Solve 1/x-6-1/x-5=1/x-3-1/x-2

$$\frac { 1 } { x - 6 } - \frac { 1 } { x - 5 } = \frac { 1 } { x - 3 } - \frac { 1 } { x - 2 }$$

Solve for x

$$x=4$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 2,3,5,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-5\right)\left(x-3\right)\left(x-2\right), the least common multiple of x-6,x-5,x-3,x-2.

$$\left(x-5\right)\left(x-3\right)\left(x-2\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x-5 by x-3 and combine like terms.

$$\left(x^{2}-8x+15\right)\left(x-2\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-8x+15 by x-2 and combine like terms.

$$x^{3}-10x^{2}+31x-30-\left(x-6\right)\left(x-3\right)\left(x-2\right)=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x-6 by x-3 and combine like terms.

$$x^{3}-10x^{2}+31x-30-\left(x^{2}-9x+18\right)\left(x-2\right)=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-9x+18 by x-2 and combine like terms.

$$x^{3}-10x^{2}+31x-30-\left(x^{3}-11x^{2}+36x-36\right)=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

To find the opposite of x^{3}-11x^{2}+36x-36, find the opposite of each term.

$$x^{3}-10x^{2}+31x-30-x^{3}+11x^{2}-36x+36=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

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

$$-10x^{2}+31x-30+11x^{2}-36x+36=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Combine -10x^{2} and 11x^{2} to get x^{2}.

$$x^{2}+31x-30-36x+36=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Combine 31x and -36x to get -5x.

$$x^{2}-5x-30+36=\left(x-6\right)\left(x-5\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Add -30 and 36 to get 6.

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

Use the distributive property to multiply x-6 by x-5 and combine like terms.

$$x^{2}-5x+6=\left(x^{2}-11x+30\right)\left(x-2\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-11x+30 by x-2 and combine like terms.

$$x^{2}-5x+6=x^{3}-13x^{2}+52x-60-\left(x-6\right)\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x-6 by x-5 and combine like terms.

$$x^{2}-5x+6=x^{3}-13x^{2}+52x-60-\left(x^{2}-11x+30\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-11x+30 by x-3 and combine like terms.

$$x^{2}-5x+6=x^{3}-13x^{2}+52x-60-\left(x^{3}-14x^{2}+63x-90\right)$$

To find the opposite of x^{3}-14x^{2}+63x-90, find the opposite of each term.

$$x^{2}-5x+6=x^{3}-13x^{2}+52x-60-x^{3}+14x^{2}-63x+90$$

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

$$x^{2}-5x+6=-13x^{2}+52x-60+14x^{2}-63x+90$$

Combine -13x^{2} and 14x^{2} to get x^{2}.

$$x^{2}-5x+6=x^{2}+52x-60-63x+90$$

Combine 52x and -63x to get -11x.

$$x^{2}-5x+6=x^{2}-11x-60+90$$

Add -60 and 90 to get 30.

$$x^{2}-5x+6=x^{2}-11x+30$$

Subtract x^{2} from both sides.

$$x^{2}-5x+6-x^{2}=-11x+30$$

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

$$-5x+6=-11x+30$$

Add 11x to both sides.

$$-5x+6+11x=30$$

Combine -5x and 11x to get 6x.

$$6x+6=30$$

Subtract 6 from both sides.

$$6x=30-6$$

Subtract 6 from 30 to get 24.

$$6x=24$$

Divide both sides by 6.

$$x=\frac{24}{6}$$

Divide 24 by 6 to get 4.

$$x=4$$