EXERCISE (a) Simplify the followiag $$ \begin{array}{lll} ext { (1) } \frac{2 x}{3}+\frac{1}{6} & ext { (2) } \frac{7}{6 x}-\frac{2}{9 y}+\frac{1}{3 x^{2}} & ext { (3) } \frac{3}{x}+\frac{2}{x^{2}} \ ext { (4) } 1+2-\frac{2 a-1}{2} & ext { (5) } \frac{5}{6 x y}-\frac{x+2}{2 x} & ext { (6) } \frac{5}{2 x}-\frac{x+1}{x-2} \ ext { (7) } \frac{2}{x}+\frac{5}{x+1} & ext { (8) } \frac{x+2}{x^{2}+2}-\frac{6}{x+2} & ext { (9) } \frac{x-3}{x+3}-\frac{x+3}{x-3} \ ext { (10) } \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} & ext { (11) } \frac{2}{2 x}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y}{4}+1 \ ext { (12) } \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} & ext { (13) } x-\frac{2 x}{3 x-2}\end{array} $$

Question

EXERCISE (a) Simplify the followiag $$ \begin{array}{lll}	ext { (1) } \frac{2 x}{3}+\frac{1}{6} & 	ext { (2) } \frac{7}{6 x}-\frac{2}{9 y}+\frac{1}{3 x^{2}} & 	ext { (3) } \frac{3}{x}+\frac{2}{x^{2}} \ 	ext { (4) } 1+2-\frac{2 a-1}{2} & 	ext { (5) } \frac{5}{6 x y}-\frac{x+2}{2 x} & 	ext { (6) } \frac{5}{2 x}-\frac{x+1}{x-2} \ 	ext { (7) } \frac{2}{x}+\frac{5}{x+1} & 	ext { (8) } \frac{x+2}{x^{2}+2}-\frac{6}{x+2} & 	ext { (9) } \frac{x-3}{x+3}-\frac{x+3}{x-3} \ 	ext { (10) } \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} & 	ext { (11) } \frac{2}{2 x}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y}{4}+1 \ 	ext { (12) } \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} & 	ext { (13) } x-\frac{2 x}{3 x-2}\end{array} $$

