Algebra Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$- 9 \cdot 9 = x (- x)$

Step 3

Solve the equation for $x$ .

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Step 3.1

Rewrite the equation as $x (- x) = - 9 \cdot 9$ .

$x (- x) = - 9 \cdot 9$

Step 3.2

Divide each term in $x (- x) = - 9 \cdot 9$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $x (- x) = - 9 \cdot 9$ by $- 1$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Move $x$ .

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

Step 3.2.2.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} \cdot - 1}{- 1} = \frac{- 9 \cdot 9}{- 1}$

Step 3.2.2.2

Move the negative one from the denominator of $\frac{x^{2} \cdot - 1}{- 1}$ .

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

Step 3.2.2.3

Rewrite $- 1 \cdot (x^{2} \cdot - 1)$ as $- (x^{2} \cdot - 1)$ .

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

Step 3.2.2.4

Move $- 1$ to the left of $x^{2}$ .

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

Step 3.2.2.5

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$- - x^{2} = \frac{- 9 \cdot 9}{- 1}$

Step 3.2.2.6

Multiply $- - x^{2}$ .

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Step 3.2.2.6.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = \frac{- 9 \cdot 9}{- 1}$

Step 3.2.2.6.2

Multiply $x^{2}$ by $1$ .

$x^{2} = \frac{- 9 \cdot 9}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative one from the denominator of $\frac{- 9 \cdot 9}{- 1}$ .

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

Step 3.2.3.2

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

$x^{2} = - (- 9 \cdot 9)$

Step 3.2.3.3

Multiply $- (- 9 \cdot 9)$ .

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Step 3.2.3.3.1

Multiply $- 9$ by $9$ .

$x^{2} = - - 81$

Step 3.2.3.3.2

Multiply $- 1$ by $- 81$ .

$x^{2} = 81$

Step 3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{81}$

Step 3.4

Simplify $\pm \sqrt{81}$ .

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Step 3.4.1

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

$x = \pm \sqrt{9^{2}}$

Step 3.4.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 9$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 9$

Step 3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 9$

Step 3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 9, - 9$