Solve by Factoring (4x)/(x+3)-4=(x-10)/(x-3)

$\frac{4 x}{x + 3} - 4 = \frac{x - 10}{x - 3}$

Step 1

Subtract $\frac{x - 10}{x - 3}$ from both sides of the equation.

$\frac{4 x}{x + 3} - 4 - \frac{x - 10}{x - 3} = 0$

Step 2

Simplify $\frac{4 x}{x + 3} - 4 - \frac{x - 10}{x - 3}$ .

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

$\frac{4 x}{x + 3} - 4 \cdot \frac{x + 3}{x + 3} - \frac{x - 10}{x - 3} = 0$

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

$\frac{4 x}{x + 3} + \frac{- 4 (x + 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

Combine the numerators over the common denominator.

$\frac{4 x - 4 (x + 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

Simplify each term.

Simplify the numerator.

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

Factor $4$ out of $4 x$ .

$\frac{4 (x) - 4 (x + 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

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

$\frac{4 (x) + 4 (- (x + 3))}{x + 3} - \frac{x - 10}{x - 3} = 0$

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

$\frac{4 (x - (x + 3))}{x + 3} - \frac{x - 10}{x - 3} = 0$

Apply the distributive property.

$\frac{4 (x - x - 1 \cdot 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

Multiply $- 1$ by $3$ .

$\frac{4 (x - x - 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

Subtract $x$ from $x$ .

$\frac{4 (0 - 3)}{x + 3} - \frac{x - 10}{x - 3} = 0$

Subtract $3$ from $0$ .

$\frac{4 \cdot - 3}{x + 3} - \frac{x - 10}{x - 3} = 0$

Multiply $4$ by $- 3$ .

$\frac{- 12}{x + 3} - \frac{x - 10}{x - 3} = 0$

Move the negative in front of the fraction.

$- \frac{12}{x + 3} - \frac{x - 10}{x - 3} = 0$

To write $- \frac{12}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$- \frac{12}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{x - 10}{x - 3} = 0$

To write $- \frac{x - 10}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$- \frac{12}{x + 3} \cdot \frac{x - 3}{x - 3} - \frac{x - 10}{x - 3} \cdot \frac{x + 3}{x + 3} = 0$

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

Multiply $\frac{12}{x + 3}$ by $\frac{x - 3}{x - 3}$ .

$- \frac{12 (x - 3)}{(x + 3) (x - 3)} - \frac{x - 10}{x - 3} \cdot \frac{x + 3}{x + 3} = 0$

Multiply $\frac{x - 10}{x - 3}$ by $\frac{x + 3}{x + 3}$ .

$- \frac{12 (x - 3)}{(x + 3) (x - 3)} - \frac{(x - 10) (x + 3)}{(x - 3) (x + 3)} = 0$

Reorder the factors of $(x - 3) (x + 3)$ .

$- \frac{12 (x - 3)}{(x + 3) (x - 3)} - \frac{(x - 10) (x + 3)}{(x + 3) (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{- 12 (x - 3) - (x - 10) (x + 3)}{(x + 3) (x - 3)} = 0$

Simplify the numerator.

Factor $- 1$ out of $- 12 (x - 3) - (x - 10) (x + 3)$ .

Reorder $- 12 (x - 3)$ and $- (x - 10) (x + 3)$ .

$\frac{- (x - 10) (x + 3) - 12 (x - 3)}{(x + 3) (x - 3)} = 0$

Factor $- 1$ out of $- (x - 10) (x + 3)$ .

$\frac{- ((x - 10) (x + 3)) - 12 (x - 3)}{(x + 3) (x - 3)} = 0$

Factor $- 1$ out of $- 12 (x - 3)$ .

$\frac{- ((x - 10) (x + 3)) - (12 (x - 3))}{(x + 3) (x - 3)} = 0$

Factor $- 1$ out of $- ((x - 10) (x + 3)) - (12 (x - 3))$ .

$\frac{- ((x - 10) (x + 3) + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Expand $(x - 10) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{- (x (x + 3) - 10 (x + 3) + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot 3 - 10 (x + 3) + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{- (x \cdot x + x \cdot 3 - 10 x - 10 \cdot 3 + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{- (x^{2} + x \cdot 3 - 10 x - 10 \cdot 3 + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Move $3$ to the left of $x$ .

$\frac{- (x^{2} + 3 \cdot x - 10 x - 10 \cdot 3 + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Multiply $- 10$ by $3$ .

$\frac{- (x^{2} + 3 x - 10 x - 30 + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Subtract $10 x$ from $3 x$ .

$\frac{- (x^{2} - 7 x - 30 + 12 (x - 3))}{(x + 3) (x - 3)} = 0$

Apply the distributive property.

$\frac{- (x^{2} - 7 x - 30 + 12 x + 12 \cdot - 3)}{(x + 3) (x - 3)} = 0$

Multiply $12$ by $- 3$ .

$\frac{- (x^{2} - 7 x - 30 + 12 x - 36)}{(x + 3) (x - 3)} = 0$

Add $- 7 x$ and $12 x$ .

$\frac{- (x^{2} + 5 x - 30 - 36)}{(x + 3) (x - 3)} = 0$

Subtract $36$ from $- 30$ .

$\frac{- (x^{2} + 5 x - 66)}{(x + 3) (x - 3)} = 0$

Factor $x^{2} + 5 x - 66$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 66$ and whose sum is $5$ .

$- 6, 11$

Write the factored form using these integers.

$\frac{- ((x - 6) (x + 11))}{(x + 3) (x - 3)} = 0$

$\frac{- (x - 6) (x + 11)}{(x + 3) (x - 3)} = 0$

Move the negative in front of the fraction.

$- \frac{(x - 6) (x + 11)}{(x + 3) (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 6) (x + 11) = 0$

Step 4

Solve the equation for $x$ .

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 11 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

The final solution is all the values that make $(x - 6) (x + 11) = 0$ true.

$x = 6, - 11$

Solve by Factoring (4x)/(x+3)-4=(x-10)/(x-3)

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