Solve 9/x-3-3x/(x-3)^2:3x/x^2-9-2x-3/x-3

$$\frac { 9 } { x - 3 } - \frac { 3 x } { ( x - 3 ) ^ { 2 } } : \frac { 3 x } { x ^ { 2 } - 9 } - \frac { 2 x - 3 } { x - 3 }$$

Evaluate

$$-3$$

Short Solution Steps

Divide \frac{3x}{\left(x-3\right)^{2}} by \frac{3x}{x^{2}-9} by multiplying \frac{3x}{\left(x-3\right)^{2}} by the reciprocal of \frac{3x}{x^{2}-9}.

$$\frac{9}{x-3}-\frac{3x\left(x^{2}-9\right)}{\left(x-3\right)^{2}\times 3x}-\frac{2x-3}{x-3}$$

Cancel out 3x in both numerator and denominator.

$$\frac{9}{x-3}-\frac{x^{2}-9}{\left(x-3\right)^{2}}-\frac{2x-3}{x-3}$$

Factor the expressions that are not already factored in \frac{x^{2}-9}{\left(x-3\right)^{2}}.

$$\frac{9}{x-3}-\frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)^{2}}-\frac{2x-3}{x-3}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{9}{x-3}-\frac{x+3}{x-3}-\frac{2x-3}{x-3}$$

Since \frac{9}{x-3} and \frac{x+3}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9-\left(x+3\right)}{x-3}-\frac{2x-3}{x-3}$$

Do the multiplications in 9-\left(x+3\right).

$$\frac{9-x-3}{x-3}-\frac{2x-3}{x-3}$$

Combine like terms in 9-x-3.

$$\frac{6-x}{x-3}-\frac{2x-3}{x-3}$$

Since \frac{6-x}{x-3} and \frac{2x-3}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6-x-\left(2x-3\right)}{x-3}$$

Do the multiplications in 6-x-\left(2x-3\right).

$$\frac{6-x-2x+3}{x-3}$$

Combine like terms in 6-x-2x+3.

$$\frac{9-3x}{x-3}$$

Factor the expressions that are not already factored in \frac{9-3x}{x-3}.

$$\frac{3\left(-x+3\right)}{x-3}$$

Extract the negative sign in 3-x.

$$\frac{-3\left(x-3\right)}{x-3}$$

Cancel out x-3 in both numerator and denominator.

$$-3$$

Factor

$$-3$$