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

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

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

Steps for Solving Linear Equation

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

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

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

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

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

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

Combine x and 3x to get 4x.

$$x^{2}+x-2=4x+1+x^{2}+2$$

Add 1 and 2 to get 3.

$$x^{2}+x-2=4x+3+x^{2}$$

Subtract 4x from both sides.

$$x^{2}+x-2-4x=3+x^{2}$$

Combine x and -4x to get -3x.

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

Subtract x^{2} from both sides.

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

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

$$-3x-2=3$$

Add 2 to both sides.

$$-3x=3+2$$

Add 3 and 2 to get 5.

$$-3x=5$$

Divide both sides by -3.

$$x=\frac{5}{-3}$$

Fraction \frac{5}{-3} can be rewritten as -\frac{5}{3} by extracting the negative sign.

$$x=-\frac{5}{3}$$