UpStudy AI · Accepted Answer

Simplify the expression by following steps:
- step0: Solution:
  $$\frac{3}{x}+\frac{2}{x^{2}}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{3x}{x	imes x}+\frac{2}{x^{2}}$$
- step2: Multiply the terms:
  $$\frac{3x}{x^{2}}+\frac{2}{x^{2}}$$
- step3: Transform the expression:
  $$\frac{3x+2}{x^{2}}$$
Calculate or simplify the expression $$ 5/(2*x) - (x + 1)/(x - 2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{5}{2x}-\frac{\left(x+1ight)}{\left(x-2ight)}$$
- step1: Remove the parentheses:
  $$\frac{5}{2x}-\frac{x+1}{x-2}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{5\left(x-2ight)}{2x\left(x-2ight)}-\frac{\left(x+1ight)	imes 2x}{\left(x-2ight)	imes 2x}$$
- step3: Reorder the terms:
  $$\frac{5\left(x-2ight)}{2x\left(x-2ight)}-\frac{\left(x+1ight)	imes 2x}{2\left(x-2ight)x}$$
- step4: Rewrite the expression:
  $$\frac{5\left(x-2ight)}{2x\left(x-2ight)}-\frac{\left(x+1ight)	imes 2x}{2x\left(x-2ight)}$$
- step5: Transform the expression:
  $$\frac{5\left(x-2ight)-\left(x+1ight)	imes 2x}{2x\left(x-2ight)}$$
- step6: Multiply the terms:
  $$\frac{5x-10-\left(x+1ight)	imes 2x}{2x\left(x-2ight)}$$
- step7: Multiply the terms:
  $$\frac{5x-10-\left(2x^{2}+2xight)}{2x\left(x-2ight)}$$
- step8: Subtract the terms:
  $$\frac{3x-10-2x^{2}}{2x\left(x-2ight)}$$
- step9: Multiply the terms:
  $$\frac{3x-10-2x^{2}}{2x^{2}-4x}$$
Calculate or simplify the expression $$ x - (2*x)/(3*x - 2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$x-\frac{2x}{\left(3x-2ight)}$$
- step1: Remove the parentheses:
  $$x-\frac{2x}{3x-2}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{x\left(3x-2ight)}{3x-2}-\frac{2x}{3x-2}$$
- step3: Transform the expression:
  $$\frac{x\left(3x-2ight)-2x}{3x-2}$$
- step4: Multiply the terms:
  $$\frac{3x^{2}-2x-2x}{3x-2}$$
- step5: Subtract the terms:
  $$\frac{3x^{2}-4x}{3x-2}$$
Calculate or simplify the expression $$ 2/x + 5/(x + 1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2}{x}+\frac{5}{\left(x+1ight)}$$
- step1: Remove the parentheses:
  $$\frac{2}{x}+\frac{5}{x+1}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{2\left(x+1ight)}{x\left(x+1ight)}+\frac{5x}{\left(x+1ight)x}$$
- step3: Rewrite the expression:
  $$\frac{2\left(x+1ight)}{x\left(x+1ight)}+\frac{5x}{x\left(x+1ight)}$$
- step4: Transform the expression:
  $$\frac{2\left(x+1ight)+5x}{x\left(x+1ight)}$$
- step5: Multiply the terms:
  $$\frac{2x+2+5x}{x\left(x+1ight)}$$
- step6: Add the terms:
  $$\frac{7x+2}{x\left(x+1ight)}$$
- step7: Multiply the terms:
  $$\frac{7x+2}{x^{2}+x}$$
Calculate or simplify the expression $$ (x - 3)/(x + 3) - (x + 3)/(x - 3) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(x-3ight)}{\left(x+3ight)}-\frac{\left(x+3ight)}{\left(x-3ight)}$$
- step1: Remove the parentheses:
  $$\frac{x-3}{x+3}-\frac{x+3}{x-3}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{\left(x-3ight)\left(x-3ight)}{\left(x+3ight)\left(x-3ight)}-\frac{\left(x+3ight)\left(x+3ight)}{\left(x-3ight)\left(x+3ight)}$$
- step3: Rewrite the expression:
  $$\frac{\left(x-3ight)\left(x-3ight)}{\left(x+3ight)\left(x-3ight)}-\frac{\left(x+3ight)\left(x+3ight)}{\left(x+3ight)\left(x-3ight)}$$
- step4: Transform the expression:
  $$\frac{\left(x-3ight)\left(x-3ight)-\left(x+3ight)\left(x+3ight)}{\left(x+3ight)\left(x-3ight)}$$
- step5: Multiply the terms:
  $$\frac{x^{2}-6x+9-\left(x+3ight)\left(x+3ight)}{\left(x+3ight)\left(x-3ight)}$$
- step6: Multiply the terms:
  $$\frac{x^{2}-6x+9-\left(x^{2}+6x+9ight)}{\left(x+3ight)\left(x-3ight)}$$
- step7: Calculate:
  $$\frac{-12x}{\left(x+3ight)\left(x-3ight)}$$
- step8: Rewrite the fraction:
  $$-\frac{12x}{\left(x+3ight)\left(x-3ight)}$$
