Algebra Examples

$\frac{15}{x} + \frac{9 x - 7}{x + 2} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$\frac{15}{x} + \frac{9 x - 7}{x + 2} - 9 = 0$

Step 2

Simplify $\frac{15}{x} + \frac{9 x - 7}{x + 2} - 9$ .

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Step 2.1

To write $\frac{15}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{15}{x} \cdot \frac{x + 2}{x + 2} + \frac{9 x - 7}{x + 2} - 9 = 0$

Step 2.2

To write $\frac{9 x - 7}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{15}{x} \cdot \frac{x + 2}{x + 2} + \frac{9 x - 7}{x + 2} \cdot \frac{x}{x} - 9 = 0$

Step 2.3

Write each expression with a common denominator of $x (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{15}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{15 (x + 2)}{x (x + 2)} + \frac{9 x - 7}{x + 2} \cdot \frac{x}{x} - 9 = 0$

Step 2.3.2

Multiply $\frac{9 x - 7}{x + 2}$ by $\frac{x}{x}$ .

$\frac{15 (x + 2)}{x (x + 2)} + \frac{(9 x - 7) x}{(x + 2) x} - 9 = 0$

Step 2.3.3

Reorder the factors of $(x + 2) x$ .

$\frac{15 (x + 2)}{x (x + 2)} + \frac{(9 x - 7) x}{x (x + 2)} - 9 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{15 (x + 2) + (9 x - 7) x}{x (x + 2)} - 9 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Apply the distributive property.

$\frac{15 x + 15 \cdot 2 + (9 x - 7) x}{x (x + 2)} - 9 = 0$

Step 2.5.2

Multiply $15$ by $2$ .

$\frac{15 x + 30 + (9 x - 7) x}{x (x + 2)} - 9 = 0$

Step 2.5.3

Apply the distributive property.

$\frac{15 x + 30 + 9 x \cdot x - 7 x}{x (x + 2)} - 9 = 0$

Step 2.5.4

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

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Step 2.5.4.1

Move $x$ .

$\frac{15 x + 30 + 9 (x \cdot x) - 7 x}{x (x + 2)} - 9 = 0$

Step 2.5.4.2

Multiply $x$ by $x$ .

$\frac{15 x + 30 + 9 x^{2} - 7 x}{x (x + 2)} - 9 = 0$

Step 2.5.5

Subtract $7 x$ from $15 x$ .

$\frac{8 x + 30 + 9 x^{2}}{x (x + 2)} - 9 = 0$

Step 2.5.6

Reorder terms.

$\frac{9 x^{2} + 8 x + 30}{x (x + 2)} - 9 = 0$

Step 2.6

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{9 x^{2} + 8 x + 30}{x (x + 2)} - 9 \cdot \frac{x (x + 2)}{x (x + 2)} = 0$

Step 2.7

Combine $- 9$ and $\frac{x (x + 2)}{x (x + 2)}$ .

$\frac{9 x^{2} + 8 x + 30}{x (x + 2)} + \frac{- 9 (x (x + 2))}{x (x + 2)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{9 x^{2} + 8 x + 30 - 9 (x (x + 2))}{x (x + 2)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{9 x^{2} + 8 x + 30 - 9 x \cdot x - 9 x \cdot 2}{x (x + 2)} = 0$

Step 2.9.2

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

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Step 2.9.2.1

Move $x$ .

$\frac{9 x^{2} + 8 x + 30 - 9 (x \cdot x) - 9 x \cdot 2}{x (x + 2)} = 0$

Step 2.9.2.2

Multiply $x$ by $x$ .

$\frac{9 x^{2} + 8 x + 30 - 9 x^{2} - 9 x \cdot 2}{x (x + 2)} = 0$

Step 2.9.3

Multiply $2$ by $- 9$ .

$\frac{9 x^{2} + 8 x + 30 - 9 x^{2} - 18 x}{x (x + 2)} = 0$

Step 2.9.4

Subtract $9 x^{2}$ from $9 x^{2}$ .

$\frac{8 x + 30 + 0 - 18 x}{x (x + 2)} = 0$

Step 2.9.5

Add $8 x$ and $0$ .

$\frac{8 x + 30 - 18 x}{x (x + 2)} = 0$

Step 2.9.6

Subtract $18 x$ from $8 x$ .

$\frac{- 10 x + 30}{x (x + 2)} = 0$

Step 2.9.7

Factor $10$ out of $- 10 x + 30$ .

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Step 2.9.7.1

Factor $10$ out of $- 10 x$ .

$\frac{10 (- x) + 30}{x (x + 2)} = 0$

Step 2.9.7.2

Factor $10$ out of $30$ .

$\frac{10 (- x) + 10 (3)}{x (x + 2)} = 0$

Step 2.9.7.3

Factor $10$ out of $10 (- x) + 10 (3)$ .

$\frac{10 (- x + 3)}{x (x + 2)} = 0$

Step 2.10

Factor $- 1$ out of $- x$ .

$\frac{10 (- (x) + 3)}{x (x + 2)} = 0$

Step 2.11

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{10 (- (x) - 1 (- 3))}{x (x + 2)} = 0$

Step 2.12

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

$\frac{10 (- (x - 3))}{x (x + 2)} = 0$

Step 2.13

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{10 (- 1 (x - 3))}{x (x + 2)} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{10 (x - 3)}{x (x + 2)} = 0$

Step 3

Set the numerator equal to zero.

$10 (x - 3) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $10 (x - 3) = 0$ by $10$ and simplify.

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Step 4.1.1

Divide each term in $10 (x - 3) = 0$ by $10$ .

$\frac{10 (x - 3)}{10} = \frac{0}{10}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $10$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{10 (x - 3)}{10} = \frac{0}{10}$

Step 4.1.2.1.2

Divide $x - 3$ by $1$ .

$x - 3 = \frac{0}{10}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $10$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$