Solve =x-1/x+10(x-1)-1/x^2-1divx+1/(x-1)^2

$$= \frac { x - 1 } { x + 10 ( x - 1 ) } - \frac { 1 } { x ^ { 2 } - 1 } \div \frac { x + 1 } { ( x - 1 ) ^ { 2 } }$$

Evaluate

$$\frac{\left(x-1\right)\left(x^{2}-9x+11\right)}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Short Solution Steps

Use the distributive property to multiply 10 by x-1.

$$\frac{x-1}{x+10x-10}-\frac{\frac{1}{x^{2}-1}}{\frac{x+1}{\left(x-1\right)^{2}}}$$

Combine x and 10x to get 11x.

$$\frac{x-1}{11x-10}-\frac{\frac{1}{x^{2}-1}}{\frac{x+1}{\left(x-1\right)^{2}}}$$

Divide \frac{1}{x^{2}-1} by \frac{x+1}{\left(x-1\right)^{2}} by multiplying \frac{1}{x^{2}-1} by the reciprocal of \frac{x+1}{\left(x-1\right)^{2}}.

$$\frac{x-1}{11x-10}-\frac{\left(x-1\right)^{2}}{\left(x^{2}-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{\left(x-1\right)^{2}}{\left(x^{2}-1\right)\left(x+1\right)}.

$$\frac{x-1}{11x-10}-\frac{\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)^{2}}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{x-1}{11x-10}-\frac{x-1}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 11x-10 and \left(x+1\right)^{2} is \left(11x-10\right)\left(x+1\right)^{2}. Multiply \frac{x-1}{11x-10} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x-1}{\left(x+1\right)^{2}} times \frac{11x-10}{11x-10}.

$$\frac{\left(x-1\right)\left(x+1\right)^{2}}{\left(11x-10\right)\left(x+1\right)^{2}}-\frac{\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Since \frac{\left(x-1\right)\left(x+1\right)^{2}}{\left(11x-10\right)\left(x+1\right)^{2}} and \frac{\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-1\right)\left(x+1\right)^{2}-\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(x-1\right)\left(x+1\right)^{2}-\left(x-1\right)\left(11x-10\right).

$$\frac{x^{3}+2x^{2}+x-x^{2}-2x-1-11x^{2}+10x+11x-10}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{3}+2x^{2}+x-x^{2}-2x-1-11x^{2}+10x+11x-10.

$$\frac{x^{3}-10x^{2}+20x-11}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Expand \left(11x-10\right)\left(x+1\right)^{2}.

$$\frac{x^{3}-10x^{2}+20x-11}{11x^{3}+12x^{2}-9x-10}$$

Expand

$$\frac{x^{3}-10x^{2}+20x-11}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Short Solution Steps

Use the distributive property to multiply 10 by x-1.

$$\frac{x-1}{x+10x-10}-\frac{\frac{1}{x^{2}-1}}{\frac{x+1}{\left(x-1\right)^{2}}}$$

Combine x and 10x to get 11x.

$$\frac{x-1}{11x-10}-\frac{\frac{1}{x^{2}-1}}{\frac{x+1}{\left(x-1\right)^{2}}}$$

Divide \frac{1}{x^{2}-1} by \frac{x+1}{\left(x-1\right)^{2}} by multiplying \frac{1}{x^{2}-1} by the reciprocal of \frac{x+1}{\left(x-1\right)^{2}}.

$$\frac{x-1}{11x-10}-\frac{\left(x-1\right)^{2}}{\left(x^{2}-1\right)\left(x+1\right)}$$

Factor the expressions that are not already factored in \frac{\left(x-1\right)^{2}}{\left(x^{2}-1\right)\left(x+1\right)}.

$$\frac{x-1}{11x-10}-\frac{\left(x-1\right)^{2}}{\left(x-1\right)\left(x+1\right)^{2}}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{x-1}{11x-10}-\frac{x-1}{\left(x+1\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 11x-10 and \left(x+1\right)^{2} is \left(11x-10\right)\left(x+1\right)^{2}. Multiply \frac{x-1}{11x-10} times \frac{\left(x+1\right)^{2}}{\left(x+1\right)^{2}}. Multiply \frac{x-1}{\left(x+1\right)^{2}} times \frac{11x-10}{11x-10}.

$$\frac{\left(x-1\right)\left(x+1\right)^{2}}{\left(11x-10\right)\left(x+1\right)^{2}}-\frac{\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Since \frac{\left(x-1\right)\left(x+1\right)^{2}}{\left(11x-10\right)\left(x+1\right)^{2}} and \frac{\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-1\right)\left(x+1\right)^{2}-\left(x-1\right)\left(11x-10\right)}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Do the multiplications in \left(x-1\right)\left(x+1\right)^{2}-\left(x-1\right)\left(11x-10\right).

$$\frac{x^{3}+2x^{2}+x-x^{2}-2x-1-11x^{2}+10x+11x-10}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Combine like terms in x^{3}+2x^{2}+x-x^{2}-2x-1-11x^{2}+10x+11x-10.

$$\frac{x^{3}-10x^{2}+20x-11}{\left(11x-10\right)\left(x+1\right)^{2}}$$

Expand \left(11x-10\right)\left(x+1\right)^{2}.

$$\frac{x^{3}-10x^{2}+20x-11}{11x^{3}+12x^{2}-9x-10}$$