Practice: Simplify 1. $$ \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} $$ 3. $$ \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} $$ 4. $$ \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} $$

Question

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Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{x}{\left(x-1ight)}ight)-\left(\frac{12}{\left(x^{2}-1ight)}ight)-\left(\frac{6}{\left(x+1ight)}ight)$$
- step1: Remove the parentheses:
  $$\left(\frac{x}{x-1}ight)-\left(\frac{12}{x^{2}-1}ight)-\left(\frac{6}{x+1}ight)$$
- step2: Remove the parentheses:
  $$\frac{x}{x-1}-\left(\frac{12}{x^{2}-1}ight)-\left(\frac{6}{x+1}ight)$$
- step3: Remove the parentheses:
  $$\frac{x}{x-1}-\frac{12}{x^{2}-1}-\left(\frac{6}{x+1}ight)$$
- step4: Remove the parentheses:
  $$\frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1}$$
- step5: Factor the expression:
  $$\frac{x}{x-1}-\frac{12}{\left(x+1ight)\left(x-1ight)}-\frac{6}{x+1}$$
- step6: Reduce fractions to a common denominator:
  $$\frac{x\left(x+1ight)}{\left(x-1ight)\left(x+1ight)}-\frac{12}{\left(x+1ight)\left(x-1ight)}-\frac{6\left(x-1ight)}{\left(x+1ight)\left(x-1ight)}$$
- step7: Rewrite the expression:
  $$\frac{x\left(x+1ight)}{\left(x+1ight)\left(x-1ight)}-\frac{12}{\left(x+1ight)\left(x-1ight)}-\frac{6\left(x-1ight)}{\left(x+1ight)\left(x-1ight)}$$
- step8: Transform the expression:
  $$\frac{x\left(x+1ight)-12-6\left(x-1ight)}{\left(x+1ight)\left(x-1ight)}$$
- step9: Multiply the terms:
  $$\frac{x^{2}+x-12-6\left(x-1ight)}{\left(x+1ight)\left(x-1ight)}$$
- step10: Multiply the terms:
  $$\frac{x^{2}+x-12-\left(6x-6ight)}{\left(x+1ight)\left(x-1ight)}$$
- step11: Calculate:
  $$\frac{x^{2}-5x-6}{\left(x+1ight)\left(x-1ight)}$$
- step12: Factor the expression:
  $$\frac{\left(x+1ight)\left(x-6ight)}{\left(x+1ight)\left(x-1ight)}$$
- step13: Reduce the fraction:
  $$\frac{x-6}{x-1}$$
Calculate or simplify the expression $$ (3*(x-1)/(x-5)) / (2*(1-x)/(5-x)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(\frac{3\left(x-1ight)}{\left(x-5ight)}ight)}{\left(\frac{2\left(1-xight)}{\left(5-xight)}ight)}$$
- step1: Remove the parentheses:
  $$\frac{\frac{3\left(x-1ight)}{x-5}}{\frac{2\left(1-xight)}{5-x}}$$
- step2: Multiply by the reciprocal:
  $$\frac{3\left(x-1ight)}{x-5}	imes \frac{5-x}{2\left(1-xight)}$$
- step3: Rewrite the expression:
  $$\frac{3\left(x-1ight)}{x-5}	imes \frac{-x+5}{2\left(-x+1ight)}$$
- step4: Multiply the terms:
  $$\frac{3\left(x-1ight)\left(-x+5ight)}{\left(x-5ight)	imes 2\left(-x+1ight)}$$
- step5: Multiply the terms:
  $$\frac{3\left(x-1ight)\left(-x+5ight)}{2\left(x-5ight)\left(-x+1ight)}$$
- step6: Calculate:
  $$\frac{-3x^{2}+18x-15}{-2x^{2}+12x-10}$$
- step7: Calculate:
  $$\frac{-3\left(x^{2}-6x+5ight)}{-2\left(x^{2}-6x+5ight)}$$
- step8: Reduce the fraction:
  $$\frac{3}{2}$$
Calculate or simplify the expression $$ (x/(x+2)) - (2/(x-1)) - (6/(x^2)) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\left(\frac{x}{\left(x+2ight)}ight)-\left(\frac{2}{\left(x-1ight)}ight)-\frac{6}{x^{2}}$$
- step1: Remove the parentheses:
  $$\left(\frac{x}{x+2}ight)-\left(\frac{2}{x-1}ight)-\frac{6}{x^{2}}$$
- step2: Remove the parentheses:
  $$\frac{x}{x+2}-\left(\frac{2}{x-1}ight)-\frac{6}{x^{2}}$$
- step3: Remove the parentheses:
  $$\frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}}$$
- step4: Reduce fractions to a common denominator:
  $$\frac{x\left(x-1ight)x^{2}}{\left(x+2ight)\left(x-1ight)x^{2}}-\frac{2\left(x+2ight)x^{2}}{\left(x-1ight)\left(x+2ight)x^{2}}-\frac{6\left(x+2ight)\left(x-1ight)}{x^{2}\left(x+2ight)\left(x-1ight)}$$
- step5: Rewrite the expression:
  $$\frac{x\left(x-1ight)x^{2}}{\left(x+2ight)\left(x-1ight)x^{2}}-\frac{2\left(x+2ight)x^{2}}{\left(x+2ight)\left(x-1ight)x^{2}}-\frac{6\left(x+2ight)\left(x-1ight)}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step6: Transform the expression:
  $$\frac{x\left(x-1ight)x^{2}-2\left(x+2ight)x^{2}-6\left(x+2ight)\left(x-1ight)}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step7: Multiply the terms:
  $$\frac{x^{4}-x^{3}-2\left(x+2ight)x^{2}-6\left(x+2ight)\left(x-1ight)}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step8: Multiply the terms:
  $$\frac{x^{4}-x^{3}-\left(2x^{3}+4x^{2}ight)-6\left(x+2ight)\left(x-1ight)}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step9: Multiply the terms:
  $$\frac{x^{4}-x^{3}-\left(2x^{3}+4x^{2}ight)-\left(6x^{2}+6x-12ight)}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step10: Subtract the terms:
  $$\frac{x^{4}-3x^{3}-10x^{2}-6x+12}{\left(x+2ight)\left(x-1ight)x^{2}}$$
- step11: Calculate:
  $$\frac{x^{4}-3x^{3}-10x^{2}-6x+12}{x^{4}+x^{3}-2x^{2}}$$
Here are the simplified results for each expression:

1. For the expression $$ \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} $$:
   - The simplified result is $$ \frac{3}{2} $$.

2. For the expression $$ \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} $$:
   - The simplified result is $$ \frac{x-6}{x-1} $$.

3. For the expression $$ \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} $$:
   - The simplified result is $$ \frac{x^{4}-3x^{3}-10x^{2}-6x+12}{x^{4}+x^{3}-2x^{2}} $$.

If you need further assistance or explanations for any of these steps, feel free to ask!
1. $$ \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} = \frac{3}{2} $$ 2. $$ \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} = \frac{x-6}{x-1} $$ 3. $$ \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} = \frac{x^{4}-3x^{3}-10x^{2}-6x+12}{x^{4}+x^{3}-2x^{2}} $$

Practice: Simplify 1. \( \frac{3(x-1)}{x-5} \div \frac{2(1-x)}{5-x} \) 3. \( \frac{x}{x-1}-\frac{12}{x^{2}-1}-\frac{6}{x+1} \) 4. \( \frac{x}{x+2}-\frac{2}{x-1}-\frac{6}{x^{2}} \)

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