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

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

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps for Solving Linear Equation

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

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

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

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

Use the distributive property to multiply x+1 by 2x-3 and combine like terms.

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

Combine x^{2} and 2x^{2} to get 3x^{2}.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

Use the distributive property to multiply x^{2}-2x+1 by 3.

$$3x^{2}-2x-3=3x^{2}-6x+3$$

Subtract 3x^{2} from both sides.

$$3x^{2}-2x-3-3x^{2}=-6x+3$$

Combine 3x^{2} and -3x^{2} to get 0.

$$-2x-3=-6x+3$$

Add 6x to both sides.

$$-2x-3+6x=3$$

Combine -2x and 6x to get 4x.

$$4x-3=3$$

Add 3 to both sides.

$$4x=3+3$$

Add 3 and 3 to get 6.

$$4x=6$$

Divide both sides by 4.

$$x=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

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