Solve x/x+1-2(x+1)/x+3+1=0

$$\frac { x } { x + 1 } - \frac { 2 ( x + 1 ) } { x + 3 } + 1 = 0$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

Steps for Solving Linear Equation

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

$$\left(x+3\right)x-\left(x+1\right)\times 2\left(x+1\right)+\left(x+1\right)\left(x+3\right)=0$$

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

$$x^{2}+3x-\left(x+1\right)\times 2\left(x+1\right)+\left(x+1\right)\left(x+3\right)=0$$

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

$$x^{2}+3x-\left(x+1\right)^{2}\times 2+\left(x+1\right)\left(x+3\right)=0$$

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

$$x^{2}+3x-\left(x^{2}+2x+1\right)\times 2+\left(x+1\right)\left(x+3\right)=0$$

Use the distributive property to multiply x^{2}+2x+1 by 2.

$$x^{2}+3x-\left(2x^{2}+4x+2\right)+\left(x+1\right)\left(x+3\right)=0$$

To find the opposite of 2x^{2}+4x+2, find the opposite of each term.

$$x^{2}+3x-2x^{2}-4x-2+\left(x+1\right)\left(x+3\right)=0$$

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

$$-x^{2}+3x-4x-2+\left(x+1\right)\left(x+3\right)=0$$

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

$$-x^{2}-x-2+\left(x+1\right)\left(x+3\right)=0$$

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

$$-x^{2}-x-2+x^{2}+4x+3=0$$

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

$$-x-2+4x+3=0$$

Combine -x and 4x to get 3x.

$$3x-2+3=0$$

Add -2 and 3 to get 1.

$$3x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$3x=-1$$

Divide both sides by 3.

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

Fraction \frac{-1}{3} can be rewritten as -\frac{1}{3} by extracting the negative sign.

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