Let's simplify the expression step by step:
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Start with the original expression: \[ \frac{1}{2}(z + 4) - 3\left(\frac{1}{4}z + 1\right) \]
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Distribute \(\frac{1}{2}\) in the first term: \[ \frac{1}{2}z + \frac{1}{2} \cdot 4 = \frac{1}{2}z + 2 \]
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Distribute \(-3\) in the second term: \[ -3 \cdot \frac{1}{4}z - 3 \cdot 1 = -\frac{3}{4}z - 3 \]
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Combine the two results: \[ \left(\frac{1}{2}z + 2\right) + \left(-\frac{3}{4}z - 3\right) \]
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Combine like terms: \[ \frac{1}{2}z - \frac{3}{4}z + 2 - 3 \]
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Convert \(\frac{1}{2}z\) to a fraction with a denominator of 4: \[ \frac{1}{2}z = \frac{2}{4}z \] So it becomes: \[ \frac{2}{4}z - \frac{3}{4}z = \frac{2 - 3}{4}z = -\frac{1}{4}z \]
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Combine the constant terms: \[ 2 - 3 = -1 \]
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Therefore, the simplified expression is: \[ -\frac{1}{4}z - 1 \]
So, the final result is: \[ -\frac{1}{4}z - 1 \]