- step9: Multiply the terms:
  $$-\frac{12x}{x^{2}-9}$$
Calculate or simplify the expression $$ (x + 2)/(x^2 + 2) - 6/(x + 2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(x+2ight)}{\left(x^{2}+2ight)}-\frac{6}{\left(x+2ight)}$$
- step1: Remove the parentheses:
  $$\frac{x+2}{x^{2}+2}-\frac{6}{x+2}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{\left(x+2ight)\left(x+2ight)}{\left(x^{2}+2ight)\left(x+2ight)}-\frac{6\left(x^{2}+2ight)}{\left(x+2ight)\left(x^{2}+2ight)}$$
- step3: Rewrite the expression:
  $$\frac{\left(x+2ight)\left(x+2ight)}{\left(x^{2}+2ight)\left(x+2ight)}-\frac{6\left(x^{2}+2ight)}{\left(x^{2}+2ight)\left(x+2ight)}$$
- step4: Transform the expression:
  $$\frac{\left(x+2ight)\left(x+2ight)-6\left(x^{2}+2ight)}{\left(x^{2}+2ight)\left(x+2ight)}$$
- step5: Multiply the terms:
  $$\frac{x^{2}+4x+4-6\left(x^{2}+2ight)}{\left(x^{2}+2ight)\left(x+2ight)}$$
- step6: Multiply the terms:
  $$\frac{x^{2}+4x+4-\left(6x^{2}+12ight)}{\left(x^{2}+2ight)\left(x+2ight)}$$
- step7: Calculate:
  $$\frac{-5x^{2}+4x-8}{\left(x^{2}+2ight)\left(x+2ight)}$$
- step8: Multiply the terms:
  $$\frac{-5x^{2}+4x-8}{x^{3}+2x^{2}+2x+4}$$
Calculate or simplify the expression $$ 2/(2*x) + (7*x + 1)/(4*x^2*y) - (5*x*y)/4 + 1 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2}{2x}+\frac{\left(7x+1ight)}{4x^{2}y}-\frac{5xy}{4}+1$$
- step1: Remove the parentheses:
  $$\frac{2}{2x}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1$$
- step2: Calculate:
  $$\frac{1}{x}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1$$
- step3: Reduce fractions to a common denominator:
  $$\frac{4xy}{x	imes 4xy}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}$$
- step4: Reorder the terms:
  $$\frac{4xy}{4x	imes xy}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}$$
- step5: Multiply the terms:
  $$\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4xyx}+\frac{4xyx}{4xyx}$$
- step6: Multiply the terms:
  $$\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4x^{2}y}+\frac{4xyx}{4xyx}$$
- step7: Multiply the terms:
  $$\frac{4xy}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyxyx}{4x^{2}y}+\frac{4xyx}{4x^{2}y}$$
- step8: Transform the expression:
  $$\frac{4xy+7x+1-5xyxyx+4xyx}{4x^{2}y}$$
- step9: Multiply the terms:
  $$\frac{4xy+7x+1-5x^{3}y^{2}+4xyx}{4x^{2}y}$$
- step10: Multiply the terms:
  $$\frac{4xy+7x+1-5x^{3}y^{2}+4x^{2}y}{4x^{2}y}$$
- step11: Simplify:
  $$\frac{4xy+7x+1-5x^{3}y^{2}+4x^{2}y}{4yx^{2}}$$
Calculate or simplify the expression $$ (2*x)/3 + 1/6 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{2x}{3}+\frac{1}{6}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{2x	imes 2}{3	imes 2}+\frac{1}{6}$$
- step2: Multiply the numbers:
  $$\frac{2x	imes 2}{6}+\frac{1}{6}$$
- step3: Transform the expression:
  $$\frac{2x	imes 2+1}{6}$$
- step4: Multiply the terms:
  $$\frac{4x+1}{6}$$
Calculate or simplify the expression $$ 4/((2*x + 1)^2) - (x + 1)/(2*x + 1) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{4}{\left(2x+1ight)^{2}}-\frac{\left(x+1ight)}{\left(2x+1ight)}$$
- step1: Remove the parentheses:
  $$\frac{4}{\left(2x+1ight)^{2}}-\frac{x+1}{2x+1}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{4}{\left(2x+1ight)^{2}}-\frac{\left(x+1ight)\left(2x+1ight)}{\left(2x+1ight)\left(2x+1ight)}$$
- step3: Multiply:
  $$\frac{4}{\left(2x+1ight)^{2}}-\frac{\left(x+1ight)\left(2x+1ight)}{\left(2x+1ight)^{2}}$$
- step4: Transform the expression:
  $$\frac{4-\left(x+1ight)\left(2x+1ight)}{\left(2x+1ight)^{2}}$$
- step5: Multiply the terms:
  $$\frac{4-\left(2x^{2}+3x+1ight)}{\left(2x+1ight)^{2}}$$
- step6: Subtract the terms:
  $$\frac{3-2x^{2}-3x}{\left(2x+1ight)^{2}}$$
- step7: Calculate:
