) 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
\( \quad = \dfrac{x}{5} \)
) 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
\( \quad = \dfrac{2}{7} \)
) 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
\( \quad = 9x \)
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 \( 2 \)
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
\( \quad = \dfrac{2x - 15}{2} \)