Solve x+1/x-x+2/x+1-x-4/x-3+x-5/x-4

$$\frac { x + 1 } { x } - \frac { x + 2 } { x + 1 } - \frac { x - 4 } { x - 3 } + \frac { x - 5 } { x - 4 }$$

Evaluate

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

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{x+1}{x} times \frac{x+1}{x+1}. Multiply \frac{x+2}{x+1} times \frac{x}{x}.

$$\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}-\frac{\left(x+2\right)x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

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

$$\frac{\left(x+1\right)\left(x+1\right)-\left(x+2\right)x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

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

$$\frac{x^{2}+x+x+1-x^{2}-2x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

Combine like terms in x^{2}+x+x+1-x^{2}-2x.

$$\frac{1}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x-3 is x\left(x-3\right)\left(x+1\right). Multiply \frac{1}{x\left(x+1\right)} times \frac{x-3}{x-3}. Multiply \frac{x-4}{x-3} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

$$\frac{x-3}{x\left(x-3\right)\left(x+1\right)}-\frac{\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Since \frac{x-3}{x\left(x-3\right)\left(x+1\right)} and \frac{\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-3-\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Do the multiplications in x-3-\left(x-4\right)x\left(x+1\right).

$$\frac{x-3-x^{3}-x^{2}+4x^{2}+4x}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Combine like terms in x-3-x^{3}-x^{2}+4x^{2}+4x.

$$\frac{5x-3-x^{3}+3x^{2}}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+1\right) and x-4 is x\left(x-4\right)\left(x-3\right)\left(x+1\right). Multiply \frac{5x-3-x^{3}+3x^{2}}{x\left(x-3\right)\left(x+1\right)} times \frac{x-4}{x-4}. Multiply \frac{x-5}{x-4} times \frac{x\left(x-3\right)\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}.

$$\frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}+\frac{\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Since \frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)} and \frac{\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)+\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Do the multiplications in \left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)+\left(x-5\right)x\left(x-3\right)\left(x+1\right).

$$\frac{5x^{2}-20x-3x+12-x^{4}+4x^{3}+3x^{3}-12x^{2}+x^{4}-2x^{3}-3x^{2}-5x^{3}+10x^{2}+15x}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Combine like terms in 5x^{2}-20x-3x+12-x^{4}+4x^{3}+3x^{3}-12x^{2}+x^{4}-2x^{3}-3x^{2}-5x^{3}+10x^{2}+15x.

$$\frac{-8x+12}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Expand x\left(x-4\right)\left(x-3\right)\left(x+1\right).

$$\frac{-8x+12}{x^{4}-6x^{3}+5x^{2}+12x}$$

Expand

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

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{x+1}{x} times \frac{x+1}{x+1}. Multiply \frac{x+2}{x+1} times \frac{x}{x}.

$$\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}-\frac{\left(x+2\right)x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

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

$$\frac{\left(x+1\right)\left(x+1\right)-\left(x+2\right)x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

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

$$\frac{x^{2}+x+x+1-x^{2}-2x}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

Combine like terms in x^{2}+x+x+1-x^{2}-2x.

$$\frac{1}{x\left(x+1\right)}-\frac{x-4}{x-3}+\frac{x-5}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+1\right) and x-3 is x\left(x-3\right)\left(x+1\right). Multiply \frac{1}{x\left(x+1\right)} times \frac{x-3}{x-3}. Multiply \frac{x-4}{x-3} times \frac{x\left(x+1\right)}{x\left(x+1\right)}.

$$\frac{x-3}{x\left(x-3\right)\left(x+1\right)}-\frac{\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Since \frac{x-3}{x\left(x-3\right)\left(x+1\right)} and \frac{\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-3-\left(x-4\right)x\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Do the multiplications in x-3-\left(x-4\right)x\left(x+1\right).

$$\frac{x-3-x^{3}-x^{2}+4x^{2}+4x}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

Combine like terms in x-3-x^{3}-x^{2}+4x^{2}+4x.

$$\frac{5x-3-x^{3}+3x^{2}}{x\left(x-3\right)\left(x+1\right)}+\frac{x-5}{x-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-3\right)\left(x+1\right) and x-4 is x\left(x-4\right)\left(x-3\right)\left(x+1\right). Multiply \frac{5x-3-x^{3}+3x^{2}}{x\left(x-3\right)\left(x+1\right)} times \frac{x-4}{x-4}. Multiply \frac{x-5}{x-4} times \frac{x\left(x-3\right)\left(x+1\right)}{x\left(x-3\right)\left(x+1\right)}.

$$\frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}+\frac{\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Since \frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)} and \frac{\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)+\left(x-5\right)x\left(x-3\right)\left(x+1\right)}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Do the multiplications in \left(5x-3-x^{3}+3x^{2}\right)\left(x-4\right)+\left(x-5\right)x\left(x-3\right)\left(x+1\right).

$$\frac{5x^{2}-20x-3x+12-x^{4}+4x^{3}+3x^{3}-12x^{2}+x^{4}-2x^{3}-3x^{2}-5x^{3}+10x^{2}+15x}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Combine like terms in 5x^{2}-20x-3x+12-x^{4}+4x^{3}+3x^{3}-12x^{2}+x^{4}-2x^{3}-3x^{2}-5x^{3}+10x^{2}+15x.

$$\frac{-8x+12}{x\left(x-4\right)\left(x-3\right)\left(x+1\right)}$$

Expand x\left(x-4\right)\left(x-3\right)\left(x+1\right).

$$\frac{-8x+12}{x^{4}-6x^{3}+5x^{2}+12x}$$