To simplify the expression \((\frac{1}{3}x + \frac{4}{7}) + (\frac{2}{3}x + \frac{5}{6})\), we can start by combining like terms.
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Combine the \(x\) terms: \[ \frac{1}{3}x + \frac{2}{3}x = \left(\frac{1}{3} + \frac{2}{3}\right)x = \frac{3}{3}x = 1x = x \]
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Now, combine the constant terms: \[ \frac{4}{7} + \frac{5}{6} \] To do this, we need a common denominator. The least common multiple of 7 and 6 is 42.
Rewrite each fraction with a denominator of 42: \[ \frac{4}{7} = \frac{4 \times 6}{7 \times 6} = \frac{24}{42} \] \[ \frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42} \]
Now, we can add the two fractions: \[ \frac{24}{42} + \frac{35}{42} = \frac{24 + 35}{42} = \frac{59}{42} \]
Putting it all together, we have: \[ x + \frac{59}{42} \]
Thus, the simplified expression is: \[ \boxed{x + \frac{59}{42}} \]