Solve 1/lambda-1-2/x+1=5/x^2-1

$$\frac { 1 } { \lambda - 1 } - \frac { 2 } { x + 1 } = \frac { 5 } { x ^ { 2 } - 1 }$$

Solve for λ (complex solution)

$$\lambda =\frac{x^{2}+2x+2}{2x+3}$$

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

Steps for Solving Linear Equation

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

$$x^{2}-1-\left(x-1\right)\left(\lambda -1\right)\times 2=\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply x-1 by \lambda -1.

$$x^{2}-1-\left(x\lambda -x-\lambda +1\right)\times 2=\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply x\lambda -x-\lambda +1 by 2.

$$x^{2}-1-\left(2x\lambda -2x-2\lambda +2\right)=\left(\lambda -1\right)\times 5$$

To find the opposite of 2x\lambda -2x-2\lambda +2, find the opposite of each term.

$$x^{2}-1-2x\lambda +2x+2\lambda -2=\left(\lambda -1\right)\times 5$$

Subtract 2 from -1 to get -3.

$$x^{2}-3-2x\lambda +2x+2\lambda =\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply \lambda -1 by 5.

$$x^{2}-3-2x\lambda +2x+2\lambda =5\lambda -5$$

Subtract 5\lambda from both sides.

$$x^{2}-3-2x\lambda +2x+2\lambda -5\lambda =-5$$

Combine 2\lambda and -5\lambda to get -3\lambda .

$$x^{2}-3-2x\lambda +2x-3\lambda =-5$$

Subtract x^{2} from both sides.

$$-3-2x\lambda +2x-3\lambda =-5-x^{2}$$

Add 3 to both sides.

$$-2x\lambda +2x-3\lambda =-5-x^{2}+3$$

Add -5 and 3 to get -2.

$$-2x\lambda +2x-3\lambda =-2-x^{2}$$

Subtract 2x from both sides.

$$-2x\lambda -3\lambda =-2-x^{2}-2x$$

Combine all terms containing \lambda .

$$\left(-2x-3\right)\lambda =-2-x^{2}-2x$$

The equation is in standard form.

$$\left(-2x-3\right)\lambda =-x^{2}-2x-2$$

Divide both sides by -3-2x.

$$\frac{\left(-2x-3\right)\lambda }{-2x-3}=\frac{-x^{2}-2x-2}{-2x-3}$$

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

$$\lambda =\frac{-x^{2}-2x-2}{-2x-3}$$

Divide -x^{2}-2x-2 by -3-2x.

$$\lambda =\frac{x^{2}+2x+2}{2x+3}$$

Variable \lambda cannot be equal to 1.

$$\lambda =\frac{x^{2}+2x+2}{2x+3}\text{, }\lambda \neq 1$$

Solve for λ

$$\lambda =\frac{x^{2}+2x+2}{2x+3}$$

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

Steps for Solving Linear Equation

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

$$x^{2}-1-\left(x-1\right)\left(\lambda -1\right)\times 2=\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply x-1 by \lambda -1.

$$x^{2}-1-\left(x\lambda -x-\lambda +1\right)\times 2=\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply x\lambda -x-\lambda +1 by 2.

$$x^{2}-1-\left(2x\lambda -2x-2\lambda +2\right)=\left(\lambda -1\right)\times 5$$

To find the opposite of 2x\lambda -2x-2\lambda +2, find the opposite of each term.

$$x^{2}-1-2x\lambda +2x+2\lambda -2=\left(\lambda -1\right)\times 5$$

Subtract 2 from -1 to get -3.

$$x^{2}-3-2x\lambda +2x+2\lambda =\left(\lambda -1\right)\times 5$$

Use the distributive property to multiply \lambda -1 by 5.

$$x^{2}-3-2x\lambda +2x+2\lambda =5\lambda -5$$

Subtract 5\lambda from both sides.

$$x^{2}-3-2x\lambda +2x+2\lambda -5\lambda =-5$$

Combine 2\lambda and -5\lambda to get -3\lambda .

$$x^{2}-3-2x\lambda +2x-3\lambda =-5$$

Subtract x^{2} from both sides.

$$-3-2x\lambda +2x-3\lambda =-5-x^{2}$$

Add 3 to both sides.

$$-2x\lambda +2x-3\lambda =-5-x^{2}+3$$

Add -5 and 3 to get -2.

$$-2x\lambda +2x-3\lambda =-2-x^{2}$$

Subtract 2x from both sides.

$$-2x\lambda -3\lambda =-2-x^{2}-2x$$

Combine all terms containing \lambda .

$$\left(-2x-3\right)\lambda =-2-x^{2}-2x$$

The equation is in standard form.

$$\left(-2x-3\right)\lambda =-x^{2}-2x-2$$

Divide both sides by -3-2x.

$$\frac{\left(-2x-3\right)\lambda }{-2x-3}=\frac{-x^{2}-2x-2}{-2x-3}$$

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

$$\lambda =\frac{-x^{2}-2x-2}{-2x-3}$$

Divide -x^{2}-2x-2 by -3-2x.

$$\lambda =\frac{x^{2}+2x+2}{2x+3}$$

Variable \lambda cannot be equal to 1.

$$\lambda =\frac{x^{2}+2x+2}{2x+3}\text{, }\lambda \neq 1$$

Solve for x (complex solution)

$$x=\sqrt{\lambda ^{2}+\lambda -1}+\lambda -1$$

$$x=-\sqrt{\lambda ^{2}+\lambda -1}+\lambda -1\text{, }\lambda \neq 1$$

Solve for x

$$x=\sqrt{\lambda ^{2}+\lambda -1}+\lambda -1$$

$$x=-\sqrt{\lambda ^{2}+\lambda -1}+\lambda -1\text{, }\lambda \leq \frac{-\sqrt{5}-1}{2}\text{ or }\left(\lambda \neq 1\text{ and }\lambda \geq \frac{\sqrt{5}-1}{2}\right)$$