Solve (a/a-1-2/a^2-1)div(1-1/a+1)

$$( \frac { a } { a - 1 } - \frac { 2 } { a ^ { 2 } - 1 } ) \div ( 1 - \frac { 1 } { a + 1 } )$$

Evaluate

$$\frac{a+2}{a}$$

Short Solution Steps

Factor a^{2}-1.

$$\frac{\frac{a}{a-1}-\frac{2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

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

$$\frac{\frac{a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Since \frac{a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{2}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a\left(a+1\right)-2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Do the multiplications in a\left(a+1\right)-2.

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

Factor the expressions that are not already factored in \frac{a^{2}+a-2}{\left(a-1\right)\left(a+1\right)}.

$$\frac{\frac{\left(a-1\right)\left(a+2\right)}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{\frac{a+2}{a+1}}{1-\frac{1}{a+1}}$$

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

$$\frac{\frac{a+2}{a+1}}{\frac{a+1}{a+1}-\frac{1}{a+1}}$$

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

$$\frac{\frac{a+2}{a+1}}{\frac{a+1-1}{a+1}}$$

Combine like terms in a+1-1.

$$\frac{\frac{a+2}{a+1}}{\frac{a}{a+1}}$$

Divide \frac{a+2}{a+1} by \frac{a}{a+1} by multiplying \frac{a+2}{a+1} by the reciprocal of \frac{a}{a+1}.

$$\frac{\left(a+2\right)\left(a+1\right)}{\left(a+1\right)a}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{a+2}{a}$$

Expand

$$\frac{a+2}{a}$$

Short Solution Steps

Factor a^{2}-1.

$$\frac{\frac{a}{a-1}-\frac{2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

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

$$\frac{\frac{a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)}-\frac{2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Since \frac{a\left(a+1\right)}{\left(a-1\right)\left(a+1\right)} and \frac{2}{\left(a-1\right)\left(a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a\left(a+1\right)-2}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Do the multiplications in a\left(a+1\right)-2.

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

Factor the expressions that are not already factored in \frac{a^{2}+a-2}{\left(a-1\right)\left(a+1\right)}.

$$\frac{\frac{\left(a-1\right)\left(a+2\right)}{\left(a-1\right)\left(a+1\right)}}{1-\frac{1}{a+1}}$$

Cancel out a-1 in both numerator and denominator.

$$\frac{\frac{a+2}{a+1}}{1-\frac{1}{a+1}}$$

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

$$\frac{\frac{a+2}{a+1}}{\frac{a+1}{a+1}-\frac{1}{a+1}}$$

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

$$\frac{\frac{a+2}{a+1}}{\frac{a+1-1}{a+1}}$$

Combine like terms in a+1-1.

$$\frac{\frac{a+2}{a+1}}{\frac{a}{a+1}}$$

Divide \frac{a+2}{a+1} by \frac{a}{a+1} by multiplying \frac{a+2}{a+1} by the reciprocal of \frac{a}{a+1}.

$$\frac{\left(a+2\right)\left(a+1\right)}{\left(a+1\right)a}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{a+2}{a}$$