Solve 3-a/2a-4div(5/a-2-a-2)

$$\frac { 3 - a } { 2 a - 4 } \div ( \frac { 5 } { a - 2 } - a - 2 )$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-2 times \frac{a-2}{a-2}.

$$\frac{\frac{3-a}{2a-4}}{\frac{5}{a-2}+\frac{\left(-a-2\right)\left(a-2\right)}{a-2}}$$

Since \frac{5}{a-2} and \frac{\left(-a-2\right)\left(a-2\right)}{a-2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3-a}{2a-4}}{\frac{5+\left(-a-2\right)\left(a-2\right)}{a-2}}$$

Do the multiplications in 5+\left(-a-2\right)\left(a-2\right).

$$\frac{\frac{3-a}{2a-4}}{\frac{5-a^{2}+2a-2a+4}{a-2}}$$

Combine like terms in 5-a^{2}+2a-2a+4.

$$\frac{\frac{3-a}{2a-4}}{\frac{9-a^{2}}{a-2}}$$

Divide \frac{3-a}{2a-4} by \frac{9-a^{2}}{a-2} by multiplying \frac{3-a}{2a-4} by the reciprocal of \frac{9-a^{2}}{a-2}.

$$\frac{\left(3-a\right)\left(a-2\right)}{\left(2a-4\right)\left(9-a^{2}\right)}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 3-a.

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

Cancel out \left(a-3\right)\left(a-2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{-1}{-2a-6}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-2 times \frac{a-2}{a-2}.

$$\frac{\frac{3-a}{2a-4}}{\frac{5}{a-2}+\frac{\left(-a-2\right)\left(a-2\right)}{a-2}}$$

Since \frac{5}{a-2} and \frac{\left(-a-2\right)\left(a-2\right)}{a-2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3-a}{2a-4}}{\frac{5+\left(-a-2\right)\left(a-2\right)}{a-2}}$$

Do the multiplications in 5+\left(-a-2\right)\left(a-2\right).

$$\frac{\frac{3-a}{2a-4}}{\frac{5-a^{2}+2a-2a+4}{a-2}}$$

Combine like terms in 5-a^{2}+2a-2a+4.

$$\frac{\frac{3-a}{2a-4}}{\frac{9-a^{2}}{a-2}}$$

Divide \frac{3-a}{2a-4} by \frac{9-a^{2}}{a-2} by multiplying \frac{3-a}{2a-4} by the reciprocal of \frac{9-a^{2}}{a-2}.

$$\frac{\left(3-a\right)\left(a-2\right)}{\left(2a-4\right)\left(9-a^{2}\right)}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 3-a.

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

Cancel out \left(a-3\right)\left(a-2\right) in both numerator and denominator.

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

Expand the expression.

$$\frac{-1}{-2a-6}$$