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

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

Solve for x

$$x=-\frac{1}{4}=-0.25$$

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-2,x+1.

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

Multiply x+1 and x+1 to get \left(x+1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

Combine 2x and x to get 3x.

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

Subtract 2 from 1 to get -1.

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

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

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

Subtract x^{2} from both sides.

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

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

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

Add x to both sides.

$$3x-1+x=-2$$

Combine 3x and x to get 4x.

$$4x-1=-2$$

Add 1 to both sides.

$$4x=-2+1$$

Add -2 and 1 to get -1.

$$4x=-1$$

Divide both sides by 4.

$$x=\frac{-1}{4}$$

Fraction \frac{-1}{4} can be rewritten as -\frac{1}{4} by extracting the negative sign.

$$x=-\frac{1}{4}$$