Solve x+1/x-1+x+2/x-2=22x+30/11x-18

$$\frac { x + 1 } { x - 1 } + \frac { x + 2 } { x - 2 } = \frac { 22 x + 30 } { 11 x - 18 }$$

Solve for x

$$x=-6$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 1,\frac{18}{11},2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right)\left(11x-18\right), the least common multiple of x-1,x-2,11x-18.

$$\left(x-2\right)\left(11x-18\right)\left(x+1\right)+\left(x-1\right)\left(11x-18\right)\left(x+2\right)=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Use the distributive property to multiply x-2 by 11x-18 and combine like terms.

$$\left(11x^{2}-40x+36\right)\left(x+1\right)+\left(x-1\right)\left(11x-18\right)\left(x+2\right)=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Use the distributive property to multiply 11x^{2}-40x+36 by x+1 and combine like terms.

$$11x^{3}-29x^{2}-4x+36+\left(x-1\right)\left(11x-18\right)\left(x+2\right)=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Use the distributive property to multiply x-1 by 11x-18 and combine like terms.

$$11x^{3}-29x^{2}-4x+36+\left(11x^{2}-29x+18\right)\left(x+2\right)=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Use the distributive property to multiply 11x^{2}-29x+18 by x+2 and combine like terms.

$$11x^{3}-29x^{2}-4x+36+11x^{3}-7x^{2}-40x+36=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Combine 11x^{3} and 11x^{3} to get 22x^{3}.

$$22x^{3}-29x^{2}-4x+36-7x^{2}-40x+36=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Combine -29x^{2} and -7x^{2} to get -36x^{2}.

$$22x^{3}-36x^{2}-4x+36-40x+36=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Combine -4x and -40x to get -44x.

$$22x^{3}-36x^{2}-44x+36+36=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

Add 36 and 36 to get 72.

$$22x^{3}-36x^{2}-44x+72=\left(x-2\right)\left(x-1\right)\left(22x+30\right)$$

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

$$22x^{3}-36x^{2}-44x+72=\left(x^{2}-3x+2\right)\left(22x+30\right)$$

Use the distributive property to multiply x^{2}-3x+2 by 22x+30 and combine like terms.

$$22x^{3}-36x^{2}-44x+72=22x^{3}-36x^{2}-46x+60$$

Subtract 22x^{3} from both sides.

$$22x^{3}-36x^{2}-44x+72-22x^{3}=-36x^{2}-46x+60$$

Combine 22x^{3} and -22x^{3} to get 0.

$$-36x^{2}-44x+72=-36x^{2}-46x+60$$

Add 36x^{2} to both sides.

$$-36x^{2}-44x+72+36x^{2}=-46x+60$$

Combine -36x^{2} and 36x^{2} to get 0.

$$-44x+72=-46x+60$$

Add 46x to both sides.

$$-44x+72+46x=60$$

Combine -44x and 46x to get 2x.

$$2x+72=60$$

Subtract 72 from both sides.

$$2x=60-72$$

Subtract 72 from 60 to get -12.

$$2x=-12$$

Divide both sides by 2.

$$x=\frac{-12}{2}$$

Divide -12 by 2 to get -6.

$$x=-6$$