Solve 3/2x-8+1/x-4+3/2x

$$\frac { 3 } { 2 x - 8 } + \frac { 1 } { x - 4 } + \frac { 3 } { 2 x }$$

Evaluate

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

Short Solution Steps

Factor 2x-8.

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

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

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

Since \frac{3}{2\left(x-4\right)} and \frac{2}{2\left(x-4\right)} have the same denominator, add them by adding their numerators. Add 3 and 2 to get 5.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-4\right) and 2x is 2x\left(x-4\right). Multiply \frac{5}{2\left(x-4\right)} times \frac{x}{x}. Multiply \frac{3}{2x} times \frac{x-4}{x-4}.

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

Since \frac{5x}{2x\left(x-4\right)} and \frac{3\left(x-4\right)}{2x\left(x-4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 5x+3\left(x-4\right).

$$\frac{5x+3x-12}{2x\left(x-4\right)}$$

Combine like terms in 5x+3x-12.

$$\frac{8x-12}{2x\left(x-4\right)}$$

Factor the expressions that are not already factored in \frac{8x-12}{2x\left(x-4\right)}.

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

Cancel out 2 in both numerator and denominator.

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

Expand x\left(x-4\right).

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

Use the distributive property to multiply 2 by 2x-3.

$$\frac{4x-6}{x^{2}-4x}$$

Factor

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