Algebra Examples

$\frac{1}{x} = \frac{x}{6} - \frac{5}{6}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{x}{6}$ from both sides of the equation.

$\frac{1}{x} - \frac{x}{6} = - \frac{5}{6}$

Step 1.2

Add $\frac{5}{6}$ to both sides of the equation.

$\frac{1}{x} - \frac{x}{6} + \frac{5}{6} = 0$

Step 2

Reorder terms.

$- \frac{x}{6} + \frac{1}{x} + \frac{5}{6} = 0$

Step 3

To write $- \frac{x}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{1}{x} + \frac{5}{6} = 0$

Step 4

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{x}{6} \cdot \frac{x}{x} + \frac{1}{x} \cdot \frac{6}{6} + \frac{5}{6} = 0$

Step 5

Write each expression with a common denominator of $6 x$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{x}{6}$ by $\frac{x}{x}$ .

$- \frac{x \cdot x}{6 x} + \frac{1}{x} \cdot \frac{6}{6} + \frac{5}{6} = 0$

Step 5.2

Multiply $\frac{1}{x}$ by $\frac{6}{6}$ .

$- \frac{x \cdot x}{6 x} + \frac{6}{x \cdot 6} + \frac{5}{6} = 0$

Step 5.3

Reorder the factors of $x \cdot 6$ .

$- \frac{x \cdot x}{6 x} + \frac{6}{6 x} + \frac{5}{6} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- x \cdot x + 6}{6 x} + \frac{5}{6} = 0$

Step 7

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

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Step 7.1

Move $x$ .

$\frac{- (x \cdot x) + 6}{6 x} + \frac{5}{6} = 0$

Step 7.2

Multiply $x$ by $x$ .

$\frac{- x^{2} + 6}{6 x} + \frac{5}{6} = 0$

Step 8

To write $\frac{5}{6}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{- x^{2} + 6}{6 x} + \frac{5}{6} \cdot \frac{x}{x} = 0$

Step 9

Multiply $\frac{5}{6}$ by $\frac{x}{x}$ .

$\frac{- x^{2} + 6}{6 x} + \frac{5 x}{6 x} = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{- x^{2} + 6 + 5 x}{6 x} = 0$

Step 11

Factor by grouping.

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Step 11.1

Reorder terms.

$\frac{- x^{2} + 5 x + 6}{6 x} = 0$

Step 11.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 6 = - 6$ and whose sum is $b = 5$ .

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Step 11.2.1

Factor $5$ out of $5 x$ .

$\frac{- x^{2} + 5 (x) + 6}{6 x} = 0$

Step 11.2.2

Rewrite $5$ as $- 1$ plus $6$

$\frac{- x^{2} + (- 1 + 6) x + 6}{6 x} = 0$

Step 11.2.3

Apply the distributive property.

$\frac{- x^{2} - 1 x + 6 x + 6}{6 x} = 0$

Step 11.3

Factor out the greatest common factor from each group.

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Step 11.3.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) + 6 x + 6}{6 x} = 0$

Step 11.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x - 1) - 6 (- x - 1)}{6 x} = 0$

Step 11.4

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

$\frac{(- x - 1) (x - 6)}{6 x} = 0$

Step 12

Set the numerator equal to zero.

$(- x - 1) (x - 6) = 0$

Step 13

Solve the equation for $x$ .

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Step 13.1

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

$- x - 1 = 0$

$x - 6 = 0$

Step 13.2

Set $- x - 1$ equal to $0$ and solve for $x$ .

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Step 13.2.1

Set $- x - 1$ equal to $0$ .

$- x - 1 = 0$

Step 13.2.2

Solve $- x - 1 = 0$ for $x$ .

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Step 13.2.2.1

Add $1$ to both sides of the equation.

$- x = 1$

Step 13.2.2.2

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

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Step 13.2.2.2.1

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

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

Step 13.2.2.2.2

Simplify the left side.

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Step 13.2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 13.2.2.2.2.2

Divide $x$ by $1$ .

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

Step 13.2.2.2.3

Simplify the right side.

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Step 13.2.2.2.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 13.3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 13.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 13.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 13.4

The final solution is all the values that make $(- x - 1) (x - 6) = 0$ true.

$x = - 1, 6$