Solve by Factoring (3x+4)/3-(4x)/(x-4)=x

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

Step 1

Subtract $x$ from both sides of the equation.

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

Step 2

Simplify $\frac{3 x + 4}{3} - \frac{4 x}{x - 4} - x$ .

To write $\frac{3 x + 4}{3}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

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

To write $- \frac{4 x}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Write each expression with a common denominator of $3 (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x + 4}{3}$ by $\frac{x - 4}{x - 4}$ .

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

Multiply $\frac{4 x}{x - 4}$ by $\frac{3}{3}$ .

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

Reorder the factors of $(x - 4) \cdot 3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Expand $(3 x + 4) (x - 4)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 4$ by $3$ .

$\frac{3 x^{2} - 12 x + 4 x + 4 \cdot - 4 - 4 x \cdot 3}{3 (x - 4)} - x = 0$

Multiply $4$ by $- 4$ .

$\frac{3 x^{2} - 12 x + 4 x - 16 - 4 x \cdot 3}{3 (x - 4)} - x = 0$

Add $- 12 x$ and $4 x$ .

$\frac{3 x^{2} - 8 x - 16 - 4 x \cdot 3}{3 (x - 4)} - x = 0$

Multiply $3$ by $- 4$ .

$\frac{3 x^{2} - 8 x - 16 - 12 x}{3 (x - 4)} - x = 0$

Subtract $12 x$ from $- 8 x$ .

$\frac{3 x^{2} - 20 x - 16}{3 (x - 4)} - x = 0$

To write $- x$ as a fraction with a common denominator, multiply by $\frac{3 (x - 4)}{3 (x - 4)}$ .

$\frac{3 x^{2} - 20 x - 16}{3 (x - 4)} - x \cdot \frac{3 (x - 4)}{3 (x - 4)} = 0$

Combine $- x$ and $\frac{3 (x - 4)}{3 (x - 4)}$ .

$\frac{3 x^{2} - 20 x - 16}{3 (x - 4)} + \frac{- x (3 (x - 4))}{3 (x - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{3 x^{2} - 20 x - 16 - x (3 (x - 4))}{3 (x - 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{3 x^{2} - 20 x - 16 - x (3 x + 3 \cdot - 4)}{3 (x - 4)} = 0$

Multiply $3$ by $- 4$ .

$\frac{3 x^{2} - 20 x - 16 - x (3 x - 12)}{3 (x - 4)} = 0$

Apply the distributive property.

$\frac{3 x^{2} - 20 x - 16 - x (3 x) - x \cdot - 12}{3 (x - 4)} = 0$

Rewrite using the commutative property of multiplication.

$\frac{3 x^{2} - 20 x - 16 - 1 \cdot 3 x \cdot x - x \cdot - 12}{3 (x - 4)} = 0$

Multiply $- 12$ by $- 1$ .

$\frac{3 x^{2} - 20 x - 16 - 1 \cdot 3 x \cdot x + 12 x}{3 (x - 4)} = 0$

Simplify each term.

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

Move $x$ .

$\frac{3 x^{2} - 20 x - 16 - 1 \cdot 3 (x \cdot x) + 12 x}{3 (x - 4)} = 0$

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 20 x - 16 - 1 \cdot 3 x^{2} + 12 x}{3 (x - 4)} = 0$

Multiply $- 1$ by $3$ .

$\frac{3 x^{2} - 20 x - 16 - 3 x^{2} + 12 x}{3 (x - 4)} = 0$

Subtract $3 x^{2}$ from $3 x^{2}$ .

$\frac{- 20 x - 16 + 0 + 12 x}{3 (x - 4)} = 0$

Add $- 20 x$ and $0$ .

$\frac{- 20 x - 16 + 12 x}{3 (x - 4)} = 0$

Add $- 20 x$ and $12 x$ .

$\frac{- 8 x - 16}{3 (x - 4)} = 0$

Factor $8$ out of $- 8 x - 16$ .

Factor $8$ out of $- 8 x$ .

$\frac{8 (- x) - 16}{3 (x - 4)} = 0$

Factor $8$ out of $- 16$ .

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

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

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

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

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

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{8 (- (x) - 1 (2))}{3 (x - 4)} = 0$

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

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

Rewrite $- (x + 2)$ as $- 1 (x + 2)$ .

$\frac{8 (- 1 (x + 2))}{3 (x - 4)} = 0$

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$8 (x + 2) = 0$

Step 4

Solve the equation for $x$ .

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

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

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

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

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

Divide $x + 2$ by $1$ .

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

Simplify the right side.

Divide $0$ by $8$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Solve by Factoring (3x+4)/3-(4x)/(x-4)=x

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