Solutions to the fraction rules question are presented with steps and explanations.

1. )
   \( \quad \dfrac{0}{3} + \dfrac{1}{3} = \dfrac{0+1}{3} = \dfrac{1}{3} \)
   
   
2. )
   \( \quad \dfrac{2}{0} + 5 \) undefined because the fraction \( \dfrac{2}{0} \) has a denominator equal to zero and is therefore undefined.
   
   
3. )
   given:   \( \dfrac{3}{5} + 2 \)
   Rewrite the integer \( 2 \) as a fraction with denominator \( 5 \)
   \( \quad \dfrac{3}{5} + 2 = \dfrac{3}{5} + 2 \times \dfrac{5}{5} \)
   Add the fractions and simplify.
   \( \quad = \dfrac{3 + 2 \times 5}{5} = \dfrac{13}{5} \)
   
   
   
4. )
   given:   \( \dfrac{3}{2} + 2.1 \)
   Rewrite the decimal number \( 2.1 \) as a fraction
   \( \quad \dfrac{3}{2} + 2.1 = \dfrac{3}{2} + \dfrac{2.1}{1} \)
   \( \quad = \dfrac{3}{2} + \dfrac{2.1 \times 10}{1 \times 10} \)
   \( \quad = \dfrac{3}{2} + \dfrac{21}{10} \)
   Rewrite the two fractions with a common denominator
   \( \quad = \dfrac{3}{2} \times \dfrac{10}{10} + \dfrac{21}{10} \times \dfrac{2}{2} \)
   Multiply fractions and simplify
   \( \quad = \dfrac{30}{20} + \dfrac{42}{20} \)
   Add the fractions
   \( \quad = \dfrac{30 + 42}{20} = \dfrac{72}{20} \)
   Which may be reduced to
   \( \quad = \dfrac{18}{5} \)
   
   
   
5. )
   given:   \( 0.1 x + \dfrac{2x}{3}\)
   Rewrite the term \( 0.1 x \) as a fraction
   \( \quad 0.1 x + \dfrac{2x}{3} = \dfrac{1}{10} x + \dfrac{2x}{3} = \dfrac{x}{10} + \dfrac{2x}{3} \)
   Rewrite the fractions with common denominator
   \( \quad = \dfrac{x}{10} \times \dfrac{3}{3} + \dfrac{2x}{3} \times \dfrac{10}{10} \)
   Simplify
   \( \quad = \dfrac{3 x}{30} + \dfrac{20 x}{30} \)
   Add the above fractions
   \( \quad = \dfrac{3 x + 20 x}{30} = \dfrac{23 x}{30} \)
   
   
   
6. )
   given:   \( 3 x + \dfrac{x}{4} \)
   Rewrite the fractions with common denominator
   \( \quad 3 x + \dfrac{x}{4} = 3 x \times \dfrac{4}{4} + \dfrac{x}{4} \)
   Simplify
   \( \quad = \dfrac{12 x}{4} + \dfrac{x}{4} \)
   Add the fractions
   \( \quad = \dfrac{12x + x}{4} = \dfrac{13 x }{4} \)
   
   
   
7. )
   given:   \( 3x - \dfrac{5 x}{4} \)
   Rewrite the fractions with common denominator
   \( \quad 3x - \dfrac{5 x}{4} = 3 x \times \dfrac{4}{4} - \dfrac{5 x}{4} \)
   Simplify
   \( \quad = \dfrac{12 x}{4} - \dfrac{5 x}{4} \)
   Subtract the fractions
   \( \quad = \dfrac{12x - 5 x}{4} = \dfrac{7 x }{4} \)
   
   
   
8. )
   given:   \( \dfrac{3}{5} \times \dfrac{4}{9} \)
   Apply multiplication rule of fractions
   \( \quad \dfrac{3}{5} \times \dfrac{4}{9} = \dfrac{3 \times 4}{5 \times 9} = \dfrac{3 \times 4}{5 \times 3 \times 3} \)
   The numerator and denominator have a common factor \( 3 \) and therefore the fraction may be reduced by dividing numerator and denominator by \( 3 \)
   \( \quad = \dfrac{ (3 \times 4) \div 3 }{ (5 \times 9) \div 3} \)
   Simplify
   \( \quad = \dfrac{ 4 }{ 15 } \)
   
   
   