  $$\frac{3-2x^{2}-3x}{4x^{2}+4x+1}$$
Calculate or simplify the expression $$ 7/(6*x) - 2/(9*y) + 1/(3*x^2) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{7}{6x}-\frac{2}{9y}+\frac{1}{3x^{2}}$$
- step1: Reduce fractions to a common denominator:
  $$\frac{7	imes 3yx}{6x	imes 3yx}-\frac{2	imes 2x	imes x}{9y	imes 2x	imes x}+\frac{2	imes 3y}{3x^{2}	imes 2	imes 3y}$$
- step2: Multiply the numbers:
  $$\frac{7	imes 3yx}{18xyx}-\frac{2	imes 2x	imes x}{9y	imes 2x	imes x}+\frac{2	imes 3y}{3x^{2}	imes 2	imes 3y}$$
- step3: Multiply the numbers:
  $$\frac{7	imes 3yx}{18xyx}-\frac{2	imes 2x	imes x}{18yx	imes x}+\frac{2	imes 3y}{3x^{2}	imes 2	imes 3y}$$
- step4: Multiply the terms:
  $$\frac{7	imes 3yx}{18xyx}-\frac{2	imes 2x	imes x}{18yx	imes x}+\frac{2	imes 3y}{18x^{2}y}$$
- step5: Multiply the terms:
  $$\frac{7	imes 3yx}{18x^{2}y}-\frac{2	imes 2x	imes x}{18yx	imes x}+\frac{2	imes 3y}{18x^{2}y}$$
- step6: Multiply the terms:
  $$\frac{7	imes 3yx}{18x^{2}y}-\frac{2	imes 2x	imes x}{18yx^{2}}+\frac{2	imes 3y}{18x^{2}y}$$
- step7: Rewrite the expression:
  $$\frac{7	imes 3yx}{18x^{2}y}-\frac{2	imes 2x	imes x}{18x^{2}y}+\frac{2	imes 3y}{18x^{2}y}$$
- step8: Transform the expression:
  $$\frac{7	imes 3yx-2	imes 2x	imes x+2	imes 3y}{18x^{2}y}$$
- step9: Multiply the terms:
  $$\frac{21yx-2	imes 2x	imes x+2	imes 3y}{18x^{2}y}$$
- step10: Multiply the terms:
  $$\frac{21yx-4x^{2}+2	imes 3y}{18x^{2}y}$$
- step11: Multiply the terms:
  $$\frac{21yx-4x^{2}+6y}{18x^{2}y}$$
- step12: Simplify:
  $$\frac{21yx-4x^{2}+6y}{18yx^{2}}$$
Calculate or simplify the expression $$ 1 + 2 - (2*a - 1)/2 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$1+2-\frac{\left(2a-1ight)}{2}$$
- step1: Remove the parentheses:
  $$1+2-\frac{2a-1}{2}$$
- step2: Add the numbers:
  $$3-\frac{2a-1}{2}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{3	imes 2}{2}-\frac{2a-1}{2}$$
- step4: Transform the expression:
  $$\frac{3	imes 2-\left(2a-1ight)}{2}$$
- step5: Multiply the numbers:
  $$\frac{6-\left(2a-1ight)}{2}$$
- step6: Subtract the terms:
  $$\frac{7-2a}{2}$$
Calculate or simplify the expression $$ 5/(6*x*y) - (x + 2)/(2*x) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{5}{6xy}-\frac{\left(x+2ight)}{2x}$$
- step1: Remove the parentheses:
  $$\frac{5}{6xy}-\frac{x+2}{2x}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{5}{6xy}-\frac{\left(x+2ight)	imes 3y}{2x	imes 3y}$$
- step3: Multiply the numbers:
  $$\frac{5}{6xy}-\frac{\left(x+2ight)	imes 3y}{6xy}$$
- step4: Transform the expression:
  $$\frac{5-\left(x+2ight)	imes 3y}{6xy}$$
- step5: Multiply the terms:
  $$\frac{5-\left(3xy+6yight)}{6xy}$$
- step6: Remove the parentheses:
  $$\frac{5-3xy-6y}{6xy}$$
Calculate or simplify the expression $$ (x - 3)/3 - (x - 2)/2 + (x + 1)/6 $$.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{\left(x-3ight)}{3}-\frac{\left(x-2ight)}{2}+\frac{\left(x+1ight)}{6}$$
- step1: Remove the parentheses:
  $$\frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6}$$
- step2: Reduce fractions to a common denominator:
  $$\frac{\left(x-3ight)	imes 2}{3	imes 2}-\frac{\left(x-2ight)	imes 3}{2	imes 3}+\frac{x+1}{6}$$
- step3: Multiply the numbers:
  $$\frac{\left(x-3ight)	imes 2}{6}-\frac{\left(x-2ight)	imes 3}{2	imes 3}+\frac{x+1}{6}$$
- step4: Multiply the numbers:
  $$\frac{\left(x-3ight)	imes 2}{6}-\frac{\left(x-2ight)	imes 3}{6}+\frac{x+1}{6}$$
- step5: Transform the expression:
  $$\frac{\left(x-3ight)	imes 2-\left(x-2ight)	imes 3+x+1}{6}$$
- step6: Multiply the terms:
  $$\frac{2x-6-\left(x-2ight)	imes 3+x+1}{6}$$
- step7: Multiply the terms:
  $$\frac{2x-6-\left(3x-6ight)+x+1}{6}$$
- step8: Calculate:
  $$\frac{1}{6}$$
Here are the simplified results for each expression:

