Solve by Factoring 24/(2x-6)+1/(x-3)=13/2

$\frac{24}{2 x - 6} + \frac{1}{x - 3} = \frac{13}{2}$

Step 1

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

$\frac{24}{2 x - 6} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Step 2

Simplify $\frac{24}{2 x - 6} + \frac{1}{x - 3} - \frac{13}{2}$ .

Cancel the common factor of $24$ and $2 x - 6$ .

Factor $2$ out of $24$ .

$\frac{2 \cdot 12}{2 x - 6} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Cancel the common factors.

Factor $2$ out of $2 x$ .

$\frac{2 \cdot 12}{2 (x) - 6} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Factor $2$ out of $- 6$ .

$\frac{2 \cdot 12}{2 (x) + 2 (- 3)} + \frac{1}{x - 3} - \frac{13}{2} = 0$

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

$\frac{2 \cdot 12}{2 (x - 3)} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Cancel the common factor.

$\frac{2 \cdot 12}{2 (x - 3)} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Rewrite the expression.

$\frac{12}{x - 3} + \frac{1}{x - 3} - \frac{13}{2} = 0$

Combine the numerators over the common denominator.

$\frac{12 + 1}{x - 3} + \frac{- 13}{2} = 0$

Add $12$ and $1$ .

$\frac{13}{x - 3} + \frac{- 13}{2} = 0$

Move the negative in front of the fraction.

$\frac{13}{x - 3} - \frac{13}{2} = 0$

To write $\frac{13}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{13}{x - 3} \cdot \frac{2}{2} - \frac{13}{2} = 0$

To write $- \frac{13}{2}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{13}{x - 3} \cdot \frac{2}{2} - \frac{13}{2} \cdot \frac{x - 3}{x - 3} = 0$

Write each expression with a common denominator of $(x - 3) \cdot 2$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{13}{x - 3}$ by $\frac{2}{2}$ .

$\frac{13 \cdot 2}{(x - 3) \cdot 2} - \frac{13}{2} \cdot \frac{x - 3}{x - 3} = 0$

Multiply $\frac{13}{2}$ by $\frac{x - 3}{x - 3}$ .

$\frac{13 \cdot 2}{(x - 3) \cdot 2} - \frac{13 (x - 3)}{2 (x - 3)} = 0$

Reorder the factors of $(x - 3) \cdot 2$ .

$\frac{13 \cdot 2}{2 (x - 3)} - \frac{13 (x - 3)}{2 (x - 3)} = 0$

Combine the numerators over the common denominator.

$\frac{13 \cdot 2 - 13 (x - 3)}{2 (x - 3)} = 0$

Simplify the numerator.

Factor $13$ out of $13 \cdot 2 - 13 (x - 3)$ .

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

$\frac{13 (2) - 13 (x - 3)}{2 (x - 3)} = 0$

Factor $13$ out of $- 13 (x - 3)$ .

$\frac{13 (2) + 13 (- (x - 3))}{2 (x - 3)} = 0$

Factor $13$ out of $13 (2) + 13 (- (x - 3))$ .

$\frac{13 (2 - (x - 3))}{2 (x - 3)} = 0$

Apply the distributive property.

$\frac{13 (2 - x - - 3)}{2 (x - 3)} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{13 (2 - x + 3)}{2 (x - 3)} = 0$

Add $2$ and $3$ .

$\frac{13 (- x + 5)}{2 (x - 3)} = 0$

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

$\frac{13 (- (x) + 5)}{2 (x - 3)} = 0$

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{13 (- (x) - 1 (- 5))}{2 (x - 3)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 5)$ .

$\frac{13 (- (x - 5))}{2 (x - 3)} = 0$

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{13 (- 1 (x - 5))}{2 (x - 3)} = 0$

Move the negative in front of the fraction.

$- \frac{13 (x - 5)}{2 (x - 3)} = 0$

Step 3

Set the numerator equal to zero.

$13 (x - 5) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $13 (x - 5) = 0$ by $13$ and simplify.

Divide each term in $13 (x - 5) = 0$ by $13$ .

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

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

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

Divide $x - 5$ by $1$ .

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

Simplify the right side.

Divide $0$ by $13$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Solve by Factoring 24/(2x-6)+1/(x-3)=13/2

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