Algebra Examples

$30 x - 11 = \frac{30}{x}$

Step 1

Subtract $\frac{30}{x}$ from both sides of the equation.

$30 x - 11 - \frac{30}{x} = 0$

Step 2

Reorder terms.

$30 x - \frac{30}{x} - 11 = 0$

Step 3

To write $30 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{30 x \cdot x}{x} - \frac{30}{x} - 11 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{30 x \cdot x - 30}{x} - 11 = 0$

Step 5

Simplify the numerator.

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Step 5.1

Factor $30$ out of $30 x \cdot x - 30$ .

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Step 5.1.1

Factor $30$ out of $30 x \cdot x$ .

$\frac{30 (x \cdot x) - 30}{x} - 11 = 0$

Step 5.1.2

Factor $30$ out of $- 30$ .

$\frac{30 (x \cdot x) + 30 (- 1)}{x} - 11 = 0$

Step 5.1.3

Factor $30$ out of $30 (x \cdot x) + 30 (- 1)$ .

$\frac{30 (x \cdot x - 1)}{x} - 11 = 0$

Step 5.2

Raise $x$ to the power of $1$ .

$\frac{30 (x \cdot x - 1)}{x} - 11 = 0$

Step 5.3

Raise $x$ to the power of $1$ .

$\frac{30 (x \cdot x - 1)}{x} - 11 = 0$

Step 5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{30 (x^{1 + 1} - 1)}{x} - 11 = 0$

Step 5.5

Add $1$ and $1$ .

$\frac{30 (x^{2} - 1)}{x} - 11 = 0$

Step 5.6

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

$\frac{30 (x^{2} - 1^{2})}{x} - 11 = 0$

Step 5.7

Factor.

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Step 5.7.1

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

$\frac{30 ((x + 1) (x - 1))}{x} - 11 = 0$

Step 5.7.2

Remove unnecessary parentheses.

$\frac{30 (x + 1) (x - 1)}{x} - 11 = 0$

Step 6

To write $- 11$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{30 (x + 1) (x - 1)}{x} - 11 \cdot \frac{x}{x} = 0$

Step 7

Combine $- 11$ and $\frac{x}{x}$ .

$\frac{30 (x + 1) (x - 1)}{x} + \frac{- 11 x}{x} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{30 (x + 1) (x - 1) - 11 x}{x} = 0$

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

$\frac{(30 x + 30 \cdot 1) (x - 1) - 11 x}{x} = 0$

Step 9.2

Multiply $30$ by $1$ .

$\frac{(30 x + 30) (x - 1) - 11 x}{x} = 0$

Step 9.3

Expand $(30 x + 30) (x - 1)$ using the FOIL Method.

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Step 9.3.1

Apply the distributive property.

$\frac{30 x (x - 1) + 30 (x - 1) - 11 x}{x} = 0$

Step 9.3.2

Apply the distributive property.

$\frac{30 x \cdot x + 30 x \cdot - 1 + 30 (x - 1) - 11 x}{x} = 0$

Step 9.3.3

Apply the distributive property.

$\frac{30 x \cdot x + 30 x \cdot - 1 + 30 x + 30 \cdot - 1 - 11 x}{x} = 0$

Step 9.4

Simplify and combine like terms.

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Step 9.4.1

Simplify each term.

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Step 9.4.1.1

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

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Step 9.4.1.1.1

Move $x$ .

$\frac{30 (x \cdot x) + 30 x \cdot - 1 + 30 x + 30 \cdot - 1 - 11 x}{x} = 0$

Step 9.4.1.1.2

Multiply $x$ by $x$ .

$\frac{30 x^{2} + 30 x \cdot - 1 + 30 x + 30 \cdot - 1 - 11 x}{x} = 0$

Step 9.4.1.2

Multiply $- 1$ by $30$ .

$\frac{30 x^{2} - 30 x + 30 x + 30 \cdot - 1 - 11 x}{x} = 0$

Step 9.4.1.3

Multiply $30$ by $- 1$ .

$\frac{30 x^{2} - 30 x + 30 x - 30 - 11 x}{x} = 0$

Step 9.4.2

Add $- 30 x$ and $30 x$ .

$\frac{30 x^{2} + 0 - 30 - 11 x}{x} = 0$

Step 9.4.3

Add $30 x^{2}$ and $0$ .

$\frac{30 x^{2} - 30 - 11 x}{x} = 0$

Step 9.5

Reorder terms.

$\frac{30 x^{2} - 11 x - 30}{x} = 0$

Step 9.6

Factor by grouping.

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Step 9.6.1

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 $a \cdot c = 30 \cdot - 30 = - 900$ and whose sum is $b = - 11$ .

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Step 9.6.1.1

Factor $- 11$ out of $- 11 x$ .

$\frac{30 x^{2} - 11 x - 30}{x} = 0$

Step 9.6.1.2

Rewrite $- 11$ as $25$ plus $- 36$

$\frac{30 x^{2} + (25 - 36) x - 30}{x} = 0$

Step 9.6.1.3

Apply the distributive property.

$\frac{30 x^{2} + 25 x - 36 x - 30}{x} = 0$

Step 9.6.2

Factor out the greatest common factor from each group.

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Step 9.6.2.1

Group the first two terms and the last two terms.

$\frac{(30 x^{2} + 25 x) - 36 x - 30}{x} = 0$

Step 9.6.2.2

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

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

Step 9.6.3

Factor the polynomial by factoring out the greatest common factor, $6 x + 5$ .

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

Step 10

Set the numerator equal to zero.

$(6 x + 5) (5 x - 6) = 0$

Step 11

Solve the equation for $x$ .

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Step 11.1

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

$6 x + 5 = 0$

$5 x - 6 = 0$

Step 11.2

Set $6 x + 5$ equal to $0$ and solve for $x$ .

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Step 11.2.1

Set $6 x + 5$ equal to $0$ .

$6 x + 5 = 0$

Step 11.2.2

Solve $6 x + 5 = 0$ for $x$ .

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Step 11.2.2.1

Subtract $5$ from both sides of the equation.

$6 x = - 5$

Step 11.2.2.2

Divide each term in $6 x = - 5$ by $6$ and simplify.

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Step 11.2.2.2.1

Divide each term in $6 x = - 5$ by $6$ .

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

Step 11.2.2.2.2

Simplify the left side.

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Step 11.2.2.2.2.1

Cancel the common factor of $6$ .

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Step 11.2.2.2.2.1.1

Cancel the common factor.

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

Step 11.2.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 11.2.2.2.3

Simplify the right side.

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Step 11.2.2.2.3.1

Move the negative in front of the fraction.

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

Step 11.3

Set $5 x - 6$ equal to $0$ and solve for $x$ .

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Step 11.3.1

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Step 11.3.2

Solve $5 x - 6 = 0$ for $x$ .

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Step 11.3.2.1

Add $6$ to both sides of the equation.

$5 x = 6$

Step 11.3.2.2

Divide each term in $5 x = 6$ by $5$ and simplify.

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Step 11.3.2.2.1

Divide each term in $5 x = 6$ by $5$ .

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

Step 11.3.2.2.2

Simplify the left side.

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Step 11.3.2.2.2.1

Cancel the common factor of $5$ .

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Step 11.3.2.2.2.1.1

Cancel the common factor.

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

Step 11.3.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 11.4

The final solution is all the values that make $(6 x + 5) (5 x - 6) = 0$ true.

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