Solve 1-1/x^2/1-frac{2{x}-3/x^2}

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}}{x^{2}}.

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

Since \frac{x^{2}}{x^{2}} and \frac{1}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{x-2}{x} times \frac{x}{x}.

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

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

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

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

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

Divide \frac{x^{2}-1}{x^{2}} by \frac{x^{2}-2x-3}{x^{2}} by multiplying \frac{x^{2}-1}{x^{2}} by the reciprocal of \frac{x^{2}-2x-3}{x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x^{2}}{x^{2}}.

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

Since \frac{x^{2}}{x^{2}} and \frac{1}{x^{2}} have the same denominator, subtract them by subtracting their numerators.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x}{x}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{x-2}{x} times \frac{x}{x}.

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

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

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

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

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

Divide \frac{x^{2}-1}{x^{2}} by \frac{x^{2}-2x-3}{x^{2}} by multiplying \frac{x^{2}-1}{x^{2}} by the reciprocal of \frac{x^{2}-2x-3}{x^{2}}.

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

Cancel out x^{2} in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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