Solve by Factoring 3/(x+2)-1/x=1/(5x)

$\frac{3}{x + 2} - \frac{1}{x} = \frac{1}{5 x}$

Step 1

Subtract $\frac{1}{5 x}$ from both sides of the equation.

$\frac{3}{x + 2} - \frac{1}{x} - \frac{1}{5 x} = 0$

Step 2

Simplify $\frac{3}{x + 2} - \frac{1}{x} - \frac{1}{5 x}$ .

To write $\frac{3}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{3}{x + 2} \cdot \frac{x}{x} - \frac{1}{x} - \frac{1}{5 x} = 0$

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{3}{x + 2} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{x + 2}{x + 2} - \frac{1}{5 x} = 0$

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

Multiply $\frac{3}{x + 2}$ by $\frac{x}{x}$ .

$\frac{3 x}{(x + 2) x} - \frac{1}{x} \cdot \frac{x + 2}{x + 2} - \frac{1}{5 x} = 0$

Multiply $\frac{1}{x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{3 x}{(x + 2) x} - \frac{x + 2}{x (x + 2)} - \frac{1}{5 x} = 0$

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

$\frac{3 x}{x (x + 2)} - \frac{x + 2}{x (x + 2)} - \frac{1}{5 x} = 0$

Combine the numerators over the common denominator.

$\frac{3 x - (x + 2)}{x (x + 2)} - \frac{1}{5 x} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x - x - 1 \cdot 2}{x (x + 2)} - \frac{1}{5 x} = 0$

Multiply $- 1$ by $2$ .

$\frac{3 x - x - 2}{x (x + 2)} - \frac{1}{5 x} = 0$

Subtract $x$ from $3 x$ .

$\frac{2 x - 2}{x (x + 2)} - \frac{1}{5 x} = 0$

Factor $2$ out of $2 x - 2$ .

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 2}{x (x + 2)} - \frac{1}{5 x} = 0$

Factor $2$ out of $- 2$ .

$\frac{2 (x) + 2 (- 1)}{x (x + 2)} - \frac{1}{5 x} = 0$

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{2 (x - 1)}{x (x + 2)} - \frac{1}{5 x} = 0$

To write $\frac{2 (x - 1)}{x (x + 2)}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 (x - 1)}{x (x + 2)} \cdot \frac{5}{5} - \frac{1}{5 x} = 0$

To write $- \frac{1}{5 x}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{2 (x - 1)}{x (x + 2)} \cdot \frac{5}{5} - \frac{1}{5 x} \cdot \frac{x + 2}{x + 2} = 0$

Write each expression with a common denominator of $x (x + 2) \cdot 5$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 (x - 1)}{x (x + 2)}$ by $\frac{5}{5}$ .

$\frac{2 (x - 1) \cdot 5}{x (x + 2) \cdot 5} - \frac{1}{5 x} \cdot \frac{x + 2}{x + 2} = 0$

Multiply $\frac{1}{5 x}$ by $\frac{x + 2}{x + 2}$ .

$\frac{2 (x - 1) \cdot 5}{x (x + 2) \cdot 5} - \frac{x + 2}{5 x (x + 2)} = 0$

Reorder the factors of $x (x + 2) \cdot 5$ .

$\frac{2 (x - 1) \cdot 5}{5 x (x + 2)} - \frac{x + 2}{5 x (x + 2)} = 0$

Combine the numerators over the common denominator.

$\frac{2 (x - 1) \cdot 5 - (x + 2)}{5 x (x + 2)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 1) \cdot 5 - (x + 2)}{5 x (x + 2)} = 0$

Multiply $2$ by $- 1$ .

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

Apply the distributive property.

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

Multiply $5$ by $2$ .

$\frac{10 x - 2 \cdot 5 - (x + 2)}{5 x (x + 2)} = 0$

Multiply $- 2$ by $5$ .

$\frac{10 x - 10 - (x + 2)}{5 x (x + 2)} = 0$

Apply the distributive property.

$\frac{10 x - 10 - x - 1 \cdot 2}{5 x (x + 2)} = 0$

Multiply $- 1$ by $2$ .

$\frac{10 x - 10 - x - 2}{5 x (x + 2)} = 0$

Subtract $x$ from $10 x$ .

$\frac{9 x - 10 - 2}{5 x (x + 2)} = 0$

Subtract $2$ from $- 10$ .

$\frac{9 x - 12}{5 x (x + 2)} = 0$

Factor $3$ out of $9 x - 12$ .

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 12}{5 x (x + 2)} = 0$

Factor $3$ out of $- 12$ .

$\frac{3 (3 x) + 3 (- 4)}{5 x (x + 2)} = 0$

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

$\frac{3 (3 x - 4)}{5 x (x + 2)} = 0$

Step 3

Set the numerator equal to zero.

$3 (3 x - 4) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (3 x - 4) = 0$ by $3$ and simplify.

Divide each term in $3 (3 x - 4) = 0$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $3 x - 4$ by $1$ .

$3 x - 4 = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$3 x - 4 = 0$

Add $4$ to both sides of the equation.

$3 x = 4$

Divide each term in $3 x = 4$ by $3$ and simplify.

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Solve by Factoring 3/(x+2)-1/x=1/(5x)

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