Solve by Factoring 24/(2x-10)+1/(x-5)=13/2

$\frac{24}{2 x - 10} + \frac{1}{x - 5} = \frac{13}{2}$

Step 1

Subtract $\frac{13}{2}$ from both sides of the equation.

$\frac{24}{2 x - 10} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Step 2

Simplify $\frac{24}{2 x - 10} + \frac{1}{x - 5} - \frac{13}{2}$ .

Cancel the common factor of $24$ and $2 x - 10$ .

Factor $2$ out of $24$ .

$\frac{2 \cdot 12}{2 x - 10} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Cancel the common factors.

Factor $2$ out of $2 x$ .

$\frac{2 \cdot 12}{2 (x) - 10} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Factor $2$ out of $- 10$ .

$\frac{2 \cdot 12}{2 (x) + 2 (- 5)} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Factor $2$ out of $2 (x) + 2 (- 5)$ .

$\frac{2 \cdot 12}{2 (x - 5)} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Cancel the common factor.

$\frac{2 \cdot 12}{2 (x - 5)} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Rewrite the expression.

$\frac{12}{x - 5} + \frac{1}{x - 5} - \frac{13}{2} = 0$

Combine the numerators over the common denominator.

$\frac{12 + 1}{x - 5} + \frac{- 13}{2} = 0$

Add $12$ and $1$ .

$\frac{13}{x - 5} + \frac{- 13}{2} = 0$

Move the negative in front of the fraction.

$\frac{13}{x - 5} - \frac{13}{2} = 0$

To write $\frac{13}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{13}{x - 5} \cdot \frac{2}{2} - \frac{13}{2} = 0$

To write $- \frac{13}{2}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{13}{x - 5} \cdot \frac{2}{2} - \frac{13}{2} \cdot \frac{x - 5}{x - 5} = 0$

Write each expression with a common denominator of $(x - 5) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{13}{x - 5}$ by $\frac{2}{2}$ .

$\frac{13 \cdot 2}{(x - 5) \cdot 2} - \frac{13}{2} \cdot \frac{x - 5}{x - 5} = 0$

Multiply $\frac{13}{2}$ by $\frac{x - 5}{x - 5}$ .

$\frac{13 \cdot 2}{(x - 5) \cdot 2} - \frac{13 (x - 5)}{2 (x - 5)} = 0$

Reorder the factors of $(x - 5) \cdot 2$ .

$\frac{13 \cdot 2}{2 (x - 5)} - \frac{13 (x - 5)}{2 (x - 5)} = 0$

Combine the numerators over the common denominator.

$\frac{13 \cdot 2 - 13 (x - 5)}{2 (x - 5)} = 0$

Simplify the numerator.

Factor $13$ out of $13 \cdot 2 - 13 (x - 5)$ .

Factor $13$ out of $13 \cdot 2$ .

$\frac{13 (2) - 13 (x - 5)}{2 (x - 5)} = 0$

Factor $13$ out of $- 13 (x - 5)$ .

$\frac{13 (2) + 13 (- (x - 5))}{2 (x - 5)} = 0$

Factor $13$ out of $13 (2) + 13 (- (x - 5))$ .

$\frac{13 (2 - (x - 5))}{2 (x - 5)} = 0$

Apply the distributive property.

$\frac{13 (2 - x - - 5)}{2 (x - 5)} = 0$

Multiply $- 1$ by $- 5$ .

$\frac{13 (2 - x + 5)}{2 (x - 5)} = 0$

Add $2$ and $5$ .

$\frac{13 (- x + 7)}{2 (x - 5)} = 0$

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

$\frac{13 (- (x) + 7)}{2 (x - 5)} = 0$

Rewrite $7$ as $- 1 (- 7)$ .

$\frac{13 (- (x) - 1 (- 7))}{2 (x - 5)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 7)$ .

$\frac{13 (- (x - 7))}{2 (x - 5)} = 0$

Rewrite $- (x - 7)$ as $- 1 (x - 7)$ .

$\frac{13 (- 1 (x - 7))}{2 (x - 5)} = 0$

Move the negative in front of the fraction.

$- \frac{13 (x - 7)}{2 (x - 5)} = 0$

Step 3

Set the numerator equal to zero.

$13 (x - 7) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $13 (x - 7) = 0$ by $13$ and simplify.

Divide each term in $13 (x - 7) = 0$ by $13$ .

$\frac{13 (x - 7)}{13} = \frac{0}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 (x - 7)}{13} = \frac{0}{13}$

Divide $x - 7$ by $1$ .

$x - 7 = \frac{0}{13}$

Simplify the right side.

Divide $0$ by $13$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Solve by Factoring 24/(2x-10)+1/(x-5)=13/2

Related Questions

Support LCMGCF.com by disabling your adblocker.