Solve x+1/x^2+5x+6=1/x

$$\frac { x + 1 } { x ^ { 2 } + 5 x + 6 } = \frac { 1 } { x }$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,-2,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+2\right)\left(x+3\right), the least common multiple of x^{2}+5x+6,x.

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

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

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

Use the distributive property to multiply x+2 by x+3 and combine like terms.

$$x^{2}+x=x^{2}+5x+6$$

Subtract x^{2} from both sides.

$$x^{2}+x-x^{2}=5x+6$$

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

$$x=5x+6$$

Subtract 5x from both sides.

$$x-5x=6$$

Combine x and -5x to get -4x.

$$-4x=6$$

Divide both sides by -4.

$$x=\frac{6}{-4}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$