\frac{3 (2 (x + 1) - 3 (x - 1))}{(x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{3 (2 x + 2 \cdot 1 - 3 (x - 1))}{(x + 1) (x - 1)} = 0$

Multiply $2$ by $1$ .

$\frac{3 (2 x + 2 - 3 (x - 1))}{(x + 1) (x - 1)} = 0$

Apply the distributive property.

$\frac{3 (2 x + 2 - 3 x - 3 \cdot - 1)}{(x + 1) (x - 1)} = 0$

Multiply $- 3$ by $- 1$ .

$\frac{3 (2 x + 2 - 3 x + 3)}{(x + 1) (x - 1)} = 0$

Subtract $3 x$ from $2 x$ .

$\frac{3 (- x + 2 + 3)}{(x + 1) (x - 1)} = 0$

Add $2$ and $3$ .

$\frac{3 (- x + 5)}{(x + 1) (x - 1)} = 0$

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

$\frac{3 (- (x) + 5)}{(x + 1) (x - 1)} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{3 (- (x) - 1 (- 5))}{(x + 1) (x - 1)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{3 (- (x - 5))}{(x + 1) (x - 1)} = 0$

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{3 (- 1 (x - 5))}{(x + 1) (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{3 (x - 5)}{(x + 1) (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$3 (x - 5) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (x - 5) = 0$ by $3$ and simplify.

Divide each term in $3 (x - 5) = 0$ by $3$ .

$\frac{3 (x - 5)}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 (x - 5)}{3} = \frac{0}{3}$

Divide $x - 5$ by $1$ .

$x - 5 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Solve by Factoring 6/(x-1)=9/(x+1)

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