Solve by Factoring 1/(2x-10)-2/(x-5)=3/4

$\frac{1}{2 x - 10} - \frac{2}{x - 5} = \frac{3}{4}$

Step 1

Subtract $\frac{3}{4}$ from both sides of the equation.

$\frac{1}{2 x - 10} - \frac{2}{x - 5} - \frac{3}{4} = 0$

Step 2

Simplify $\frac{1}{2 x - 10} - \frac{2}{x - 5} - \frac{3}{4}$ .

Factor $2$ out of $2 x - 10$ .

Factor $2$ out of $2 x$ .

$\frac{1}{2 (x) - 10} - \frac{2}{x - 5} - \frac{3}{4} = 0$

Factor $2$ out of $- 10$ .

$\frac{1}{2 x + 2 \cdot - 5} - \frac{2}{x - 5} - \frac{3}{4} = 0$

Factor $2$ out of $2 x + 2 \cdot - 5$ .

$\frac{1}{2 (x - 5)} - \frac{2}{x - 5} - \frac{3}{4} = 0$

Find the common denominator.

Multiply $\frac{1}{2 (x - 5)}$ by $\frac{2}{2}$ .

$\frac{1}{2 (x - 5)} \cdot \frac{2}{2} - \frac{2}{x - 5} - \frac{3}{4} = 0$

Multiply $\frac{1}{2 (x - 5)}$ by $\frac{2}{2}$ .

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

Multiply $\frac{2}{x - 5}$ by $\frac{4}{4}$ .

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

Multiply $\frac{2}{x - 5}$ by $\frac{4}{4}$ .

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

Multiply $\frac{3}{4}$ by $\frac{x - 5}{x - 5}$ .

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

Multiply $\frac{3}{4}$ by $\frac{x - 5}{x - 5}$ .

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

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

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

Multiply $2$ by $2$ .

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

Reorder the factors of $(x - 5) \cdot 4$ .

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

Combine the numerators over the common denominator.

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

Simplify each term.

Multiply $- 2$ by $4$ .

$\frac{2 - 8 - 3 (x - 5)}{4 (x - 5)} = 0$

Apply the distributive property.

$\frac{2 - 8 - 3 x - 3 \cdot - 5}{4 (x - 5)} = 0$

Multiply $- 3$ by $- 5$ .

$\frac{2 - 8 - 3 x + 15}{4 (x - 5)} = 0$

Subtract $8$ from $2$ .

$\frac{- 6 - 3 x + 15}{4 (x - 5)} = 0$

Add $- 6$ and $15$ .

$\frac{- 3 x + 9}{4 (x - 5)} = 0$

Factor $3$ out of $- 3 x + 9$ .

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

$\frac{3 (- x) + 9}{4 (x - 5)} = 0$

Factor $3$ out of $9$ .

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

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

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

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

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

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{3 (- (x) - 1 (- 3))}{4 (x - 5)} = 0$

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

$\frac{3 (- 1 (x - 3))}{4 (x - 5)} = 0$

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$3 (x - 3) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $3 (x - 3) = 0$ by $3$ and simplify.

Divide each term in $3 (x - 3) = 0$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x - 3$ by $1$ .

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

Simplify the right side.

Divide $0$ by $3$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Solve by Factoring 1/(2x-10)-2/(x-5)=3/4

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