Solve (2/a+1-3/a+3):(2/a^2-a-2-3/a^2+a-6)

$$( \frac { 2 } { a + 1 } - \frac { 3 } { a + 3 } ) : ( \frac { 2 } { a ^ { 2 } - a - 2 } - \frac { 3 } { a ^ { 2 } + a - 6 } )$$

Evaluate

$$a-2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a+3 is \left(a+1\right)\left(a+3\right). Multiply \frac{2}{a+1} times \frac{a+3}{a+3}. Multiply \frac{3}{a+3} times \frac{a+1}{a+1}.

$$\frac{\frac{2\left(a+3\right)}{\left(a+1\right)\left(a+3\right)}-\frac{3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Since \frac{2\left(a+3\right)}{\left(a+1\right)\left(a+3\right)} and \frac{3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2\left(a+3\right)-3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Do the multiplications in 2\left(a+3\right)-3\left(a+1\right).

$$\frac{\frac{2a+6-3a-3}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Combine like terms in 2a+6-3a-3.

$$\frac{\frac{-a+3}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Factor a^{2}-a-2. Factor a^{2}+a-6.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a+1\right) and \left(a-2\right)\left(a+3\right) is \left(a-2\right)\left(a+1\right)\left(a+3\right). Multiply \frac{2}{\left(a-2\right)\left(a+1\right)} times \frac{a+3}{a+3}. Multiply \frac{3}{\left(a-2\right)\left(a+3\right)} times \frac{a+1}{a+1}.

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

Since \frac{2\left(a+3\right)}{\left(a-2\right)\left(a+1\right)\left(a+3\right)} and \frac{3\left(a+1\right)}{\left(a-2\right)\left(a+1\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a+3\right)-3\left(a+1\right).

$$\frac{\frac{-a+3}{\left(a+1\right)\left(a+3\right)}}{\frac{2a+6-3a-3}{\left(a-2\right)\left(a+1\right)\left(a+3\right)}}$$

Combine like terms in 2a+6-3a-3.

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

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

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

Cancel out \left(a+1\right)\left(a+3\right)\left(-a+3\right) in both numerator and denominator.

$$a-2$$

Expand

$$a-2$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+1 and a+3 is \left(a+1\right)\left(a+3\right). Multiply \frac{2}{a+1} times \frac{a+3}{a+3}. Multiply \frac{3}{a+3} times \frac{a+1}{a+1}.

$$\frac{\frac{2\left(a+3\right)}{\left(a+1\right)\left(a+3\right)}-\frac{3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Since \frac{2\left(a+3\right)}{\left(a+1\right)\left(a+3\right)} and \frac{3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2\left(a+3\right)-3\left(a+1\right)}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Do the multiplications in 2\left(a+3\right)-3\left(a+1\right).

$$\frac{\frac{2a+6-3a-3}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Combine like terms in 2a+6-3a-3.

$$\frac{\frac{-a+3}{\left(a+1\right)\left(a+3\right)}}{\frac{2}{a^{2}-a-2}-\frac{3}{a^{2}+a-6}}$$

Factor a^{2}-a-2. Factor a^{2}+a-6.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-2\right)\left(a+1\right) and \left(a-2\right)\left(a+3\right) is \left(a-2\right)\left(a+1\right)\left(a+3\right). Multiply \frac{2}{\left(a-2\right)\left(a+1\right)} times \frac{a+3}{a+3}. Multiply \frac{3}{\left(a-2\right)\left(a+3\right)} times \frac{a+1}{a+1}.

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

Since \frac{2\left(a+3\right)}{\left(a-2\right)\left(a+1\right)\left(a+3\right)} and \frac{3\left(a+1\right)}{\left(a-2\right)\left(a+1\right)\left(a+3\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2\left(a+3\right)-3\left(a+1\right).

$$\frac{\frac{-a+3}{\left(a+1\right)\left(a+3\right)}}{\frac{2a+6-3a-3}{\left(a-2\right)\left(a+1\right)\left(a+3\right)}}$$

Combine like terms in 2a+6-3a-3.

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

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

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

Cancel out \left(a+1\right)\left(a+3\right)\left(-a+3\right) in both numerator and denominator.

$$a-2$$