Solve (a+1-3/a-1)-a-2/2a-2

$$( a + 1 - \frac { 3 } { a - 1 } ) - \frac { a - 2 } { 2 a - 2 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

Factor 2a-2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and 2\left(a-1\right) is 2\left(a-1\right). Multiply \frac{a^{2}-4}{a-1} times \frac{2}{2}.

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

Since \frac{2\left(a^{2}-4\right)}{2\left(a-1\right)} and \frac{a-2}{2\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a^{2}-4\right)-\left(a-2\right).

$$\frac{2a^{2}-8-a+2}{2\left(a-1\right)}$$

Combine like terms in 2a^{2}-8-a+2.

$$\frac{2a^{2}-6-a}{2\left(a-1\right)}$$

Expand 2\left(a-1\right).

$$\frac{2a^{2}-6-a}{2a-2}$$

Expand

$$\frac{2a^{2}-a-6}{2\left(a-1\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

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

Factor 2a-2.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and 2\left(a-1\right) is 2\left(a-1\right). Multiply \frac{a^{2}-4}{a-1} times \frac{2}{2}.

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

Since \frac{2\left(a^{2}-4\right)}{2\left(a-1\right)} and \frac{a-2}{2\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a^{2}-4\right)-\left(a-2\right).

$$\frac{2a^{2}-8-a+2}{2\left(a-1\right)}$$

Combine like terms in 2a^{2}-8-a+2.

$$\frac{2a^{2}-6-a}{2\left(a-1\right)}$$

Expand 2\left(a-1\right).

$$\frac{2a^{2}-6-a}{2a-2}$$