To add the two fractions \( \frac{1}{2}(x + 4) + \frac{1}{4}(4x - 2) \), we'll first simplify each term.
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Simplify \( \frac{1}{2}(x + 4) \): \[ \frac{1}{2}(x + 4) = \frac{x + 4}{2} \]
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Simplify \( \frac{1}{4}(4x - 2) \): \[ \frac{1}{4}(4x - 2) = \frac{4x - 2}{4} \] We can further simplify \( \frac{4x - 2}{4} = x - \frac{1}{2} \).
Now we can combine the two terms: \[ \frac{x + 4}{2} + \left( x - \frac{1}{2} \right) \]
To add these, we need a common denominator. The common denominator is 2: \[ x - \frac{1}{2} = \frac{2x}{2} - \frac{1}{2} = \frac{2x - 1}{2} \]
Now add: \[ \frac{x + 4}{2} + \frac{2x - 1}{2} = \frac{x + 4 + 2x - 1}{2} = \frac{3x + 3}{2} \]
This can be factored further: \[ \frac{3(x + 1)}{2} \]
Now we can express this in mixed number form if necessary. Since \( 3(x+1) \) is a polynomial, to convert this to mixed number format would only apply if we had a fractional component. Here, we can leave it as a fraction.
Thus, the final answer is: \[ \frac{3(x + 1)}{2} \]
This expression does not convert neatly into a mixed number, but is simplified fully. If you need it evaluated for a specific value of \( x \), please provide that value!
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