Simplify the following: (1) \( \frac{2 x}{3}+\frac{1}{6} \) (2) \( \frac{7}{6 x}-\frac{2}{9 x y}+\frac{1}{3 x^{3}} \) (4) \( 1+a-\frac{2 a-1}{2} \) (5) \( \frac{5}{6 x y}-\frac{x+2}{2 x} \) (7) \( \frac{2}{x}+\frac{5}{x+1} \) (8) \( \frac{x+2}{x^{2}+2}-\frac{6}{x+2} \) (10) \( \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} \) (11) \( \frac{2}{2 x y}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y}{4}+1 \) (12) \( \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \) (13) \( x-\frac{2 x}{3 x-2} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{2}{x}+\frac{5}{\left(x+1\right)}\) - step1: Remove the parentheses: \(\frac{2}{x}+\frac{5}{x+1}\) - step2: Reduce fractions to a common denominator: \(\frac{2\left(x+1\right)}{x\left(x+1\right)}+\frac{5x}{\left(x+1\right)x}\) - step3: Rewrite the expression: \(\frac{2\left(x+1\right)}{x\left(x+1\right)}+\frac{5x}{x\left(x+1\right)}\) - step4: Transform the expression: \(\frac{2\left(x+1\right)+5x}{x\left(x+1\right)}\) - step5: Multiply the terms: \(\frac{2x+2+5x}{x\left(x+1\right)}\) - step6: Add the terms: \(\frac{7x+2}{x\left(x+1\right)}\) - step7: Multiply the terms: \(\frac{7x+2}{x^{2}+x}\) Calculate or simplify the expression \( x - (2*x)/(3*x - 2) \). Simplify the expression by following steps: - step0: Solution: \(x-\frac{2x}{\left(3x-2\right)}\) - step1: Remove the parentheses: \(x-\frac{2x}{3x-2}\) - step2: Reduce fractions to a common denominator: \(\frac{x\left(3x-2\right)}{3x-2}-\frac{2x}{3x-2}\) - step3: Transform the expression: \(\frac{x\left(3x-2\right)-2x}{3x-2}\) - 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 \( 4/((2*x + 1)^2) - (x + 1)/(2*x + 1) \). Simplify the expression by following steps: - step0: Solution: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)}{\left(2x+1\right)}\) - step1: Remove the parentheses: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{x+1}{2x+1}\) - step2: Reduce fractions to a common denominator: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)\left(2x+1\right)}\) - step3: Multiply: \(\frac{4}{\left(2x+1\right)^{2}}-\frac{\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^{2}}\) - step4: Transform the expression: \(\frac{4-\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^{2}}\) - step5: Multiply the terms: \(\frac{4-\left(2x^{2}+3x+1\right)}{\left(2x+1\right)^{2}}\) - step6: Subtract the terms: \(\frac{3-2x^{2}-3x}{\left(2x+1\right)^{2}}\) - step7: Calculate: \(\frac{3-2x^{2}-3x}{4x^{2}+4x+1}\) Calculate or simplify the expression \( (x + 2)/(x^2 + 2) - 6/(x + 2) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x+2\right)}{\left(x^{2}+2\right)}-\frac{6}{\left(x+2\right)}\) - step1: Remove the parentheses: \(\frac{x+2}{x^{2}+2}-\frac{6}{x+2}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x+2\right)\left(x+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}-\frac{6\left(x^{2}+2\right)}{\left(x+2\right)\left(x^{2}+2\right)}\) - step3: Rewrite the expression: \(\frac{\left(x+2\right)\left(x+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}-\frac{6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step4: Transform the expression: \(\frac{\left(x+2\right)\left(x+2\right)-6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step5: Multiply the terms: \(\frac{x^{2}+4x+4-6\left(x^{2}+2\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step6: Multiply the terms: \(\frac{x^{2}+4x+4-\left(6x^{2}+12\right)}{\left(x^{2}+2\right)\left(x+2\right)}\) - step7: Calculate: \(\frac{-5x^{2}+4x-8}{\left(x^{2}+2\right)\left(x+2\right)}\) - step8: Multiply the terms: \(\frac{-5x^{2}+4x-8}{x^{3}+2x^{2}+2x+4}\) Calculate or simplify the expression \( 2/(2*x*y) + (7*x + 1)/(4*x^2*y) - (5*x*y)/4 + 1 \). Simplify the expression by following steps: - step0: Solution: \(\frac{2}{2xy}+\frac{\left(7x+1\right)}{4x^{2}y}-\frac{5xy}{4}+1\) - step1: Remove the parentheses: \(\frac{2}{2xy}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1\) - step2: Calculate: \(\frac{1}{xy}+\frac{7x+1}{4x^{2}y}-\frac{5xy}{4}+1\) - step3: Reduce fractions to a common denominator: \(\frac{4x}{xy\times 4x}+\frac{7x+1}{4x^{2}y}-\frac{5xyx\times xy}{4x\times xy}+\frac{4x\times xy}{4x\times xy}\) - step4: Reorder the terms: \(\frac{4x}{4xyx}+\frac{7x+1}{4x^{2}y}-\frac{5xyx\times xy}{4x\times xy}+\frac{4x\times xy}{4x\times xy}\) - step5: Multiply the terms: \(\frac{4x}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyx\times xy}{4x\times xy}+\frac{4x\times xy}{4x\times xy}\) - step6: Multiply the terms: \(\frac{4x}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyx\times xy}{4x^{2}y}+\frac{4x\times xy}{4x\times xy}\) - step7: Multiply the terms: \(\frac{4x}{4x^{2}y}+\frac{7x+1}{4x^{2}y}-\frac{5xyx\times xy}{4x^{2}y}+\frac{4x\times xy}{4x^{2}y}\) - step8: Transform the expression: \(\frac{4x+7x+1-5xyx\times xy+4x\times xy}{4x^{2}y}\) - step9: Multiply the terms: \(\frac{4x+7x+1-5x^{3}y^{2}+4x\times xy}{4x^{2}y}\) - step10: Multiply the terms: \(\frac{4x+7x+1-5x^{3}y^{2}+4x^{2}y}{4x^{2}y}\) - step11: Add the terms: \(\frac{11x+1-5x^{3}y^{2}+4x^{2}y}{4x^{2}y}\) - step12: Simplify: \(\frac{11x+1-5x^{3}y^{2}+4x^{2}y}{4yx^{2}}\) 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: \(\frac{2x\times 2}{3\times 2}+\frac{1}{6}\) - step2: Multiply the numbers: \(\frac{2x\times 2}{6}+\frac{1}{6}\) - step3: Transform the expression: \(\frac{2x\times 2+1}{6}\) - step4: Multiply the terms: \(\frac{4x+1}{6}\) Calculate or simplify the expression \( 7/(6*x) - 2/(9*x*y) + 1/(3*x^3) \). Simplify the expression by following steps: - step0: Solution: \(\frac{7}{6x}-\frac{2}{9xy}+\frac{1}{3x^{3}}\) - step1: Reduce fractions to a common denominator: \(\frac{7\times 3yx^{2}}{6x\times 3yx^{2}}-\frac{2\times 2x^{2}}{9xy\times 2x^{2}}+\frac{2\times 3y}{3x^{3}\times 2\times 3y}\) - step2: Multiply the numbers: \(\frac{7\times 3yx^{2}}{18xyx^{2}}-\frac{2\times 2x^{2}}{9xy\times 2x^{2}}+\frac{2\times 3y}{3x^{3}\times 2\times 3y}\) - step3: Multiply the numbers: \(\frac{7\times 3yx^{2}}{18xyx^{2}}-\frac{2\times 2x^{2}}{18xyx^{2}}+\frac{2\times 3y}{3x^{3}\times 2\times 3y}\) - step4: Multiply the terms: \(\frac{7\times 3yx^{2}}{18xyx^{2}}-\frac{2\times 2x^{2}}{18xyx^{2}}+\frac{2\times 3y}{18x^{3}y}\) - step5: Multiply: \(\frac{7\times 3yx^{2}}{18x^{3}y}-\frac{2\times 2x^{2}}{18xyx^{2}}+\frac{2\times 3y}{18x^{3}y}\) - step6: Multiply: \(\frac{7\times 3yx^{2}}{18x^{3}y}-\frac{2\times 2x^{2}}{18x^{3}y}+\frac{2\times 3y}{18x^{3}y}\) - step7: Transform the expression: \(\frac{7\times 3yx^{2}-2\times 2x^{2}+2\times 3y}{18x^{3}y}\) - step8: Multiply the terms: \(\frac{21yx^{2}-2\times 2x^{2}+2\times 3y}{18x^{3}y}\) - step9: Multiply the terms: \(\frac{21yx^{2}-4x^{2}+2\times 3y}{18x^{3}y}\) - step10: Multiply the terms: \(\frac{21yx^{2}-4x^{2}+6y}{18x^{3}y}\) - step11: Simplify: \(\frac{21yx^{2}-4x^{2}+6y}{18yx^{3}}\) Calculate or simplify the expression \( (x - 3)/3 - (x - 2)/2 + (x + 1)/6 \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x-3\right)}{3}-\frac{\left(x-2\right)}{2}+\frac{\left(x+1\right)}{6}\) - step1: Remove the parentheses: \(\frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6}\) - step2: Reduce fractions to a common denominator: \(\frac{\left(x-3\right)\times 2}{3\times 2}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step3: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{2\times 3}+\frac{x+1}{6}\) - step4: Multiply the numbers: \(\frac{\left(x-3\right)\times 2}{6}-\frac{\left(x-2\right)\times 3}{6}+\frac{x+1}{6}\) - step5: Transform the expression: \(\frac{\left(x-3\right)\times 2-\left(x-2\right)\times 3+x+1}{6}\) - step6: Multiply the terms: \(\frac{2x-6-\left(x-2\right)\times 3+x+1}{6}\) - step7: Multiply the terms: \(\frac{2x-6-\left(3x-6\right)+x+1}{6}\) - step8: Calculate: \(\frac{1}{6}\) Calculate or simplify the expression \( 1 + a - (2*a - 1)/2 \). Simplify the expression by following steps: - step0: Solution: \(1+a-\frac{\left(2a-1\right)}{2}\) - step1: Remove the parentheses: \(1+a-\frac{2a-1}{2}\) - step2: Reduce fractions to a common denominator: \(\frac{2}{2}+\frac{a\times 2}{2}-\frac{2a-1}{2}\) - step3: Transform the expression: \(\frac{2+a\times 2-\left(2a-1\right)}{2}\) - step4: Reorder the terms: \(\frac{2+2a-\left(2a-1\right)}{2}\) - step5: Calculate: \(\frac{3}{2}\) Calculate or simplify the expression \( 5/(6*x*y) - (x + 2)/(2*x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{5}{6xy}-\frac{\left(x+2\right)}{2x}\) - step1: Remove the parentheses: \(\frac{5}{6xy}-\frac{x+2}{2x}\) - step2: Reduce fractions to a common denominator: \(\frac{5}{6xy}-\frac{\left(x+2\right)\times 3y}{2x\times 3y}\) - step3: Multiply the numbers: \(\frac{5}{6xy}-\frac{\left(x+2\right)\times 3y}{6xy}\) - step4: Transform the expression: \(\frac{5-\left(x+2\right)\times 3y}{6xy}\) - step5: Multiply the terms: \(\frac{5-\left(3xy+6y\right)}{6xy}\) - step6: Remove the parentheses: \(\frac{5-3xy-6y}{6xy}\) Here are the simplified results for each expression: 1. **Expression:** \( \frac{2 x}{3}+\frac{1}{6} \) **Simplified Result:** \( \frac{4x+1}{6} \) 2. **Expression:** \( \frac{7}{6 x}-\frac{2}{9 x y}+\frac{1}{3 x^{3}} \) **Simplified Result:** \( \frac{21yx^{2}-4x^{2}+6y}{18yx^{3}} \) 3. **Expression:** \( 1+a-\frac{2 a-1}{2} \) **Simplified Result:** \( \frac{3}{2} \) (or \( 1.5 \)) 4. **Expression:** \( \frac{5}{6 x y}-\frac{x+2}{2 x} \) **Simplified Result:** \( \frac{5-3xy-6y}{6xy} \) 5. **Expression:** \( \frac{2}{x}+\frac{5}{x+1} \) **Simplified Result:** \( \frac{7x+2}{x^{2}+x} \) 6. **Expression:** \( \frac{x+2}{x^{2}+2}-\frac{6}{x+2} \) **Simplified Result:** \( \frac{-5x^{2}+4x-8}{x^{3}+2x^{2}+2x+4} \) 7. **Expression:** \( \frac{4}{(2 x+1)^{2}}-\frac{x+1}{2 x+1} \) **Simplified Result:** \( \frac{3-2x^{2}-3x}{4x^{2}+4x+1} \) 8. **Expression:** \( \frac{2}{2 x y}+\frac{7 x+1}{4 x^{2} y}-\frac{5 x y}{4}+1 \) **Simplified Result:** \( \frac{11x+1-5x^{3}y^{2}+4x^{2}y}{4yx^{2}} \) 9. **Expression:** \( \frac{x-3}{3}-\frac{x-2}{2}+\frac{x+1}{6} \) **Simplified Result:** \( \frac{1}{6} \) (or \( 0.1\dot{6} \)) 10. **Expression:** \( x-\frac{2 x}{3 x-2} \) **Simplified Result:** \( \frac{3x^{2}-4x}{3x-2} \) If you need further assistance or explanations for any of these results, feel free to ask!