9. )
   given:   \( 6 \times \dfrac{3}{7} \)
   Rewrite \( 6 \) as a fraction
   \( \quad 6 \times \dfrac{3}{7} = \dfrac{6}{1} \times \dfrac{3}{7} \)
   Apply multiplication rule of fractions
   \( \quad = \dfrac{6 \times 3}{ 1 \times 7 } \)
   Simplify
   \( \quad = \dfrac{18}{7} \)
   
   
   
10. )
    given:   \( \dfrac{2x}{5} \times \dfrac{1}{2} \)
    Apply rule of multiplication of fractions
    \( \quad \dfrac{2x}{5} \times \dfrac{1}{2} = \dfrac{2x \times 1}{5 \times 2} \)
    The numerator and denominator have a common factor \( 2 \) and the fraction may then be reduced by dividing numerator and denominator by \( 2 \)
    \( \quad = \dfrac{(2x \times 1) \div 2}{(5 \times 2)\div 2} \)
    Simplify
    \( \quad = \dfrac{x}{5} \)
    
    
    
11. )
    given:   \( \dfrac{6}{7} \div 3 \)
    Rewrite \( 3 \) as a fraction
    \( \quad \dfrac{6}{7} \div 3 = \dfrac{6}{7} \div \dfrac{3}{1} \)
    Apply division rule of fraction (change into a multiplication of the first fraction by the reciprocal of the second fraction)
    \( \quad = \dfrac{6}{7} \times \dfrac{1}{3} \)
    Apply the multiplication rule of fractions
    \( \quad = \dfrac{6 \times 1}{7 \times 3} \)
    Factor the numerator
    \( \quad = \dfrac{3 \times 2 \times 1}{7 \times 3} \)
    Divide numerator and denominator by the common factor \( 3 \) to reduce the fraction
    \( \quad = \dfrac{ (3 \times 2 \times 1) \div 3}{ (7 \times 3) \div 3} \)
    Simplify
    \( \quad = \dfrac{2}{7} \)
    
    
    
12. )
    given:   \( x \div \dfrac{1}{9} \)
    Use division rule of fractions
    \( \quad x \div \dfrac{1}{9} = x \times \dfrac{9}{1} \)
    Rewrite \( x \) as a fraction
    \( \quad = \dfrac{x}{1} \times \dfrac{9}{1} \)
    Apply the multiplication rule of fractions
    \( \quad = \dfrac{x \times 9}{1 \times 1} \)
    Simplify
    \( \quad = \dfrac{9 x}{1} \)
    Simplify fraction with denominator equal to 1
    \( \quad = 9x \)
    
    
    
13. )
    given:   \( \dfrac{2x}{5} \div \dfrac{1}{9} \)
    Use division rule of fractions
    \( \quad \dfrac{2x}{5} \div \dfrac{1}{9} = \dfrac{2x}{5} \times \dfrac{9}{1} \)
    Apply the multiplication rule of fractions
    \( \quad = \dfrac{2x \times 9}{5 \times 1} \)
    Simplify
    \( \quad = \dfrac{18 x}{5} \)
    
    
    
14. )
    given:   \( - \dfrac{-3}{5} + \dfrac{-3}{5} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{-3}{5} \) included in the given expression as \( - \dfrac{3}{5} \)
    \( \quad - \dfrac{-3}{5} + \dfrac{-3}{5} = - ( - \dfrac{3}{5}) + \dfrac{-3}{5} \)
    Simplify
    \( \quad = \dfrac{3}{5} + \dfrac{-3}{5} \)
    Add the fractions and simplify
    \( \quad = \dfrac{3 +(- 3)}{5} = \dfrac{0}{5} = 0 \)
    
    
    
15. )
    given:   \( \dfrac{2}{-9} + \dfrac{7}{9} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{2}{-9} \) included in the given expression as \( \dfrac{-2}{9} \)
    \( \quad \dfrac{2}{-9} + \dfrac{7}{9} = \dfrac{- 2}{9} + \dfrac{7}{9} \)
    Add the fractions and simplify
    \( \quad = \dfrac{-2 + 7}{9} = \dfrac {5}{9} \)
    
    
    
16. )
    given:   \( \dfrac{-5}{-2} - \dfrac{7}{2} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{-5}{-2} \) included in the given expression as \( \dfrac{5}{2} \)
    \( \quad \dfrac{-5}{-2} - \dfrac{7}{2} = \dfrac{5}{2} - \dfrac{7}{2} \)
    Subtract the fractions
    \( \quad = \dfrac{5 - 7}{2} \)
    Apply rule of signs to simplify
    \( \quad = \dfrac{-2}{2} = - \dfrac{2}{2} = -1\)
    
