Solve by Factoring 1/(x+9)+18/(x^2-81)=1

$\frac{1}{x + 9} + \frac{18}{x^{2} - 81} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{1}{x + 9} + \frac{18}{x^{2} - 81} - 1 = 0$

Step 2

Simplify $\frac{1}{x + 9} + \frac{18}{x^{2} - 81} - 1$ .

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

$\frac{1}{x + 9} + \frac{18}{x^{2} - 9^{2}} - 1 = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$\frac{1}{x + 9} + \frac{18}{(x + 9) (x - 9)} - 1 = 0$

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

$\frac{1}{x + 9} \cdot \frac{x - 9}{x - 9} + \frac{18}{(x + 9) (x - 9)} - 1 = 0$

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

$\frac{x - 9}{(x + 9) (x - 9)} + \frac{18}{(x + 9) (x - 9)} - 1 = 0$

Combine the numerators over the common denominator.

$\frac{x - 9 + 18}{(x + 9) (x - 9)} - 1 = 0$

Add $- 9$ and $18$ .

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

Cancel the common factor of $x + 9$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Combine $- 1$ and $\frac{x - 9}{x - 9}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 9$ .

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

Add $1$ and $9$ .

$\frac{- x + 10}{x - 9} = 0$

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

$\frac{- (x) + 10}{x - 9} = 0$

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (x) - 1 (- 10)}{x - 9} = 0$

Factor $- 1$ out of $- (x) - 1 (- 10)$ .

$\frac{- (x - 10)}{x - 9} = 0$

Rewrite $- (x - 10)$ as $- 1 (x - 10)$ .

$\frac{- 1 (x - 10)}{x - 9} = 0$

Move the negative in front of the fraction.

$- \frac{x - 10}{x - 9} = 0$

Step 3

Set the numerator equal to zero.

$x - 10 = 0$

Step 4

Add $10$ to both sides of the equation.

$x = 10$

Solve by Factoring 1/(x+9)+18/(x^2-81)=1

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