To simplify the expression \( \frac{1}{2} ( z + 4 ) - 3 \left( \frac{1}{4} z + 1 \right) \), we will follow these steps:
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Distribute the \( \frac{1}{2} \) and the \( -3 \) across the terms: \[ \frac{1}{2} (z + 4) = \frac{1}{2} z + \frac{1}{2} \cdot 4 = \frac{1}{2} z + 2 \] \[ -3 \left( \frac{1}{4} z + 1 \right) = -3 \cdot \frac{1}{4} z - 3 \cdot 1 = -\frac{3}{4} z - 3 \]
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Now combine these results: \[ \frac{1}{2} z + 2 - \frac{3}{4} z - 3 \]
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Combine the \( z \) terms:
- Convert \( \frac{1}{2} z \) into fourths: \[ \frac{1}{2} z = \frac{2}{4} z \]
- Now combine: \[ \frac{2}{4} z - \frac{3}{4} z = \frac{2 - 3}{4} z = -\frac{1}{4} z \]
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Now combine the constant terms: \[ 2 - 3 = -1 \]
So, putting it all together, we get: \[ -\frac{1}{4} z - 1 \]
Thus, the final result is: \[ -\frac{1}{4} z - 1 \]