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

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

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 1,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-1,x-2,x-3.

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

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

$$\left(x^{2}-5x+6\right)\left(x+1\right)+\left(x-3\right)\left(x-1\right)\left(x+2\right)-\left(x-2\right)\left(x-1\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-5x+6 by x+1 and combine like terms.

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

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

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

Use the distributive property to multiply x^{2}-4x+3 by x+2 and combine like terms.

$$x^{3}-4x^{2}+x+6+x^{3}-2x^{2}-5x+6-\left(x-2\right)\left(x-1\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$2x^{3}-4x^{2}+x+6-2x^{2}-5x+6-\left(x-2\right)\left(x-1\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Combine -4x^{2} and -2x^{2} to get -6x^{2}.

$$2x^{3}-6x^{2}+x+6-5x+6-\left(x-2\right)\left(x-1\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Combine x and -5x to get -4x.

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

Add 6 and 6 to get 12.

$$2x^{3}-6x^{2}-4x+12-\left(x-2\right)\left(x-1\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$2x^{3}-6x^{2}-4x+12-\left(x^{2}-3x+2\right)\left(x+3\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$2x^{3}-6x^{2}-4x+12-\left(x^{3}-7x+6\right)=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

To find the opposite of x^{3}-7x+6, find the opposite of each term.

$$2x^{3}-6x^{2}-4x+12-x^{3}+7x-6=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$x^{3}-6x^{2}-4x+12+7x-6=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Combine -4x and 7x to get 3x.

$$x^{3}-6x^{2}+3x+12-6=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Subtract 6 from 12 to get 6.

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

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

$$x^{3}-6x^{2}+3x+6=\left(x^{2}-5x+6\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-5x+6 by x-1 and combine like terms.

$$x^{3}-6x^{2}+3x+6=x^{3}-6x^{2}+11x-6$$

Subtract x^{3} from both sides.

$$x^{3}-6x^{2}+3x+6-x^{3}=-6x^{2}+11x-6$$

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

$$-6x^{2}+3x+6=-6x^{2}+11x-6$$

Add 6x^{2} to both sides.

$$-6x^{2}+3x+6+6x^{2}=11x-6$$

Combine -6x^{2} and 6x^{2} to get 0.

$$3x+6=11x-6$$

Subtract 11x from both sides.

$$3x+6-11x=-6$$

Combine 3x and -11x to get -8x.

$$-8x+6=-6$$

Subtract 6 from both sides.

$$-8x=-6-6$$

Subtract 6 from -6 to get -12.

$$-8x=-12$$

Divide both sides by -8.

$$x=\frac{-12}{-8}$$

Reduce the fraction \frac{-12}{-8} to lowest terms by extracting and canceling out -4.

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