Solve 2/(x-7)(x-5)-2/(5-x)(x-3)+2/(x-3)(x-1)-2/(1-x)(x+1)+frac{2}{(x+1)(x+3)}

$$\frac{ 2 }{ (x-7)(x-5) } - \frac{ 2 }{ (5-x)(x-3) } + \frac{ 2 }{ (x-3)(x-1) } - \frac{ 2 }{ (1-x)(x+1) } + \frac{ 2 }{ (x+1)(x+3) }$$

Evaluate

$$\frac{10}{\left(x-7\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-5\right) and \left(5-x\right)\left(x-3\right) is \left(x-7\right)\left(x-5\right)\left(x-3\right). Multiply \frac{2}{\left(x-7\right)\left(x-5\right)} times \frac{x-3}{x-3}. Multiply \frac{2}{\left(5-x\right)\left(x-3\right)} times \frac{-\left(x-7\right)}{-\left(x-7\right)}.

$$\frac{2\left(x-3\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}-\frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{2\left(x-3\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)} and \frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(x-3\right)-2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 2\left(x-3\right)-2\left(-1\right)\left(x-7\right).

$$\frac{2x-6+2x-14}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 2x-6+2x-14.

$$\frac{4x-20}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{4x-20}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}.

$$\frac{4\left(x-5\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{4}{\left(x-7\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-3\right) and \left(x-3\right)\left(x-1\right) is \left(x-7\right)\left(x-3\right)\left(x-1\right). Multiply \frac{4}{\left(x-7\right)\left(x-3\right)} times \frac{x-1}{x-1}. Multiply \frac{2}{\left(x-3\right)\left(x-1\right)} times \frac{x-7}{x-7}.

$$\frac{4\left(x-1\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}+\frac{2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{4\left(x-1\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)} and \frac{2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x-1\right)+2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 4\left(x-1\right)+2\left(x-7\right).

$$\frac{4x-4+2x-14}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 4x-4+2x-14.

$$\frac{6x-18}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{6x-18}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}.

$$\frac{6\left(x-3\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{6}{\left(x-7\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-1\right) and \left(1-x\right)\left(x+1\right) is \left(x-7\right)\left(x-1\right)\left(x+1\right). Multiply \frac{6}{\left(x-7\right)\left(x-1\right)} times \frac{x+1}{x+1}. Multiply \frac{2}{\left(1-x\right)\left(x+1\right)} times \frac{-\left(x-7\right)}{-\left(x-7\right)}.

$$\frac{6\left(x+1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}-\frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{6\left(x+1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)} and \frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\left(x+1\right)-2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 6\left(x+1\right)-2\left(-1\right)\left(x-7\right).

$$\frac{6x+6+2x-14}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 6x+6+2x-14.

$$\frac{8x-8}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{8x-8}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{8\left(x-1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{8}{\left(x-7\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+1\right) and \left(x+1\right)\left(x+3\right) is \left(x-7\right)\left(x+1\right)\left(x+3\right). Multiply \frac{8}{\left(x-7\right)\left(x+1\right)} times \frac{x+3}{x+3}. Multiply \frac{2}{\left(x+1\right)\left(x+3\right)} times \frac{x-7}{x-7}.

$$\frac{8\left(x+3\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}+\frac{2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Since \frac{8\left(x+3\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)} and \frac{2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x+3\right)+2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 8\left(x+3\right)+2\left(x-7\right).

$$\frac{8x+24+2x-14}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 8x+24+2x-14.

$$\frac{10x+10}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{10x+10}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}.

$$\frac{10\left(x+1\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{10}{\left(x-7\right)\left(x+3\right)}$$

Expand \left(x-7\right)\left(x+3\right).

$$\frac{10}{x^{2}-4x-21}$$

Expand

$$\frac{10}{\left(x-7\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-5\right) and \left(5-x\right)\left(x-3\right) is \left(x-7\right)\left(x-5\right)\left(x-3\right). Multiply \frac{2}{\left(x-7\right)\left(x-5\right)} times \frac{x-3}{x-3}. Multiply \frac{2}{\left(5-x\right)\left(x-3\right)} times \frac{-\left(x-7\right)}{-\left(x-7\right)}.

$$\frac{2\left(x-3\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}-\frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{2\left(x-3\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)} and \frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(x-3\right)-2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 2\left(x-3\right)-2\left(-1\right)\left(x-7\right).

$$\frac{2x-6+2x-14}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 2x-6+2x-14.

$$\frac{4x-20}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{4x-20}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}.

$$\frac{4\left(x-5\right)}{\left(x-7\right)\left(x-5\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{4}{\left(x-7\right)\left(x-3\right)}+\frac{2}{\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-3\right) and \left(x-3\right)\left(x-1\right) is \left(x-7\right)\left(x-3\right)\left(x-1\right). Multiply \frac{4}{\left(x-7\right)\left(x-3\right)} times \frac{x-1}{x-1}. Multiply \frac{2}{\left(x-3\right)\left(x-1\right)} times \frac{x-7}{x-7}.

$$\frac{4\left(x-1\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}+\frac{2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{4\left(x-1\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)} and \frac{2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x-1\right)+2\left(x-7\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 4\left(x-1\right)+2\left(x-7\right).

$$\frac{4x-4+2x-14}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 4x-4+2x-14.

$$\frac{6x-18}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{6x-18}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}.

$$\frac{6\left(x-3\right)}{\left(x-7\right)\left(x-3\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-3 in both numerator and denominator.

$$\frac{6}{\left(x-7\right)\left(x-1\right)}-\frac{2}{\left(1-x\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x-1\right) and \left(1-x\right)\left(x+1\right) is \left(x-7\right)\left(x-1\right)\left(x+1\right). Multiply \frac{6}{\left(x-7\right)\left(x-1\right)} times \frac{x+1}{x+1}. Multiply \frac{2}{\left(1-x\right)\left(x+1\right)} times \frac{-\left(x-7\right)}{-\left(x-7\right)}.

$$\frac{6\left(x+1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}-\frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Since \frac{6\left(x+1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)} and \frac{2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6\left(x+1\right)-2\left(-1\right)\left(x-7\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 6\left(x+1\right)-2\left(-1\right)\left(x-7\right).

$$\frac{6x+6+2x-14}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 6x+6+2x-14.

$$\frac{8x-8}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{8x-8}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}.

$$\frac{8\left(x-1\right)}{\left(x-7\right)\left(x-1\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{8}{\left(x-7\right)\left(x+1\right)}+\frac{2}{\left(x+1\right)\left(x+3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+1\right) and \left(x+1\right)\left(x+3\right) is \left(x-7\right)\left(x+1\right)\left(x+3\right). Multiply \frac{8}{\left(x-7\right)\left(x+1\right)} times \frac{x+3}{x+3}. Multiply \frac{2}{\left(x+1\right)\left(x+3\right)} times \frac{x-7}{x-7}.

$$\frac{8\left(x+3\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}+\frac{2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Since \frac{8\left(x+3\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)} and \frac{2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x+3\right)+2\left(x-7\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Do the multiplications in 8\left(x+3\right)+2\left(x-7\right).

$$\frac{8x+24+2x-14}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Combine like terms in 8x+24+2x-14.

$$\frac{10x+10}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Factor the expressions that are not already factored in \frac{10x+10}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}.

$$\frac{10\left(x+1\right)}{\left(x-7\right)\left(x+1\right)\left(x+3\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{10}{\left(x-7\right)\left(x+3\right)}$$

Expand \left(x-7\right)\left(x+3\right).

$$\frac{10}{x^{2}-4x-21}$$