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

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

Solve for x

$$x=17$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,-1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right)\left(x+3\right), the least common multiple of x-2,x+3,x+1.

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

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

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

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

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

Use the distributive property to multiply x^{2}-x-2 by 2.

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

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

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

Combine 4x and -2x to get 2x.

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

Subtract 4 from 3 to get -1.

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

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

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

Use the distributive property to multiply x^{2}+x-6 by 3.

$$3x^{2}+2x-1=3x^{2}+3x-18$$

Subtract 3x^{2} from both sides.

$$3x^{2}+2x-1-3x^{2}=3x-18$$

Combine 3x^{2} and -3x^{2} to get 0.

$$2x-1=3x-18$$

Subtract 3x from both sides.

$$2x-1-3x=-18$$

Combine 2x and -3x to get -x.

$$-x-1=-18$$

Add 1 to both sides.

$$-x=-18+1$$

Add -18 and 1 to get -17.

$$-x=-17$$

Multiply both sides by -1.

$$x=17$$