Solve by Factoring 15/x+(9x-7)/(x+2)=9

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

Step 1

Subtract $9$ from both sides of the equation.

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

Step 2

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

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

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $15$ by $2$ .

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

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Subtract $7 x$ from $15 x$ .

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

Reorder terms.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $- 9$ .

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

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

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

Add $8 x$ and $0$ .

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

Subtract $18 x$ from $8 x$ .

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

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

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

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

Factor $10$ out of $30$ .

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

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

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

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

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

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

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

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

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

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

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

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 $x$ .

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

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

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

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

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

Divide $x - 3$ by $1$ .

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

Simplify the right side.

Divide $0$ by $10$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Solve by Factoring 15/x+(9x-7)/(x+2)=9

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