Solve 1/a-b-3/a+b/frac{2{b-a}+4/b+a}

$$\frac { \frac { 1 } { a - b } - \frac { 3 } { a + b } } { \frac { 2 } { b - a } + \frac { 4 } { b + a } }$$

Evaluate

$$-\frac{2b-a}{3b-a}$$

Short Solution Steps

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

$$\frac{\frac{a+b}{\left(a+b\right)\left(a-b\right)}-\frac{3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

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

$$\frac{\frac{a+b-3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

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

$$\frac{\frac{a+b-3a+3b}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

Combine like terms in a+b-3a+3b.

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

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

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

Since \frac{2\left(a+b\right)}{\left(a+b\right)\left(-a+b\right)} and \frac{4\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(a+b\right)+4\left(-a+b\right).

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

Combine like terms in 2a+2b-4a+4b.

$$\frac{\frac{-2a+4b}{\left(a+b\right)\left(a-b\right)}}{\frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)}}$$

Divide \frac{-2a+4b}{\left(a+b\right)\left(a-b\right)} by \frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)} by multiplying \frac{-2a+4b}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)}.

$$\frac{\left(-2a+4b\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(a-b\right)\left(-2a+6b\right)}$$

Extract the negative sign in -a+b.

$$\frac{-\left(a+b\right)\left(a-b\right)\left(-2a+4b\right)}{\left(a+b\right)\left(a-b\right)\left(-2a+6b\right)}$$

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

$$\frac{-\left(-2a+4b\right)}{-2a+6b}$$

Factor the expressions that are not already factored.

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

Cancel out 2 in both numerator and denominator.

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

Expand the expression.

$$\frac{a-2b}{-a+3b}$$

Expand

$$-\frac{2b-a}{3b-a}$$

Short Solution Steps

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

$$\frac{\frac{a+b}{\left(a+b\right)\left(a-b\right)}-\frac{3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

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

$$\frac{\frac{a+b-3\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

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

$$\frac{\frac{a+b-3a+3b}{\left(a+b\right)\left(a-b\right)}}{\frac{2}{b-a}+\frac{4}{b+a}}$$

Combine like terms in a+b-3a+3b.

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

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

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

Since \frac{2\left(a+b\right)}{\left(a+b\right)\left(-a+b\right)} and \frac{4\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(a+b\right)+4\left(-a+b\right).

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

Combine like terms in 2a+2b-4a+4b.

$$\frac{\frac{-2a+4b}{\left(a+b\right)\left(a-b\right)}}{\frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)}}$$

Divide \frac{-2a+4b}{\left(a+b\right)\left(a-b\right)} by \frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)} by multiplying \frac{-2a+4b}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{-2a+6b}{\left(a+b\right)\left(-a+b\right)}.

$$\frac{\left(-2a+4b\right)\left(a+b\right)\left(-a+b\right)}{\left(a+b\right)\left(a-b\right)\left(-2a+6b\right)}$$

Extract the negative sign in -a+b.

$$\frac{-\left(a+b\right)\left(a-b\right)\left(-2a+4b\right)}{\left(a+b\right)\left(a-b\right)\left(-2a+6b\right)}$$

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

$$\frac{-\left(-2a+4b\right)}{-2a+6b}$$

Factor the expressions that are not already factored.

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

Cancel out 2 in both numerator and denominator.

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

Expand the expression.

$$\frac{a-2b}{-a+3b}$$