3/4(1/2 x + 2/3) + 1/4 x = 6 3/4

Steps to Solve

1. Convert the mixed number to an improper fraction
   To solve the equation, first convert ( 6\frac{3}{4} ) into an improper fraction:
   $$ 6\frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{24 + 3}{4} = \frac{27}{4} $$

2. Distribute the fraction
   Now distribute ( \frac{3}{4} ) inside the parentheses:
   $$ \frac{3}{4} \left( \frac{1}{2} x + \frac{2}{3} \right) = \frac{3}{4} \cdot \frac{1}{2} x + \frac{3}{4} \cdot \frac{2}{3} $$
   Calculate the terms separately:
   $$ \frac{3}{4} \cdot \frac{1}{2} x = \frac{3}{8} x $$
   $$ \frac{3}{4} \cdot \frac{2}{3} = \frac{3 \cdot 2}{4 \cdot 3} = \frac{2}{4} = \frac{1}{2} $$
   So, the equation now is:
   $$ \frac{3}{8} x + \frac{1}{2} + \frac{1}{4} x = \frac{27}{4} $$

3. Combine like terms
   Now combine ( \frac{3}{8} x ) and ( \frac{1}{4} x ):
   Convert ( \frac{1}{4} x ) to eighths:
   $$ \frac{1}{4} x = \frac{2}{8} x $$
   Now we can add them:
   $$ \frac{3}{8} x + \frac{2}{8} x = \frac{5}{8} x $$
   The equation simplifies to:
   $$ \frac{5}{8} x + \frac{1}{2} = \frac{27}{4} $$

4. Isolate the term with x
   Subtract ( \frac{1}{2} ) from both sides of the equation:
   Convert ( \frac{1}{2} ) to fourths:
   $$ \frac{1}{2} = \frac{2}{4} $$
   So,
   $$ \frac{27}{4} - \frac{2}{4} = \frac{25}{4} $$
   This gives us:
   $$ \frac{5}{8} x = \frac{25}{4} $$

5. Solve for x
   Multiply both sides by the reciprocal of ( \frac{5}{8} ):
   $$ \frac{8}{5} \cdot \frac{5}{8} x = \frac{8}{5} \cdot \frac{25}{4} $$
   So, $x$ simplifies to:
   $$ x = \frac{8 \cdot 25}{5 \cdot 4} = \frac{200}{20} = 10 $$