Solve k/2x-4-1=x/1-2

$$\frac { k } { 2 x - 4 } - 1 = \frac { x } { 1 - 2 }$$

Solve for k

$$k=-2\left(x-2\right)\left(x-1\right)$$

$$x\neq 2$$

Solution Steps

Multiply both sides of the equation by 2\left(x-2\right).

$$k+2\left(x-2\right)\left(-1\right)=\left(-2x+4\right)x$$

Multiply 2 and -1 to get -2.

$$k-2\left(x-2\right)=\left(-2x+4\right)x$$

Use the distributive property to multiply -2 by x-2.

$$k-2x+4=\left(-2x+4\right)x$$

Use the distributive property to multiply -2x+4 by x.

$$k-2x+4=-2x^{2}+4x$$

Add 2x to both sides.

$$k+4=-2x^{2}+4x+2x$$

Combine 4x and 2x to get 6x.

$$k+4=-2x^{2}+6x$$

Subtract 4 from both sides.

$$k=-2x^{2}+6x-4$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{1-2k}+3}{2}\text{, }&\text{unconditionally}\\x=\frac{\sqrt{1-2k}+3}{2}\text{, }&k\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{-\sqrt{1-2k}+3}{2}\text{, }&k\leq \frac{1}{2}\\x=\frac{\sqrt{1-2k}+3}{2}\text{, }&k\neq 0\text{ and }k\leq \frac{1}{2}\end{matrix}\right.$$