To combine the expressions \( \frac{1}{2}(x+4) + \frac{1}{4}(4x-2) \), we will start by distributing each term.
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Distribute \( \frac{1}{2} \) to \( (x+4) \): \[ \frac{1}{2}(x + 4) = \frac{1}{2}x + \frac{1}{2} \cdot 4 = \frac{1}{2}x + 2 \]
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Distribute \( \frac{1}{4} \) to \( (4x-2) \): \[ \frac{1}{4}(4x - 2) = \frac{1}{4} \cdot 4x - \frac{1}{4} \cdot 2 = x - \frac{1}{2} \]
Now we combine both results: \[ \frac{1}{2}x + 2 + x - \frac{1}{2} \]
Next, we combine like terms. We start with the \( x \) terms: \[ \frac{1}{2}x + x = \frac{1}{2}x + \frac{2}{2}x = \frac{3}{2}x \]
Now combine the constant terms: \[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]
Putting it all together, we have: \[ \frac{3}{2}x + \frac{3}{2} \]
Since \( \frac{3}{2} \) can be expressed as a mixed number: \[ \frac{3}{2} = 1\frac{1}{2} \]
Thus, the final result in mixed number form is: \[ \frac{3}{2}x + 1\frac{1}{2} \]
So, the answer is: \[ \frac{3}{2}x + 1\frac{1}{2} \]