Solve by Factoring 1/x=x/9

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $x \cdot 9$ , by multiplying each by an appropriate factor of $1$ .

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

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

Multiply $\frac{x}{9}$ by $\frac{x}{x}$ .

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

Reorder the factors of $x \cdot 9$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x \cdot x}{9 x} = 0$

Rewrite $x \cdot x$ as $x^{2}$ .

$\frac{3^{2} - x^{2}}{9 x} = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 7

Set the numerator equal to zero.

$(3 + x) (3 - x) = 0$

Step 8

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 $0$ .

$3 + x = 0$

$3 - x = 0$

Set $3 + x$ equal to $0$ and solve for $x$ .

Set $3 + x$ equal to $0$ .

$3 + x = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Set $3 - x$ equal to $0$ and solve for $x$ .

Set $3 - x$ equal to $0$ .

$3 - x = 0$

Solve $3 - x = 0$ for $x$ .

Subtract $3$ from both sides of the equation.

$- x = - 3$

Divide each term in $- x = - 3$ by $- 1$ and simplify.

Divide each term in $- x = - 3$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 3}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 3}{- 1}$

Simplify the right side.

Divide $- 3$ by $- 1$ .

$x = 3$

The final solution is all the values that make $(3 + x) (3 - x) = 0$ true.

$x = - 3, 3$

Solve by Factoring 1/x=x/9

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