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

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

Step 1

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

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

Step 2

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

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

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

Combine $- 1$ and $\frac{x - 1}{x - 1}$ .

$\frac{x}{x - 1} + \frac{- (x - 1)}{x - 1} - \frac{x}{2} = 0$

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $x$ .

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

Add $0$ and $1$ .

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $(x - 1) \cdot 2$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- x \cdot - 1$ .

Multiply $- 1$ by $- 1$ .

$\frac{2 - x^{2} + 1 x}{2 (x - 1)} = 0$

Multiply $x$ by $1$ .

$\frac{2 - x^{2} + x}{2 (x - 1)} = 0$

Reorder terms.

$\frac{- x^{2} + x + 2}{2 (x - 1)} = 0$

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 $a \cdot c = - 1 \cdot 2 = - 2$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{- x^{2} + 1 x + 2}{2 (x - 1)} = 0$

Rewrite $1$ as $- 1$ plus $2$

$\frac{- x^{2} + (- 1 + 2) x + 2}{2 (x - 1)} = 0$

Apply the distributive property.

$\frac{- x^{2} - 1 x + 2 x + 2}{2 (x - 1)} = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(- x^{2} - 1 x) + 2 x + 2}{2 (x - 1)} = 0$

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

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

Factor the polynomial by factoring out the greatest common factor, $- x - 1$ .

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

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

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

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

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

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

$\frac{- (x + 1) (x - 2)}{2 (x - 1)} = 0$

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

$\frac{- 1 (x + 1) (x - 2)}{2 (x - 1)} = 0$

Move the negative in front of the fraction.

$- \frac{(x + 1) (x - 2)}{2 (x - 1)} = 0$

Step 3

Set the numerator equal to zero.

$(x + 1) (x - 2) = 0$

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 $0$ .

$x + 1 = 0$

$x - 2 = 0$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x + 1) (x - 2) = 0$ true.

$x = - 1, 2$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.