Solve by Factoring 6/(x+2)=8/x

$\frac{6}{x + 2} = \frac{8}{x}$

Step 1

Subtract $\frac{8}{x}$ from both sides of the equation.

$\frac{6}{x + 2} - \frac{8}{x} = 0$

Step 2

Simplify $\frac{6}{x + 2} - \frac{8}{x}$ .

To write $\frac{6}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{6}{x + 2} \cdot \frac{x}{x} - \frac{8}{x} = 0$

To write $- \frac{8}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{6}{x + 2} \cdot \frac{x}{x} - \frac{8}{x} \cdot \frac{x + 2}{x + 2} = 0$

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

Multiply $\frac{6}{x + 2}$ by $\frac{x}{x}$ .

$\frac{6 x}{(x + 2) x} - \frac{8}{x} \cdot \frac{x + 2}{x + 2} = 0$

Multiply $\frac{8}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{6 x}{(x + 2) x} - \frac{8 (x + 2)}{x (x + 2)} = 0$

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

$\frac{6 x}{x (x + 2)} - \frac{8 (x + 2)}{x (x + 2)} = 0$

Combine the numerators over the common denominator.

$\frac{6 x - 8 (x + 2)}{x (x + 2)} = 0$

Simplify the numerator.

Factor $2$ out of $6 x - 8 (x + 2)$ .

Factor $2$ out of $6 x$ .

$\frac{2 (3 x) - 8 (x + 2)}{x (x + 2)} = 0$

Factor $2$ out of $- 8 (x + 2)$ .

$\frac{2 (3 x) + 2 (- 4 (x + 2))}{x (x + 2)} = 0$

Factor $2$ out of $2 (3 x) + 2 (- 4 (x + 2))$ .

$\frac{2 (3 x - 4 (x + 2))}{x (x + 2)} = 0$

Apply the distributive property.

$\frac{2 (3 x - 4 x - 4 \cdot 2)}{x (x + 2)} = 0$

Multiply $- 4$ by $2$ .

$\frac{2 (3 x - 4 x - 8)}{x (x + 2)} = 0$

Subtract $4 x$ from $3 x$ .

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

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

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

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{2 (- (x) - 1 (8))}{x (x + 2)} = 0$

Factor $- 1$ out of $- (x) - 1 (8)$ .

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

Rewrite $- (x + 8)$ as $- 1 (x + 8)$ .

$\frac{2 (- 1 (x + 8))}{x (x + 2)} = 0$

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$2 (x + 8) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $2 (x + 8) = 0$ by $2$ and simplify.

Divide each term in $2 (x + 8) = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x + 8$ by $1$ .

$x + 8 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Solve by Factoring 6/(x+2)=8/x

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