Solve (x^2-9/x^2-6x+9-1/3-x)div(7/x-3x-3)

$$( \frac { x ^ { 2 } - 9 } { x ^ { 2 } - 6 x + 9 } - \frac { 1 } { 3 - x } ) \div ( \frac { 7 } { x - 3 } x - 3 )$$

Evaluate

$$\frac{x+4}{4x+9}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-9}{x^{2}-6x+9}.

$$\frac{\frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)^{2}}-\frac{1}{3-x}}{\frac{7}{x-3}x-3}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{\frac{x+3}{x-3}-\frac{1}{3-x}}{\frac{7}{x-3}x-3}$$

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

$$\frac{\frac{x+3}{x-3}-\frac{-1}{x-3}}{\frac{7}{x-3}x-3}$$

Since \frac{x+3}{x-3} and \frac{-1}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+3-\left(-1\right)}{x-3}}{\frac{7}{x-3}x-3}$$

Do the multiplications in x+3-\left(-1\right).

$$\frac{\frac{x+3+1}{x-3}}{\frac{7}{x-3}x-3}$$

Combine like terms in x+3+1.

$$\frac{\frac{x+4}{x-3}}{\frac{7}{x-3}x-3}$$

Express \frac{7}{x-3}x as a single fraction.

$$\frac{\frac{x+4}{x-3}}{\frac{7x}{x-3}-3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-3}{x-3}.

$$\frac{\frac{x+4}{x-3}}{\frac{7x}{x-3}-\frac{3\left(x-3\right)}{x-3}}$$

Since \frac{7x}{x-3} and \frac{3\left(x-3\right)}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+4}{x-3}}{\frac{7x-3\left(x-3\right)}{x-3}}$$

Do the multiplications in 7x-3\left(x-3\right).

$$\frac{\frac{x+4}{x-3}}{\frac{7x-3x+9}{x-3}}$$

Combine like terms in 7x-3x+9.

$$\frac{\frac{x+4}{x-3}}{\frac{4x+9}{x-3}}$$

Divide \frac{x+4}{x-3} by \frac{4x+9}{x-3} by multiplying \frac{x+4}{x-3} by the reciprocal of \frac{4x+9}{x-3}.

$$\frac{\left(x+4\right)\left(x-3\right)}{\left(x-3\right)\left(4x+9\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{x+4}{4x+9}$$

Expand

$$\frac{x+4}{4x+9}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{x^{2}-9}{x^{2}-6x+9}.

$$\frac{\frac{\left(x-3\right)\left(x+3\right)}{\left(x-3\right)^{2}}-\frac{1}{3-x}}{\frac{7}{x-3}x-3}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{\frac{x+3}{x-3}-\frac{1}{3-x}}{\frac{7}{x-3}x-3}$$

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

$$\frac{\frac{x+3}{x-3}-\frac{-1}{x-3}}{\frac{7}{x-3}x-3}$$

Since \frac{x+3}{x-3} and \frac{-1}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+3-\left(-1\right)}{x-3}}{\frac{7}{x-3}x-3}$$

Do the multiplications in x+3-\left(-1\right).

$$\frac{\frac{x+3+1}{x-3}}{\frac{7}{x-3}x-3}$$

Combine like terms in x+3+1.

$$\frac{\frac{x+4}{x-3}}{\frac{7}{x-3}x-3}$$

Express \frac{7}{x-3}x as a single fraction.

$$\frac{\frac{x+4}{x-3}}{\frac{7x}{x-3}-3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{x-3}{x-3}.

$$\frac{\frac{x+4}{x-3}}{\frac{7x}{x-3}-\frac{3\left(x-3\right)}{x-3}}$$

Since \frac{7x}{x-3} and \frac{3\left(x-3\right)}{x-3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x+4}{x-3}}{\frac{7x-3\left(x-3\right)}{x-3}}$$

Do the multiplications in 7x-3\left(x-3\right).

$$\frac{\frac{x+4}{x-3}}{\frac{7x-3x+9}{x-3}}$$

Combine like terms in 7x-3x+9.

$$\frac{\frac{x+4}{x-3}}{\frac{4x+9}{x-3}}$$

Divide \frac{x+4}{x-3} by \frac{4x+9}{x-3} by multiplying \frac{x+4}{x-3} by the reciprocal of \frac{4x+9}{x-3}.

$$\frac{\left(x+4\right)\left(x-3\right)}{\left(x-3\right)\left(4x+9\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{x+4}{4x+9}$$