To simplify the expression \( \frac{1}{2} (x + 4) + \frac{1}{4} (4x - 2) \), we can follow these steps:
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Distribute \( \frac{1}{2} \) in the first term and \( \frac{1}{4} \) in the second term: \[ \frac{1}{2}(x + 4) = \frac{1}{2}x + \frac{1}{2} \cdot 4 = \frac{1}{2}x + 2 \] \[ \frac{1}{4}(4x - 2) = \frac{1}{4} \cdot 4x - \frac{1}{4} \cdot 2 = x - \frac{1}{2} \]
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Now combine the two results: \[ \left(\frac{1}{2}x + 2\right) + \left(x - \frac{1}{2}\right) \] Combine like terms:
- For \( x \): \( \frac{1}{2}x + x = \frac{1}{2}x + \frac{2}{2}x = \frac{3}{2}x \)
- For the constant terms: \( 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \)
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The combined expression is: \[ \frac{3}{2}x + \frac{3}{2} \]
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To express it in mixed number form, we express \( \frac{3}{2} \) as \( 1 \frac{1}{2} \).
So, the final answer is: \[ \frac{3}{2}x + 1 \frac{1}{2} \]