Solve 6*f(x)=x+1/x-2

$$6 \cdot f ( x ) = \frac { x + 1 } { x - 2 }$$

Solve for f

$$f=\frac{x+1}{6x\left(x-2\right)}$$

$$x\neq 2\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-2.

$$6fx\left(x-2\right)=x+1$$

Use the distributive property to multiply 6fx by x-2.

$$6fx^{2}-12fx=x+1$$

Combine all terms containing f.

$$\left(6x^{2}-12x\right)f=x+1$$

Divide both sides by 6x^{2}-12x.

$$\frac{\left(6x^{2}-12x\right)f}{6x^{2}-12x}=\frac{x+1}{6x^{2}-12x}$$

Dividing by 6x^{2}-12x undoes the multiplication by 6x^{2}-12x.

$$f=\frac{x+1}{6x^{2}-12x}$$

Divide x+1 by 6x^{2}-12x.

$$f=\frac{x+1}{6x\left(x-2\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{144f^{2}+48f+1}+12f+1}{12f}\text{; }x=\frac{-\sqrt{144f^{2}+48f+1}+12f+1}{12f}\text{, }&f\neq 0\\x=-1\text{, }&f=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{144f^{2}+48f+1}+12f+1}{12f}\text{; }x=\frac{-\sqrt{144f^{2}+48f+1}+12f+1}{12f}\text{, }&f\leq -\frac{\sqrt{3}}{12}-\frac{1}{6}\text{ or }\left(f\neq 0\text{ and }f\geq \frac{\sqrt{3}}{12}-\frac{1}{6}\right)\\x=-1\text{, }&f=0\end{matrix}\right.$$