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

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

Step 1

Subtract $\frac{1}{5 - 2 x}$ from both sides of the equation.

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

Step 2

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

To write $\frac{3}{2 x - 3}$ as a fraction with a common denominator, multiply by $\frac{5 - 2 x}{5 - 2 x}$ .

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

To write $- \frac{1}{5 - 2 x}$ as a fraction with a common denominator, multiply by $\frac{2 x - 3}{2 x - 3}$ .

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

Write each expression with a common denominator of $(2 x - 3) (5 - 2 x)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3}{2 x - 3}$ by $\frac{5 - 2 x}{5 - 2 x}$ .

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

Multiply $\frac{1}{5 - 2 x}$ by $\frac{2 x - 3}{2 x - 3}$ .

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

Reorder the factors of $(5 - 2 x) (2 x - 3)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $5$ .

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

Multiply $- 2$ by $3$ .

$\frac{15 - 6 x - (2 x - 3)}{(2 x - 3) (5 - 2 x)} = 0$

Apply the distributive property.

$\frac{15 - 6 x - (2 x) - - 3}{(2 x - 3) (5 - 2 x)} = 0$

Multiply $2$ by $- 1$ .

$\frac{15 - 6 x - 2 x - - 3}{(2 x - 3) (5 - 2 x)} = 0$

Multiply $- 1$ by $- 3$ .

$\frac{15 - 6 x - 2 x + 3}{(2 x - 3) (5 - 2 x)} = 0$

Add $15$ and $3$ .

$\frac{- 6 x - 2 x + 18}{(2 x - 3) (5 - 2 x)} = 0$

Subtract $2 x$ from $- 6 x$ .

$\frac{- 8 x + 18}{(2 x - 3) (5 - 2 x)} = 0$

Factor $2$ out of $- 8 x + 18$ .

Factor $2$ out of $- 8 x$ .

$\frac{2 (- 4 x) + 18}{(2 x - 3) (5 - 2 x)} = 0$

Factor $2$ out of $18$ .

$\frac{2 (- 4 x) + 2 (9)}{(2 x - 3) (5 - 2 x)} = 0$

Factor $2$ out of $2 (- 4 x) + 2 (9)$ .

$\frac{2 (- 4 x + 9)}{(2 x - 3) (5 - 2 x)} = 0$

Factor $- 1$ out of $- 4 x$ .

$\frac{2 (- (4 x) + 9)}{(2 x - 3) (5 - 2 x)} = 0$

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{2 (- (4 x) - 1 (- 9))}{(2 x - 3) (5 - 2 x)} = 0$

Factor $- 1$ out of $- (4 x) - 1 (- 9)$ .

$\frac{2 (- (4 x - 9))}{(2 x - 3) (5 - 2 x)} = 0$

Rewrite $- (4 x - 9)$ as $- 1 (4 x - 9)$ .

$\frac{2 (- 1 (4 x - 9))}{(2 x - 3) (5 - 2 x)} = 0$

Move the negative in front of the fraction.

$- \frac{2 (4 x - 9)}{(2 x - 3) (5 - 2 x)} = 0$

Step 3

Set the numerator equal to zero.

$2 (4 x - 9) = 0$

Step 4

Solve the equation for $x$ .

Divide each term in $2 (4 x - 9) = 0$ by $2$ and simplify.

Divide each term in $2 (4 x - 9) = 0$ by $2$ .

$\frac{2 (4 x - 9)}{2} = \frac{0}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (4 x - 9)}{2} = \frac{0}{2}$

Divide $4 x - 9$ by $1$ .

$4 x - 9 = \frac{0}{2}$

Simplify the right side.

Divide $0$ by $2$ .

$4 x - 9 = 0$

Add $9$ to both sides of the equation.

$4 x = 9$

Divide each term in $4 x = 9$ by $4$ and simplify.

Divide each term in $4 x = 9$ by $4$ .

$\frac{4 x}{4} = \frac{9}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{9}{4}$

Divide $x$ by $1$ .

$x = \frac{9}{4}$

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

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