Solve 1/x-3-1/x-4=1/x-5-1/x-6

$$\frac { 1 } { x - 3 } - \frac { 1 } { x - 4 } = \frac { 1 } { x - 5 } - \frac { 1 } { x - 6 }$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 3,4,5,6 since division by zero is not defined. Multiply both sides of the equation by \left(x-6\right)\left(x-5\right)\left(x-4\right)\left(x-3\right), the least common multiple of x-3,x-4,x-5,x-6.

$$\left(x-6\right)\left(x-5\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

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

$$\left(x^{2}-11x+30\right)\left(x-4\right)-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-11x+30 by x-4 and combine like terms.

$$x^{3}-15x^{2}+74x-120-\left(x-6\right)\left(x-5\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

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

$$x^{3}-15x^{2}+74x-120-\left(x^{2}-11x+30\right)\left(x-3\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

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

$$x^{3}-15x^{2}+74x-120-\left(x^{3}-14x^{2}+63x-90\right)=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

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

$$x^{3}-15x^{2}+74x-120-x^{3}+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

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

$$-15x^{2}+74x-120+14x^{2}-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Combine -15x^{2} and 14x^{2} to get -x^{2}.

$$-x^{2}+74x-120-63x+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Combine 74x and -63x to get 11x.

$$-x^{2}+11x-120+90=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Add -120 and 90 to get -30.

$$-x^{2}+11x-30=\left(x-6\right)\left(x-4\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Use the distributive property to multiply x-6 by x-4 and combine like terms.

$$-x^{2}+11x-30=\left(x^{2}-10x+24\right)\left(x-3\right)-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-10x+24 by x-3 and combine like terms.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x-5\right)\left(x-4\right)\left(x-3\right)$$

Use the distributive property to multiply x-5 by x-4 and combine like terms.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{2}-9x+20\right)\left(x-3\right)$$

Use the distributive property to multiply x^{2}-9x+20 by x-3 and combine like terms.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-\left(x^{3}-12x^{2}+47x-60\right)$$

To find the opposite of x^{3}-12x^{2}+47x-60, find the opposite of each term.

$$-x^{2}+11x-30=x^{3}-13x^{2}+54x-72-x^{3}+12x^{2}-47x+60$$

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

$$-x^{2}+11x-30=-13x^{2}+54x-72+12x^{2}-47x+60$$

Combine -13x^{2} and 12x^{2} to get -x^{2}.

$$-x^{2}+11x-30=-x^{2}+54x-72-47x+60$$

Combine 54x and -47x to get 7x.

$$-x^{2}+11x-30=-x^{2}+7x-72+60$$

Add -72 and 60 to get -12.

$$-x^{2}+11x-30=-x^{2}+7x-12$$

Add x^{2} to both sides.

$$-x^{2}+11x-30+x^{2}=7x-12$$

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

$$11x-30=7x-12$$

Subtract 7x from both sides.

$$11x-30-7x=-12$$

Combine 11x and -7x to get 4x.

$$4x-30=-12$$

Add 30 to both sides.

$$4x=-12+30$$

Add -12 and 30 to get 18.

$$4x=18$$

Divide both sides by 4.

$$x=\frac{18}{4}$$

Reduce the fraction \frac{18}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{2}$$