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

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

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps for Solving Linear Equation

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

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

Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

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

Use the distributive property to multiply x+1 by 3.

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

Add -4 and 3 to get -1.

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

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

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

Subtract 2 from 3 to get 1.

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

Subtract x^{2} from both sides.

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

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

$$-1+3x=1-x$$

Add x to both sides.

$$-1+3x+x=1$$

Combine 3x and x to get 4x.

$$-1+4x=1$$

Add 1 to both sides.

$$4x=1+1$$

Add 1 and 1 to get 2.

$$4x=2$$

Divide both sides by 4.

$$x=\frac{2}{4}$$

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

$$x=\frac{1}{2}$$