Finite Math Examples

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

Step 1

Subtract $9$ from both sides of the equation.

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

Step 2

Multiply $x$ by $x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{x}$ and $b = 3$ .

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

Step 7

To write $3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 10

Combine $- 3$ and $\frac{x}{x}$ .

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\frac{1 + 3 x}{x} = 0$

$\frac{1 - 3 x}{x} = 0$

Step 13

Set $\frac{1 + 3 x}{x}$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $\frac{1 + 3 x}{x}$ equal to $0$ .

$\frac{1 + 3 x}{x} = 0$

Step 13.2

Solve $\frac{1 + 3 x}{x} = 0$ for $x$ .

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Step 13.2.1

Set the numerator equal to zero.

$1 + 3 x = 0$

Step 13.2.2

Solve the equation for $x$ .

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Step 13.2.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 13.2.2.2

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

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Step 13.2.2.2.1

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

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

Step 13.2.2.2.2

Simplify the left side.

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Step 13.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 13.2.2.2.2.1.1

Cancel the common factor.

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

Step 13.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 13.2.2.2.3

Simplify the right side.

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Step 13.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 14

Set $\frac{1 - 3 x}{x}$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $\frac{1 - 3 x}{x}$ equal to $0$ .

$\frac{1 - 3 x}{x} = 0$

Step 14.2

Solve $\frac{1 - 3 x}{x} = 0$ for $x$ .

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Step 14.2.1

Set the numerator equal to zero.

$1 - 3 x = 0$

Step 14.2.2

Solve the equation for $x$ .

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Step 14.2.2.1

Subtract $1$ from both sides of the equation.

$- 3 x = - 1$

Step 14.2.2.2

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

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Step 14.2.2.2.1

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

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

Step 14.2.2.2.2

Simplify the left side.

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Step 14.2.2.2.2.1

Cancel the common factor of $- 3$ .

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Step 14.2.2.2.2.1.1

Cancel the common factor.

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

Step 14.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 14.2.2.2.3

Simplify the right side.

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Step 14.2.2.2.3.1

Dividing two negative values results in a positive value.

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

Step 15

The final solution is all the values that make $\frac{1 + 3 x}{x} \cdot \frac{1 - 3 x}{x} = 0$ true.

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