Solve x/2k^2-2+1-2x/k^2-k-2=1/k^2-3k+2

$$\frac { x } { 2 k ^ { 2 } - 2 } + \frac { 1 - 2 x } { k ^ { 2 } - k - 2 } = \frac { 1 } { k ^ { 2 } - 3 k + 2 }$$

Solve for k (complex solution)

$$k=\frac{2}{3}-\frac{4}{3x}$$

$$x\neq \frac{4}{5}\text{ and }x\neq 0\text{ and }x\neq -4\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Variable k cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by 2\left(k-2\right)\left(k-1\right)\left(k+1\right), the least common multiple of 2k^{2}-2,k^{2}-k-2,k^{2}-3k+2.

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

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

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

Use the distributive property to multiply 2k-2 by 1-2x.

$$kx-2x+2k-4xk-2+4x=2k+2$$

Combine kx and -4xk to get -3kx.

$$-3kx-2x+2k-2+4x=2k+2$$

Combine -2x and 4x to get 2x.

$$-3kx+2x+2k-2=2k+2$$

Subtract 2k from both sides.

$$-3kx+2x+2k-2-2k=2$$

Combine 2k and -2k to get 0.

$$-3kx+2x-2=2$$

Subtract 2x from both sides.

$$-3kx-2=2-2x$$

Add 2 to both sides.

$$-3kx=2-2x+2$$

Add 2 and 2 to get 4.

$$-3kx=4-2x$$

The equation is in standard form.

$$\left(-3x\right)k=4-2x$$

Divide both sides by -3x.

$$\frac{\left(-3x\right)k}{-3x}=\frac{4-2x}{-3x}$$

Dividing by -3x undoes the multiplication by -3x.

$$k=\frac{4-2x}{-3x}$$

Divide 4-2x by -3x.

$$k=\frac{2}{3}-\frac{4}{3x}$$

Variable k cannot be equal to any of the values -1,1,2.

$$k=\frac{2}{3}-\frac{4}{3x}\text{, }k\neq -1\text{ and }k\neq 1\text{ and }k\neq 2$$

Solve for x (complex solution)

$$x=\frac{4}{2-3k}$$

$$k\neq \frac{2}{3}\text{ and }k\neq -1\text{ and }k\neq 1\text{ and }k\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(k-2\right)\left(k-1\right)\left(k+1\right), the least common multiple of 2k^{2}-2,k^{2}-k-2,k^{2}-3k+2.

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

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

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

Use the distributive property to multiply 2k-2 by 1-2x.

$$kx-2x+2k-4kx-2+4x=2k+2$$

Combine kx and -4kx to get -3kx.

$$-3kx-2x+2k-2+4x=2k+2$$

Combine -2x and 4x to get 2x.

$$-3kx+2x+2k-2=2k+2$$

Subtract 2k from both sides.

$$-3kx+2x-2=2k+2-2k$$

Combine 2k and -2k to get 0.

$$-3kx+2x-2=2$$

Add 2 to both sides.

$$-3kx+2x=2+2$$

Add 2 and 2 to get 4.

$$-3kx+2x=4$$

Combine all terms containing x.

$$\left(-3k+2\right)x=4$$

The equation is in standard form.

$$\left(2-3k\right)x=4$$

Divide both sides by 2-3k.

$$\frac{\left(2-3k\right)x}{2-3k}=\frac{4}{2-3k}$$

Dividing by 2-3k undoes the multiplication by 2-3k.

$$x=\frac{4}{2-3k}$$

Solve for k

$$k=\frac{2}{3}-\frac{4}{3x}$$

$$x\neq -4\text{ and }x\neq 0\text{ and }x\neq \frac{4}{5}\text{ and }x\neq -1$$

Steps for Solving Linear Equation

Variable k cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by 2\left(k-2\right)\left(k-1\right)\left(k+1\right), the least common multiple of 2k^{2}-2,k^{2}-k-2,k^{2}-3k+2.

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

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

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

Use the distributive property to multiply 2k-2 by 1-2x.

$$kx-2x+2k-4xk-2+4x=2k+2$$

Combine kx and -4xk to get -3kx.

$$-3kx-2x+2k-2+4x=2k+2$$

Combine -2x and 4x to get 2x.

$$-3kx+2x+2k-2=2k+2$$

Subtract 2k from both sides.

$$-3kx+2x+2k-2-2k=2$$

Combine 2k and -2k to get 0.

$$-3kx+2x-2=2$$

Subtract 2x from both sides.

$$-3kx-2=2-2x$$

Add 2 to both sides.

$$-3kx=2-2x+2$$

Add 2 and 2 to get 4.

$$-3kx=4-2x$$

The equation is in standard form.

$$\left(-3x\right)k=4-2x$$

Divide both sides by -3x.

$$\frac{\left(-3x\right)k}{-3x}=\frac{4-2x}{-3x}$$

Dividing by -3x undoes the multiplication by -3x.

$$k=\frac{4-2x}{-3x}$$

Divide 4-2x by -3x.

$$k=\frac{2}{3}-\frac{4}{3x}$$

Variable k cannot be equal to any of the values -1,1,2.

$$k=\frac{2}{3}-\frac{4}{3x}\text{, }k\neq -1\text{ and }k\neq 1\text{ and }k\neq 2$$

Solve for x

$$x=\frac{4}{2-3k}$$

$$k\neq \frac{2}{3}\text{ and }k\neq 2\text{ and }|k|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(k-2\right)\left(k-1\right)\left(k+1\right), the least common multiple of 2k^{2}-2,k^{2}-k-2,k^{2}-3k+2.

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

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

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

Use the distributive property to multiply 2k-2 by 1-2x.

$$kx-2x+2k-4kx-2+4x=2k+2$$

Combine kx and -4kx to get -3kx.

$$-3kx-2x+2k-2+4x=2k+2$$

Combine -2x and 4x to get 2x.

$$-3kx+2x+2k-2=2k+2$$

Subtract 2k from both sides.

$$-3kx+2x-2=2k+2-2k$$

Combine 2k and -2k to get 0.

$$-3kx+2x-2=2$$

Add 2 to both sides.

$$-3kx+2x=2+2$$

Add 2 and 2 to get 4.

$$-3kx+2x=4$$

Combine all terms containing x.

$$\left(-3k+2\right)x=4$$

The equation is in standard form.

$$\left(2-3k\right)x=4$$

Divide both sides by 2-3k.

$$\frac{\left(2-3k\right)x}{2-3k}=\frac{4}{2-3k}$$

Dividing by 2-3k undoes the multiplication by 2-3k.

$$x=\frac{4}{2-3k}$$