    
    
17. )
    given:   \( \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} \)
    Apply rule of signs to rewrite the fraction \( \dfrac{ - 5x}{- 3} \) included in the given expression as \( \dfrac{ 5x}{3} \)
    \( \quad \dfrac{-2x}{3} - \dfrac{ - 5x}{- 3} = \dfrac{-2x}{3} - \dfrac{ 5x}{3} \)
    Subtract
    \( \quad = \dfrac{-2x - 5x}{3} \)
    Simplify
    \( \quad = \dfrac{-7x}{3} \)
    Which may also be written as (using rule of signs)
    \(\quad = - \dfrac{7x}{3} \)
    
    
    
18. )
    given:   \( 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} \)
    
    Rewrite \( 4 \) in the numerator \( 4 + \dfrac{1}{3} \) as a fraction with denominator equal to \( 3 \) and \( 1 \) in the denominator \( 1+\dfrac{1}{2} \) as a fraction with denominator equal to \( 2 \)
    
    \( \quad 2 - \dfrac{ 4 + \dfrac{1}{3}}{1+\dfrac{1}{2}} = 2 - \dfrac{ 4 \times \dfrac{3}{3} + \dfrac{1}{3}}{1 \times \dfrac{2}{2} +\dfrac{1}{2}} \)
    Simplify
    \( \quad = 2 - \dfrac{ \dfrac{12}{3} + \dfrac{1}{3}} {\dfrac{2}{2} +\dfrac{1}{2}} \)
    Add fractions with common denominator
    \( \quad = 2 - \dfrac{ \dfrac{13}{3}} {\dfrac{3}{2}} \)
    Use rule of division of fractions
    \( \quad = 2 - \dfrac{13}{3} \times {\dfrac{2}{3}} \)
    Multiply fractions
    \( \quad = 2 - \dfrac{26}{9}\)
    Rewrite \( 2 \) as a fraction with denominator equal to \( 9 \)
    \( \quad = 2 \times \dfrac{9}{9} - \dfrac{26}{9}\)
    Simplify
    \( \quad = \dfrac{18}{9} - \dfrac{26}{9}\)
    Subtract and simplify
    \( \quad = \dfrac{18-26}{9} = \dfrac{-8}{9} = - \dfrac{8}{9} \)
    
    
    
19. )
    given:   \( x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} \)
    
    Rewrite \( 2x \) in \( 2x + \dfrac{x}{2} \) as a fraction with denominator equal to \( 2 \) and \( x \) in \( x - \dfrac{2x}{3} \) as a fraction with denominator equal to \( 3 \)
    
    \( \quad x - \dfrac{ 2x + \dfrac{x}{2}}{x - \dfrac{2x}{3}} = x - \dfrac{ 2x \dfrac {2}{2} + \dfrac{x}{2}}{x \dfrac{3}{3} - \dfrac{2x}{3}} \)
    Simplify
    \( \quad = x - \dfrac{ \dfrac {4x}{2} + \dfrac{x}{2}}{\dfrac{3 x}{3} - \dfrac{2x}{3}} \)
    Add and subtract fractions with common denominator
    
    \( \quad = x - \dfrac{ \dfrac {4x + x}{2} }{\dfrac{3 x - 2x}{3} } \)
    Simplify
    \( \quad = x - \dfrac{ \dfrac {5x}{2} }{\dfrac{x}{3} } \)
    Use rule of division of fractions
    \( \quad = x - \dfrac {5x}{2} \times \dfrac{3}{x} \)
    Simplify
    \( \quad = x - \dfrac {15 x}{2 x} \)
    Reduce the fraction \( \dfrac {15 x}{2 x} \) dividing its numerator and denominator by \( x \)
    \(\quad = x - \dfrac {15 x \div x}{2 x \div x} \)
    Simplify
    \( \quad = x - \dfrac {15}{2} \)
    Rewrite \( x \) as a fraction with denominator equal to \( 2 \)
    \( \quad = x \times \dfrac{2}{2} - \dfrac {15}{2} \)
    Simplify
    \( \quad = \dfrac{2x }{2} - \dfrac {15}{2} \)
    Subtract fractions
    \( \quad = \dfrac{2x - 15}{2} \)

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