Solve 1/x-1-1/x-2=1/x-3-1/x-4

$$\frac { 1 } { x - 1 } - \frac { 1 } { x - 2 } = \frac { 1 } { x - 3 } - \frac { 1 } { x - 4 }$$

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 1,2,3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right)\left(x-2\right)\left(x-1\right), the least common multiple of x-1,x-2,x-3,x-4.

$$\left(x-4\right)\left(x-3\right)\left(x-2\right)-\left(x-4\right)\left(x-3\right)\left(x-1\right)=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

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

Use the distributive property to multiply x^{2}-7x+12 by x-2 and combine like terms.

$$x^{3}-9x^{2}+26x-24-\left(x-4\right)\left(x-3\right)\left(x-1\right)=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$x^{3}-9x^{2}+26x-24-\left(x^{2}-7x+12\right)\left(x-1\right)=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-7x+12 by x-1 and combine like terms.

$$x^{3}-9x^{2}+26x-24-\left(x^{3}-8x^{2}+19x-12\right)=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

To find the opposite of x^{3}-8x^{2}+19x-12, find the opposite of each term.

$$x^{3}-9x^{2}+26x-24-x^{3}+8x^{2}-19x+12=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$-9x^{2}+26x-24+8x^{2}-19x+12=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Combine -9x^{2} and 8x^{2} to get -x^{2}.

$$-x^{2}+26x-24-19x+12=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Combine 26x and -19x to get 7x.

$$-x^{2}+7x-24+12=\left(x-4\right)\left(x-2\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Add -24 and 12 to get -12.

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

Use the distributive property to multiply x-4 by x-2 and combine like terms.

$$-x^{2}+7x-12=\left(x^{2}-6x+8\right)\left(x-1\right)-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

Use the distributive property to multiply x^{2}-6x+8 by x-1 and combine like terms.

$$-x^{2}+7x-12=x^{3}-7x^{2}+14x-8-\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

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

$$-x^{2}+7x-12=x^{3}-7x^{2}+14x-8-\left(x^{2}-5x+6\right)\left(x-1\right)$$

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

$$-x^{2}+7x-12=x^{3}-7x^{2}+14x-8-\left(x^{3}-6x^{2}+11x-6\right)$$

To find the opposite of x^{3}-6x^{2}+11x-6, find the opposite of each term.

$$-x^{2}+7x-12=x^{3}-7x^{2}+14x-8-x^{3}+6x^{2}-11x+6$$

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

$$-x^{2}+7x-12=-7x^{2}+14x-8+6x^{2}-11x+6$$

Combine -7x^{2} and 6x^{2} to get -x^{2}.

$$-x^{2}+7x-12=-x^{2}+14x-8-11x+6$$

Combine 14x and -11x to get 3x.

$$-x^{2}+7x-12=-x^{2}+3x-8+6$$

Add -8 and 6 to get -2.

$$-x^{2}+7x-12=-x^{2}+3x-2$$

Add x^{2} to both sides.

$$-x^{2}+7x-12+x^{2}=3x-2$$

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

$$7x-12=3x-2$$

Subtract 3x from both sides.

$$7x-12-3x=-2$$

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

$$4x-12=-2$$

Add 12 to both sides.

$$4x=-2+12$$

Add -2 and 12 to get 10.

$$4x=10$$

Divide both sides by 4.

$$x=\frac{10}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

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