Solve x/x^2+3x-4+x+1/x^2+6x+8=2x/x^2+x-2

$$\frac { x } { x ^ { 2 } + 3 x - 4 } + \frac { x + 1 } { x ^ { 2 } + 6 x + 8 } = \frac { 2 x } { x ^ { 2 } + x - 2 }$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -4,-2,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)\left(x+2\right)\left(x+4\right), the least common multiple of x^{2}+3x-4,x^{2}+6x+8,x^{2}+x-2.

$$\left(x+2\right)x+\left(x-1\right)\left(x+1\right)=\left(x+4\right)\times 2x$$

Use the distributive property to multiply x+2 by x.

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

Consider \left(x-1\right)\left(x+1\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 1.

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

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

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

Use the distributive property to multiply x+4 by 2.

$$2x^{2}+2x-1=\left(2x+8\right)x$$

Use the distributive property to multiply 2x+8 by x.

$$2x^{2}+2x-1=2x^{2}+8x$$

Subtract 2x^{2} from both sides.

$$2x^{2}+2x-1-2x^{2}=8x$$

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

$$2x-1=8x$$

Subtract 8x from both sides.

$$2x-1-8x=0$$

Combine 2x and -8x to get -6x.

$$-6x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$-6x=1$$

Divide both sides by -6.

$$x=\frac{1}{-6}$$

Fraction \frac{1}{-6} can be rewritten as -\frac{1}{6} by extracting the negative sign.

$$x=-\frac{1}{6}$$