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

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

Step 1

Subtract $\frac{x - 2}{3}$ from both sides of the equation.

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

Step 2

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $x \cdot 3$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $3$ by $2$ .

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

Multiply $- 5$ by $3$ .

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

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

Apply the distributive property.

$\frac{6 x - 15 - x \cdot x + 2 x}{3 x} = 0$

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

Move $x$ .

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

Multiply $x$ by $x$ .

$\frac{6 x - 15 - x^{2} + 2 x}{3 x} = 0$

Add $6 x$ and $2 x$ .

$\frac{8 x - 15 - x^{2}}{3 x} = 0$

Reorder terms.

$\frac{- x^{2} + 8 x - 15}{3 x} = 0$

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 $a \cdot c = - 1 \cdot - 15 = 15$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x$ .

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

Rewrite $8$ as $3$ plus $5$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

Factor the polynomial by factoring out the greatest common factor, $- x + 3$ .

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

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

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

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

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

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

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

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

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

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$(x - 3) (x - 5) = 0$

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 $0$ .

$x - 3 = 0$

$x - 5 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $(x - 3) (x - 5) = 0$ true.

$x = 3, 5$

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

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