Solve 3/4x-16+1/2/frac{x-4{2}-1/2}

$$\frac { \frac { 3 } { 4 x - 16 } + \frac { 1 } { 2 } } { \frac { x - 4 } { 2 } - \frac { 1 } { 2 } }$$

Evaluate

$$\frac{2x-5}{2\left(x-5\right)\left(x-4\right)}$$

Short Solution Steps

Factor 4x-16.

$$\frac{\frac{3}{4\left(x-4\right)}+\frac{1}{2}}{\frac{x-4}{2}-\frac{1}{2}}$$

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

$$\frac{\frac{3}{4\left(x-4\right)}+\frac{2\left(x-4\right)}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Since \frac{3}{4\left(x-4\right)} and \frac{2\left(x-4\right)}{4\left(x-4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3+2\left(x-4\right)}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Do the multiplications in 3+2\left(x-4\right).

$$\frac{\frac{3+2x-8}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Combine like terms in 3+2x-8.

$$\frac{\frac{-5+2x}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Since \frac{x-4}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-5+2x}{4\left(x-4\right)}}{\frac{x-4-1}{2}}$$

Combine like terms in x-4-1.

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

Divide \frac{-5+2x}{4\left(x-4\right)} by \frac{x-5}{2} by multiplying \frac{-5+2x}{4\left(x-4\right)} by the reciprocal of \frac{x-5}{2}.

$$\frac{\left(-5+2x\right)\times 2}{4\left(x-4\right)\left(x-5\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2x-5}{2\left(x-5\right)\left(x-4\right)}$$

Use the distributive property to multiply 2 by x-5.

$$\frac{2x-5}{\left(2x-10\right)\left(x-4\right)}$$

Apply the distributive property by multiplying each term of 2x-10 by each term of x-4.

$$\frac{2x-5}{2x^{2}-8x-10x+40}$$

Combine -8x and -10x to get -18x.

$$\frac{2x-5}{2x^{2}-18x+40}$$

Expand

$$\frac{2x-5}{2\left(x-5\right)\left(x-4\right)}$$

Short Solution Steps

Factor 4x-16.

$$\frac{\frac{3}{4\left(x-4\right)}+\frac{1}{2}}{\frac{x-4}{2}-\frac{1}{2}}$$

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

$$\frac{\frac{3}{4\left(x-4\right)}+\frac{2\left(x-4\right)}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Since \frac{3}{4\left(x-4\right)} and \frac{2\left(x-4\right)}{4\left(x-4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3+2\left(x-4\right)}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Do the multiplications in 3+2\left(x-4\right).

$$\frac{\frac{3+2x-8}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Combine like terms in 3+2x-8.

$$\frac{\frac{-5+2x}{4\left(x-4\right)}}{\frac{x-4}{2}-\frac{1}{2}}$$

Since \frac{x-4}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{-5+2x}{4\left(x-4\right)}}{\frac{x-4-1}{2}}$$

Combine like terms in x-4-1.

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

Divide \frac{-5+2x}{4\left(x-4\right)} by \frac{x-5}{2} by multiplying \frac{-5+2x}{4\left(x-4\right)} by the reciprocal of \frac{x-5}{2}.

$$\frac{\left(-5+2x\right)\times 2}{4\left(x-4\right)\left(x-5\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2x-5}{2\left(x-5\right)\left(x-4\right)}$$

Use the distributive property to multiply 2 by x-5.

$$\frac{2x-5}{\left(2x-10\right)\left(x-4\right)}$$

Apply the distributive property by multiplying each term of 2x-10 by each term of x-4.

$$\frac{2x-5}{2x^{2}-8x-10x+40}$$

Combine -8x and -10x to get -18x.

$$\frac{2x-5}{2x^{2}-18x+40}$$