Solve -f(x)=2x+3/x-4

$$- f ( x ) = \frac { 2 x + 3 } { x - 4 }$$

Solve for f

$$f=\frac{2x+3}{x\left(4-x\right)}$$

$$x\neq 4\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x-4.

$$\left(-f\right)x\left(x-4\right)=2x+3$$

Use the distributive property to multiply \left(-f\right)x by x-4.

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

Multiply -4 and -1 to get 4.

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

Reorder the terms.

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

Combine all terms containing f.

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

The equation is in standard form.

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

Divide both sides by -x^{2}+4x.

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

Dividing by -x^{2}+4x undoes the multiplication by -x^{2}+4x.

$$f=\frac{2x+3}{4x-x^{2}}$$

Divide 2x+3 by -x^{2}+4x.

$$f=\frac{2x+3}{x\left(4-x\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{4f^{2}-7f+1}-2f+1}{f}\text{; }x=\frac{\sqrt{4f^{2}-7f+1}+2f-1}{f}\text{, }&f\neq 0\\x=-\frac{3}{2}\text{, }&f=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4f^{2}-7f+1}-2f+1}{f}\text{; }x=\frac{\sqrt{4f^{2}-7f+1}+2f-1}{f}\text{, }&\left(f\neq 0\text{ and }f\leq \frac{7-\sqrt{33}}{8}\right)\text{ or }f\geq \frac{\sqrt{33}+7}{8}\\x=-\frac{3}{2}\text{, }&f=0\end{matrix}\right.$$