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

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+3x-x-1.

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

Factor the expressions that are not already factored in \frac{x^{2}+2x-3}{x^{2}-1}.

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

Cancel out x-1 in both numerator and denominator.

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

Divide \frac{x^{2}+2x-1}{x\left(x+1\right)} by \frac{x+3}{x+1} by multiplying \frac{x^{2}+2x-1}{x\left(x+1\right)} by the reciprocal of \frac{x+3}{x+1}.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply x by x+3.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+3x-x-1.

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

Factor the expressions that are not already factored in \frac{x^{2}+2x-3}{x^{2}-1}.

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

Cancel out x-1 in both numerator and denominator.

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

Divide \frac{x^{2}+2x-1}{x\left(x+1\right)} by \frac{x+3}{x+1} by multiplying \frac{x^{2}+2x-1}{x\left(x+1\right)} by the reciprocal of \frac{x+3}{x+1}.

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

Cancel out x+1 in both numerator and denominator.

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

Use the distributive property to multiply x by x+3.

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