Solve by Factoring 10/x+3=(x+9)/(x-4)

$\frac{10}{x} + 3 = \frac{x + 9}{x - 4}$

Step 1

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

$\frac{10}{x} + 3 - \frac{x + 9}{x - 4} = 0$

Step 2

Simplify $\frac{10}{x} + 3 - \frac{x + 9}{x - 4}$ .

To write $\frac{10}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{10}{x} \cdot \frac{x - 4}{x - 4} - \frac{x + 9}{x - 4} + 3 = 0$

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

$\frac{10}{x} \cdot \frac{x - 4}{x - 4} - \frac{x + 9}{x - 4} \cdot \frac{x}{x} + 3 = 0$

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

Multiply $\frac{10}{x}$ by $\frac{x - 4}{x - 4}$ .

$\frac{10 (x - 4)}{x (x - 4)} - \frac{x + 9}{x - 4} \cdot \frac{x}{x} + 3 = 0$

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

$\frac{10 (x - 4)}{x (x - 4)} - \frac{(x + 9) x}{(x - 4) x} + 3 = 0$

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

$\frac{10 (x - 4)}{x (x - 4)} - \frac{(x + 9) x}{x (x - 4)} + 3 = 0$

Combine the numerators over the common denominator.

$\frac{10 (x - 4) - (x + 9) x}{x (x - 4)} + 3 = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{10 x + 10 \cdot - 4 - (x + 9) x}{x (x - 4)} + 3 = 0$

Multiply $10$ by $- 4$ .

$\frac{10 x - 40 - (x + 9) x}{x (x - 4)} + 3 = 0$

Apply the distributive property.

$\frac{10 x - 40 + (- x - 1 \cdot 9) x}{x (x - 4)} + 3 = 0$

Multiply $- 1$ by $9$ .

$\frac{10 x - 40 + (- x - 9) x}{x (x - 4)} + 3 = 0$

Apply the distributive property.

$\frac{10 x - 40 - x \cdot x - 9 x}{x (x - 4)} + 3 = 0$

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

Move $x$ .

$\frac{10 x - 40 - (x \cdot x) - 9 x}{x (x - 4)} + 3 = 0$

Multiply $x$ by $x$ .

$\frac{10 x - 40 - x^{2} - 9 x}{x (x - 4)} + 3 = 0$

Subtract $9 x$ from $10 x$ .

$\frac{x - 40 - x^{2}}{x (x - 4)} + 3 = 0$

Reorder terms.

$\frac{- x^{2} + x - 40}{x (x - 4)} + 3 = 0$

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

$\frac{- x^{2} + x - 40}{x (x - 4)} + 3 \cdot \frac{x (x - 4)}{x (x - 4)} = 0$

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

$\frac{- x^{2} + x - 40}{x (x - 4)} + \frac{3 (x (x - 4))}{x (x - 4)} = 0$

Combine the numerators over the common denominator.

$\frac{- x^{2} + x - 40 + 3 (x (x - 4))}{x (x - 4)} = 0$

Simplify the numerator.

Apply the distributive property.

$\frac{- x^{2} + x - 40 + 3 x \cdot x + 3 x \cdot - 4}{x (x - 4)} = 0$

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

Move $x$ .

$\frac{- x^{2} + x - 40 + 3 (x \cdot x) + 3 x \cdot - 4}{x (x - 4)} = 0$

Multiply $x$ by $x$ .

$\frac{- x^{2} + x - 40 + 3 x^{2} + 3 x \cdot - 4}{x (x - 4)} = 0$

Multiply $- 4$ by $3$ .

$\frac{- x^{2} + x - 40 + 3 x^{2} - 12 x}{x (x - 4)} = 0$

Add $- x^{2}$ and $3 x^{2}$ .

$\frac{2 x^{2} + x - 40 - 12 x}{x (x - 4)} = 0$

Subtract $12 x$ from $x$ .

$\frac{2 x^{2} - 11 x - 40}{x (x - 4)} = 0$

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 $a \cdot c = 2 \cdot - 40 = - 80$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 x$ .

$\frac{2 x^{2} - 11 (x) - 40}{x (x - 4)} = 0$

Rewrite $- 11$ as $5$ plus $- 16$

$\frac{2 x^{2} + (5 - 16) x - 40}{x (x - 4)} = 0$

Apply the distributive property.

$\frac{2 x^{2} + 5 x - 16 x - 40}{x (x - 4)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 5 x) - 16 x - 40}{x (x - 4)} = 0$

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

$\frac{x (2 x + 5) - 8 (2 x + 5)}{x (x - 4)} = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

$\frac{(2 x + 5) (x - 8)}{x (x - 4)} = 0$

Step 3

Set the numerator equal to zero.

$(2 x + 5) (x - 8) = 0$

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 $0$ .

$2 x + 5 = 0$

$x - 8 = 0$

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.

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

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.

$x = 8$

The final solution is all the values that make $(2 x + 5) (x - 8) = 0$ true.

$x = - \frac{5}{2}, 8$

Solve by Factoring 10/x+3=(x+9)/(x-4)

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