Solve x^2-x+9/x^3-9x+1/x^2-9-1/x-3+1/x

$$\frac { x ^ { 2 } - x + 9 } { x ^ { 3 } - 9 x } + \frac { 1 } { x ^ { 2 } - 9 } - \frac { 1 } { x - 3 } + \frac { 1 } { x }$$

Evaluate

$$\frac{1}{x+3}$$

Short Solution Steps

Factor x^{3}-9x. Factor x^{2}-9.

$$\frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+3\right) and \left(x-3\right)\left(x+3\right) is x\left(x-3\right)\left(x+3\right). Multiply \frac{1}{\left(x-3\right)\left(x+3\right)} times \frac{x}{x}.

$$\frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)}+\frac{x}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

Since \frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)} and \frac{x}{x\left(x-3\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-x+9+x}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

Combine like terms in x^{2}-x+9+x.

$$\frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+3\right) and x-3 is x\left(x-3\right)\left(x+3\right). Multiply \frac{1}{x-3} times \frac{x\left(x+3\right)}{x\left(x+3\right)}.

$$\frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)}-\frac{x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Since \frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)} and \frac{x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+9-x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Do the multiplications in x^{2}+9-x\left(x+3\right).

$$\frac{x^{2}+9-x^{2}-3x}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Combine like terms in x^{2}+9-x^{2}-3x.

$$\frac{9-3x}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

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

$$\frac{3\left(-x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Extract the negative sign in 3-x.

$$\frac{-3\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{-3}{x\left(x+3\right)}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+3\right) and x is x\left(x+3\right). Multiply \frac{1}{x} times \frac{x+3}{x+3}.

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

Since \frac{-3}{x\left(x+3\right)} and \frac{x+3}{x\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in -3+x+3.

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

Cancel out x in both numerator and denominator.

$$\frac{1}{x+3}$$

Expand

$$\frac{1}{x+3}$$

Short Solution Steps

Factor x^{3}-9x. Factor x^{2}-9.

$$\frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+3\right) and \left(x-3\right)\left(x+3\right) is x\left(x-3\right)\left(x+3\right). Multiply \frac{1}{\left(x-3\right)\left(x+3\right)} times \frac{x}{x}.

$$\frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)}+\frac{x}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

Since \frac{x^{2}-x+9}{x\left(x-3\right)\left(x+3\right)} and \frac{x}{x\left(x-3\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-x+9+x}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

Combine like terms in x^{2}-x+9+x.

$$\frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)}-\frac{1}{x-3}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+3\right) and x-3 is x\left(x-3\right)\left(x+3\right). Multiply \frac{1}{x-3} times \frac{x\left(x+3\right)}{x\left(x+3\right)}.

$$\frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)}-\frac{x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Since \frac{x^{2}+9}{x\left(x-3\right)\left(x+3\right)} and \frac{x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+9-x\left(x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Do the multiplications in x^{2}+9-x\left(x+3\right).

$$\frac{x^{2}+9-x^{2}-3x}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Combine like terms in x^{2}+9-x^{2}-3x.

$$\frac{9-3x}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

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

$$\frac{3\left(-x+3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Extract the negative sign in 3-x.

$$\frac{-3\left(x-3\right)}{x\left(x-3\right)\left(x+3\right)}+\frac{1}{x}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{-3}{x\left(x+3\right)}+\frac{1}{x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+3\right) and x is x\left(x+3\right). Multiply \frac{1}{x} times \frac{x+3}{x+3}.

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

Since \frac{-3}{x\left(x+3\right)} and \frac{x+3}{x\left(x+3\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in -3+x+3.

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

Cancel out x in both numerator and denominator.

$$\frac{1}{x+3}$$