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

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

Step 1

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

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

Step 2

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

Find the common denominator.

Multiply $\frac{1}{x}$ by $\frac{(x - 1) \cdot 9}{(x - 1) \cdot 9}$ .

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

Multiply $\frac{1}{x}$ by $\frac{(x - 1) \cdot 9}{(x - 1) \cdot 9}$ .

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

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

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

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

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

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

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

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

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

Reorder the factors of $x ((x - 1) \cdot 9)$ .

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

Reorder the factors of $(x - 1) (x \cdot 9)$ .

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

Reorder the factors of $9 (x (x - 1))$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Move $9$ to the left of $x$ .

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

Multiply $- 1$ by $9$ .

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

Move $9$ to the left of $x$ .

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

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Add $9 x$ and $9 x$ .

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

Add $18 x$ and $x$ .

$\frac{19 x - 9 - x^{2}}{9 x (x - 1)} = 0$

Reorder terms.

$\frac{- x^{2} + 19 x - 9}{9 x (x - 1)} = 0$

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 19 x - 9}{9 x (x - 1)} = 0$

Factor $- 1$ out of $19 x$ .

$\frac{- (x^{2}) - (- 19 x) - 9}{9 x (x - 1)} = 0$

Factor $- 1$ out of $- (x^{2}) - (- 19 x)$ .

$\frac{- (x^{2} - 19 x) - 9}{9 x (x - 1)} = 0$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (x^{2} - 19 x) - 1 (9)}{9 x (x - 1)} = 0$

Factor $- 1$ out of $- (x^{2} - 19 x) - 1 (9)$ .

$\frac{- (x^{2} - 19 x + 9)}{9 x (x - 1)} = 0$

Rewrite $- (x^{2} - 19 x + 9)$ as $- 1 (x^{2} - 19 x + 9)$ .

$\frac{- 1 (x^{2} - 19 x + 9)}{9 x (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{x^{2} - 19 x + 9}{9 x (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 19 x + 9 = 0$

Step 4

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 19$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 9$ .

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36}}{2 \cdot 1}$

Subtract $36$ from $361$ .

$x = \frac{19 \pm \sqrt{325}}{2 \cdot 1}$

Rewrite $325$ as $5^{2} \cdot 13$ .

Factor $25$ out of $325$ .

$x = \frac{19 \pm \sqrt{25 (13)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{19 \pm \sqrt{5^{2} \cdot 13}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{19 \pm 5 \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{19 \pm 5 \sqrt{13}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 9$ .

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36}}{2 \cdot 1}$

Subtract $36$ from $361$ .

$x = \frac{19 \pm \sqrt{325}}{2 \cdot 1}$

Rewrite $325$ as $5^{2} \cdot 13$ .

Factor $25$ out of $325$ .

$x = \frac{19 \pm \sqrt{25 (13)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{19 \pm \sqrt{5^{2} \cdot 13}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{19 \pm 5 \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{19 \pm 5 \sqrt{13}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{19 + 5 \sqrt{13}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 9$ .

Multiply $- 4$ by $1$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 9}}{2 \cdot 1}$

Multiply $- 4$ by $9$ .

$x = \frac{19 \pm \sqrt{361 - 36}}{2 \cdot 1}$

Subtract $36$ from $361$ .

$x = \frac{19 \pm \sqrt{325}}{2 \cdot 1}$

Rewrite $325$ as $5^{2} \cdot 13$ .

Factor $25$ out of $325$ .

$x = \frac{19 \pm \sqrt{25 (13)}}{2 \cdot 1}$

Rewrite $25$ as $5^{2}$ .

$x = \frac{19 \pm \sqrt{5^{2} \cdot 13}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{19 \pm 5 \sqrt{13}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{19 \pm 5 \sqrt{13}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{19 - 5 \sqrt{13}}{2}$

The final answer is the combination of both solutions.

$x = \frac{19 + 5 \sqrt{13}}{2}, \frac{19 - 5 \sqrt{13}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{19 + 5 \sqrt{13}}{2}, \frac{19 - 5 \sqrt{13}}{2}$

Decimal Form:

$x = 18.51387818 \dots, 0.48612181 \dots$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.