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

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

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 2x-3,2x+1,x-1.

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

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

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

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

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

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

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

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

$$4x^{2}-6x-1+3=\left(2x-3\right)\left(2x+1\right)$$

Add -1 and 3 to get 2.

$$4x^{2}-6x+2=\left(2x-3\right)\left(2x+1\right)$$

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

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

Subtract 4x^{2} from both sides.

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

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

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

Add 4x to both sides.

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

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

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

Subtract 2 from both sides.

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

Subtract 2 from -3 to get -5.

$$-2x=-5$$

Divide both sides by -2.

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

Fraction \frac{-5}{-2} can be simplified to \frac{5}{2} by removing the negative sign from both the numerator and the denominator.

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