Solve 9-3a/4-2adiv(a+2-5/a-2)

$$\frac { 9 - 3 a } { 4 - 2 a } \div ( a + 2 - \frac { 5 } { a - 2 } )$$

Evaluate

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

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}.

$$\frac{\frac{9-3a}{4-2a}}{\frac{\left(a+2\right)\left(a-2\right)}{a-2}-\frac{5}{a-2}}$$

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

$$\frac{\frac{9-3a}{4-2a}}{\frac{\left(a+2\right)\left(a-2\right)-5}{a-2}}$$

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

$$\frac{\frac{9-3a}{4-2a}}{\frac{a^{2}-2a+2a-4-5}{a-2}}$$

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -2+a. Extract the negative sign in 3-a.

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

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

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

Expand the expression.

$$\frac{3}{2a+6}$$

Expand

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

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}.

$$\frac{\frac{9-3a}{4-2a}}{\frac{\left(a+2\right)\left(a-2\right)}{a-2}-\frac{5}{a-2}}$$

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

$$\frac{\frac{9-3a}{4-2a}}{\frac{\left(a+2\right)\left(a-2\right)-5}{a-2}}$$

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

$$\frac{\frac{9-3a}{4-2a}}{\frac{a^{2}-2a+2a-4-5}{a-2}}$$

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

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

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

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

Factor the expressions that are not already factored.

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

Extract the negative sign in -2+a. Extract the negative sign in 3-a.

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

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

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

Expand the expression.

$$\frac{3}{2a+6}$$