1. **Expression:** $$ \frac{2x}{3} + \frac{1}{6} $$  
   **Simplified Result:** $$ \frac{4x + 1}{6} $$

2. **Expression:** $$ \frac{7}{6x} - \frac{2}{9y} + \frac{1}{3x^2} $$  
   **Simplified Result:** $$ \frac{21yx - 4x^2 + 6y}{18yx^2} $$

3. **Expression:** $$ \frac{3}{x} + \frac{2}{x^2} $$  
   **Simplified Result:** $$ \frac{3x + 2}{x^2} $$

4. **Expression:** $$ 1 + 2 - \frac{2a - 1}{2} $$  
   **Simplified Result:** $$ \frac{7 - 2a}{2} $$

5. **Expression:** $$ \frac{5}{6xy} - \frac{x + 2}{2x} $$  
   **Simplified Result:** $$ \frac{5 - 3xy - 6y}{6xy} $$

6. **Expression:** $$ \frac{5}{2x} - \frac{x + 1}{x - 2} $$  
   **Simplified Result:** $$ \frac{3x - 10 - 2x^2}{2x^2 - 4x} $$

7. **Expression:** $$ \frac{2}{x} + \frac{5}{x + 1} $$  
   **Simplified Result:** $$ \frac{7x + 2}{x^2 + x} $$

8. **Expression:** $$ \frac{x + 2}{x^2 + 2} - \frac{6}{x + 2} $$  
   **Simplified Result:** $$ \frac{-5x^2 + 4x - 8}{x^3 + 2x^2 + 2x + 4} $$

9. **Expression:** $$ \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} $$  
   **Simplified Result:** $$ -\frac{12x}{x^2 - 9} $$

10. **Expression:** $$ \frac{4}{(2x + 1)^2} - \frac{x + 1}{2x + 1} $$  
    **Simplified Result:** $$ \frac{3 - 2x^2 - 3x}{4x^2 + 4x + 1} $$

11. **Expression:** $$ \frac{2}{2x} + \frac{7x + 1}{4x^2y} - \frac{5xy}{4} + 1 $$  
    **Simplified Result:** $$ \frac{4xy + 7x + 1 - 5x^3y^2 + 4x^2y}{4yx^2} $$

12. **Expression:** $$ \frac{x - 3}{3} - \frac{x - 2}{2} + \frac{x + 1}{6} $$  
    **Simplified Result:** $$ \frac{1}{6} $$ (or $$ 0.1\dot{6} $$)

13. **Expression:** $$ x - \frac{2x}{3x - 2} $$  
    **Simplified Result:** $$ \frac{3x^2 - 4x}{3x - 2} $$

If you need further assistance or explanations for any specific step, feel free to ask!
Here are the simplified results for each expression: 1. $$ \frac{2x}{3} + \frac{1}{6} = \frac{4x + 1}{6} $$ 2. $$ \frac{7}{6x} - \frac{2}{9y} + \frac{1}{3x^2} = \frac{21yx - 4x^2 + 6y}{18yx^2} $$ 3. $$ \frac{3}{x} + \frac{2}{x^2} = \frac{3x + 2}{x^2} $$ 4. $$ 1 + 2 - \frac{2a - 1}{2} = \frac{7 - 2a}{2} $$ 5. $$ \frac{5}{6xy} - \frac{x + 2}{2x} = \frac{5 - 3xy - 6y}{6xy} $$ 6. $$ \frac{5}{2x} - \frac{x + 1}{x - 2} = \frac{3x - 10 - 2x^2}{2x^2 - 4x} $$ 7. $$ \frac{2}{x} + \frac{5}{x + 1} = \frac{7x + 2}{x^2 + x} $$ 8. $$ \frac{x + 2}{x^2 + 2} - \frac{6}{x + 2} = \frac{-5x^2 + 4x - 8}{x^3 + 2x^2 + 2x + 4} $$ 9. $$ \frac{x - 3}{x + 3} - \frac{x + 3}{x - 3} = -\frac{12x}{x^2 - 9} $$ 10. $$ \frac{4}{(2x + 1)^2} - \frac{x + 1}{2x + 1} = \frac{3 - 2x^2 - 3x}{4x^2 + 4x + 1} $$ 11. $$ \frac{2}{2x} + \frac{7x + 1}{4x^2y} - \frac{5xy}{4} + 1 = \frac{4xy + 7x + 1 - 5x^3y^2 + 4x^2y}{4yx^2} $$ 12. $$ \frac{x - 3}{3} - \frac{x - 2}{2} + \frac{x + 1}{6} = \frac{1}{6} $$ 13. $$ x - \frac{2x}{3x - 2} = \frac{3x^2 - 4x}{3x - 2} $$ If you need further assistance or explanations for any specific step, feel free to ask!

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