Solve by Factoring 7/(x-4)=1+9/(x+4)

$\frac{7}{x - 4} = 1 + \frac{9}{x + 4}$

Step 1

Move all the expressions to the left side of the equation.

Subtract $1$ from both sides of the equation.

$\frac{7}{x - 4} - 1 = \frac{9}{x + 4}$

Subtract $\frac{9}{x + 4}$ from both sides of the equation.

$\frac{7}{x - 4} - 1 - \frac{9}{x + 4} = 0$

Step 2

Simplify $\frac{7}{x - 4} - 1 - \frac{9}{x + 4}$ .

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{7}{x - 4} - 1 \cdot \frac{x - 4}{x - 4} - \frac{9}{x + 4} = 0$

Combine $- 1$ and $\frac{x - 4}{x - 4}$ .

$\frac{7}{x - 4} + \frac{- (x - 4)}{x - 4} - \frac{9}{x + 4} = 0$

Combine the numerators over the common denominator.

$\frac{7 - (x - 4)}{x - 4} - \frac{9}{x + 4} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{7 - x - - 4}{x - 4} - \frac{9}{x + 4} = 0$

Multiply $- 1$ by $- 4$ .

$\frac{7 - x + 4}{x - 4} - \frac{9}{x + 4} = 0$

Add $7$ and $4$ .

$\frac{- x + 11}{x - 4} - \frac{9}{x + 4} = 0$

To write $\frac{- x + 11}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{- x + 11}{x - 4} \cdot \frac{x + 4}{x + 4} - \frac{9}{x + 4} = 0$

To write $- \frac{9}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{- x + 11}{x - 4} \cdot \frac{x + 4}{x + 4} - \frac{9}{x + 4} \cdot \frac{x - 4}{x - 4} = 0$

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

Multiply $\frac{- x + 11}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(- x + 11) (x + 4)}{(x - 4) (x + 4)} - \frac{9}{x + 4} \cdot \frac{x - 4}{x - 4} = 0$

Multiply $\frac{9}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(- x + 11) (x + 4)}{(x - 4) (x + 4)} - \frac{9 (x - 4)}{(x + 4) (x - 4)} = 0$

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{(- x + 11) (x + 4)}{(x + 4) (x - 4)} - \frac{9 (x - 4)}{(x + 4) (x - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{(- x + 11) (x + 4) - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Simplify the numerator.

Expand $(- x + 11) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{- x (x + 4) + 11 (x + 4) - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 4 + 11 (x + 4) - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Apply the distributive property.

$\frac{- x \cdot x - x \cdot 4 + 11 x + 11 \cdot 4 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{- (x \cdot x) - x \cdot 4 + 11 x + 11 \cdot 4 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} - x \cdot 4 + 11 x + 11 \cdot 4 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Multiply $4$ by $- 1$ .

$\frac{- x^{2} - 4 x + 11 x + 11 \cdot 4 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Multiply $11$ by $4$ .

$\frac{- x^{2} - 4 x + 11 x + 44 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Add $- 4 x$ and $11 x$ .

$\frac{- x^{2} + 7 x + 44 - 9 (x - 4)}{(x + 4) (x - 4)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 7 x + 44 - 9 x - 9 \cdot - 4}{(x + 4) (x - 4)} = 0$

Multiply $- 9$ by $- 4$ .

$\frac{- x^{2} + 7 x + 44 - 9 x + 36}{(x + 4) (x - 4)} = 0$

Subtract $9 x$ from $7 x$ .

$\frac{- x^{2} - 2 x + 44 + 36}{(x + 4) (x - 4)} = 0$

Add $44$ and $36$ .

$\frac{- x^{2} - 2 x + 80}{(x + 4) (x - 4)} = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 80 = - 80$ and whose sum is $b = - 2$ .

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

$\frac{- x^{2} - 2 (x) + 80}{(x + 4) (x - 4)} = 0$

Rewrite $- 2$ as $8$ plus $- 10$

$\frac{- x^{2} + (8 - 10) x + 80}{(x + 4) (x - 4)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 8 x - 10 x + 80}{(x + 4) (x - 4)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} + 8 x) - 10 x + 80}{(x + 4) (x - 4)} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x + 8) + 10 (- x + 8)}{(x + 4) (x - 4)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 8$ .

$\frac{(- x + 8) (x + 10)}{(x + 4) (x - 4)} = 0$

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

$\frac{(- (x) + 8) (x + 10)}{(x + 4) (x - 4)} = 0$

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

$\frac{(- (x) - 1 (- 8)) (x + 10)}{(x + 4) (x - 4)} = 0$

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

$\frac{- (x - 8) (x + 10)}{(x + 4) (x - 4)} = 0$

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

$\frac{- 1 (x - 8) (x + 10)}{(x + 4) (x - 4)} = 0$

Move the negative in front of the fraction.

$- \frac{(x - 8) (x + 10)}{(x + 4) (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(x - 8) (x + 10) = 0$

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 $0$ .

$x - 8 = 0$

$x + 10 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

The final solution is all the values that make $(x - 8) (x + 10) = 0$ true.

$x = 8, - 10$

Solve by Factoring 7/(x-4)=1+9/(x+4)

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