Solve by Factoring 5/(x+4)=4+3/(x-2)

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

Step 1

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

Subtract $4$ from both sides of the equation.

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

Subtract $\frac{3}{x - 2}$ from both sides of the equation.

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

Step 2

Simplify $\frac{5}{x + 4} - 4 - \frac{3}{x - 2}$ .

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 4$ by $4$ .

$\frac{5 - 4 x - 16}{x + 4} - \frac{3}{x - 2} = 0$

Subtract $16$ from $5$ .

$\frac{- 4 x - 11}{x + 4} - \frac{3}{x - 2} = 0$

To write $\frac{- 4 x - 11}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{- 4 x - 11}{x + 4} \cdot \frac{x - 2}{x - 2} - \frac{3}{x - 2} = 0$

To write $- \frac{3}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{- 4 x - 11}{x + 4} \cdot \frac{x - 2}{x - 2} - \frac{3}{x - 2} \cdot \frac{x + 4}{x + 4} = 0$

Write each expression with a common denominator of $(x + 4) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{- 4 x - 11}{x + 4}$ by $\frac{x - 2}{x - 2}$ .

$\frac{(- 4 x - 11) (x - 2)}{(x + 4) (x - 2)} - \frac{3}{x - 2} \cdot \frac{x + 4}{x + 4} = 0$

Multiply $\frac{3}{x - 2}$ by $\frac{x + 4}{x + 4}$ .

$\frac{(- 4 x - 11) (x - 2)}{(x + 4) (x - 2)} - \frac{3 (x + 4)}{(x - 2) (x + 4)} = 0$

Reorder the factors of $(x - 2) (x + 4)$ .

$\frac{(- 4 x - 11) (x - 2)}{(x + 4) (x - 2)} - \frac{3 (x + 4)}{(x + 4) (x - 2)} = 0$

Combine the numerators over the common denominator.

$\frac{(- 4 x - 11) (x - 2) - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Simplify the numerator.

Expand $(- 4 x - 11) (x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 4 x (x - 2) - 11 (x - 2) - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Apply the distributive property.

$\frac{- 4 x \cdot x - 4 x \cdot - 2 - 11 (x - 2) - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Apply the distributive property.

$\frac{- 4 x \cdot x - 4 x \cdot - 2 - 11 x - 11 \cdot - 2 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{- 4 (x \cdot x) - 4 x \cdot - 2 - 11 x - 11 \cdot - 2 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Multiply $x$ by $x$ .

$\frac{- 4 x^{2} - 4 x \cdot - 2 - 11 x - 11 \cdot - 2 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Multiply $- 2$ by $- 4$ .

$\frac{- 4 x^{2} + 8 x - 11 x - 11 \cdot - 2 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Multiply $- 11$ by $- 2$ .

$\frac{- 4 x^{2} + 8 x - 11 x + 22 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Subtract $11 x$ from $8 x$ .

$\frac{- 4 x^{2} - 3 x + 22 - 3 (x + 4)}{(x + 4) (x - 2)} = 0$

Apply the distributive property.

$\frac{- 4 x^{2} - 3 x + 22 - 3 x - 3 \cdot 4}{(x + 4) (x - 2)} = 0$

Multiply $- 3$ by $4$ .

$\frac{- 4 x^{2} - 3 x + 22 - 3 x - 12}{(x + 4) (x - 2)} = 0$

Subtract $3 x$ from $- 3 x$ .

$\frac{- 4 x^{2} - 6 x + 22 - 12}{(x + 4) (x - 2)} = 0$

Subtract $12$ from $22$ .

$\frac{- 4 x^{2} - 6 x + 10}{(x + 4) (x - 2)} = 0$

Rewrite $- 4 x^{2} - 6 x + 10$ in a factored form.

Factor $2$ out of $- 4 x^{2} - 6 x + 10$ .

Factor $2$ out of $- 4 x^{2}$ .

$\frac{2 (- 2 x^{2}) - 6 x + 10}{(x + 4) (x - 2)} = 0$

Factor $2$ out of $- 6 x$ .

$\frac{2 (- 2 x^{2}) + 2 (- 3 x) + 10}{(x + 4) (x - 2)} = 0$

Factor $2$ out of $10$ .

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

Factor $2$ out of $2 (- 2 x^{2}) + 2 (- 3 x)$ .

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

Factor $2$ out of $2 (- 2 x^{2} - 3 x) + 2 (5)$ .

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

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 $a \cdot c = - 2 \cdot 5 = - 10$ and whose sum is $b = - 3$ .

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

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

Rewrite $- 3$ as $2$ plus $- 5$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$\frac{2 (2 x (- x + 1) + 5 (- x + 1))}{(x + 4) (x - 2)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$\frac{2 ((- x + 1) (2 x + 5))}{(x + 4) (x - 2)} = 0$

$\frac{2 (- x + 1) (2 x + 5)}{(x + 4) (x - 2)} = 0$

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

$\frac{2 (- (x) + 1) (2 x + 5)}{(x + 4) (x - 2)} = 0$

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

$\frac{2 (- (x) - 1 (- 1)) (2 x + 5)}{(x + 4) (x - 2)} = 0$

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

$\frac{2 (- (x - 1)) (2 x + 5)}{(x + 4) (x - 2)} = 0$

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

$\frac{2 (- 1 (x - 1)) (2 x + 5)}{(x + 4) (x - 2)} = 0$

Move the negative in front of the fraction.

$- \frac{(2 (x - 1)) (2 x + 5)}{(x + 4) (x - 2)} = 0$

Reorder factors in $- \frac{(2 (x - 1)) (2 x + 5)}{(x + 4) (x - 2)}$ .

$- \frac{2 (x - 1) (2 x + 5)}{(x + 4) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$2 (x - 1) (2 x + 5) = 0$

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 $0$ .

$x - 1 = 0$

$2 x + 5 = 0$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

Set $2 x + 5$ equal to $0$ and solve for $x$ .

Set $2 x + 5$ equal to $0$ .

$2 x + 5 = 0$

Solve $2 x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$2 x = - 5$

Divide each term in $2 x = - 5$ by $2$ and simplify.

Divide each term in $2 x = - 5$ by $2$ .

$\frac{2 x}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 5}{2}$

Divide $x$ by $1$ .

$x = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

The final solution is all the values that make $2 (x - 1) (2 x + 5) = 0$ true.

$x = 1, - \frac{5}{2}$

Solve by Factoring 5/(x+4)=4+3/(x-2)

Related Questions

Support LCMGCF.com by disabling your adblocker.