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

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{x-1}{x\left(x-1\right)} and \frac{2x}{x\left(x-1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in x-1+2x.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-1\right) and x+2 is x\left(x-1\right)\left(x+2\right). Multiply \frac{3x-1}{x\left(x-1\right)} times \frac{x+2}{x+2}. Multiply \frac{3}{x+2} times \frac{x\left(x-1\right)}{x\left(x-1\right)}.

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

Since \frac{\left(3x-1\right)\left(x+2\right)}{x\left(x-1\right)\left(x+2\right)} and \frac{3x\left(x-1\right)}{x\left(x-1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(3x-1\right)\left(x+2\right)-3x\left(x-1\right).

$$\frac{3x^{2}+6x-x-2-3x^{2}+3x}{x\left(x-1\right)\left(x+2\right)}$$

Combine like terms in 3x^{2}+6x-x-2-3x^{2}+3x.

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

Expand x\left(x-1\right)\left(x+2\right).

$$\frac{8x-2}{x^{3}+x^{2}-2x}$$

Differentiate w.r.t. x

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