Solve 2/a-1div2a-4/a^2-1+1/2-a

$$\frac { 2 } { a - 1 } \div \frac { 2 a - 4 } { a ^ { 2 } - 1 } + \frac { 1 } { 2 - a }$$

Evaluate

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

Short Solution Steps

Divide \frac{2}{a-1} by \frac{2a-4}{a^{2}-1} by multiplying \frac{2}{a-1} by the reciprocal of \frac{2a-4}{a^{2}-1}.

$$\frac{2\left(a^{2}-1\right)}{\left(a-1\right)\left(2a-4\right)}+\frac{1}{2-a}$$

Factor the expressions that are not already factored in \frac{2\left(a^{2}-1\right)}{\left(a-1\right)\left(2a-4\right)}.

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

Cancel out 2\left(a-1\right) in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and 2-a is a-2. Multiply \frac{1}{2-a} times \frac{-1}{-1}.

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

Since \frac{a+1}{a-2} and \frac{-1}{a-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in a+1-1.

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

Expand

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

Short Solution Steps

Divide \frac{2}{a-1} by \frac{2a-4}{a^{2}-1} by multiplying \frac{2}{a-1} by the reciprocal of \frac{2a-4}{a^{2}-1}.

$$\frac{2\left(a^{2}-1\right)}{\left(a-1\right)\left(2a-4\right)}+\frac{1}{2-a}$$

Factor the expressions that are not already factored in \frac{2\left(a^{2}-1\right)}{\left(a-1\right)\left(2a-4\right)}.

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

Cancel out 2\left(a-1\right) in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and 2-a is a-2. Multiply \frac{1}{2-a} times \frac{-1}{-1}.

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

Since \frac{a+1}{a-2} and \frac{-1}{a-2} have the same denominator, add them by adding their numerators.

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

Combine like terms in a+1-1.

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