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

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

Solve for x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps for Solving Linear Equation

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

$$\left(2x-3\right)\left(2x+1\right)=\left(x-1\right)\left(2x-3\right)+\left(x-1\right)\left(2x+1\right)$$

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

$$4x^{2}-4x-3=\left(x-1\right)\left(2x-3\right)+\left(x-1\right)\left(2x+1\right)$$

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

$$4x^{2}-4x-3=2x^{2}-5x+3+\left(x-1\right)\left(2x+1\right)$$

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

$$4x^{2}-4x-3=2x^{2}-5x+3+2x^{2}-x-1$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4x^{2}-4x-3=4x^{2}-5x+3-x-1$$

Combine -5x and -x to get -6x.

$$4x^{2}-4x-3=4x^{2}-6x+3-1$$

Subtract 1 from 3 to get 2.

$$4x^{2}-4x-3=4x^{2}-6x+2$$

Subtract 4x^{2} from both sides.

$$4x^{2}-4x-3-4x^{2}=-6x+2$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-4x-3=-6x+2$$

Add 6x to both sides.

$$-4x-3+6x=2$$

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

$$2x-3=2$$

Add 3 to both sides.

$$2x=2+3$$

Add 2 and 3 to get 5.

$$2x=5$$

Divide both sides by 2.

$$x=\frac{5}{2}$$