To simplify the expression \( \frac{1}{2}(z + 4) - 3\left(\frac{1}{4}z + 1\right) \), we can follow these steps:
- Distribute \( \frac{1}{2} \) and \( -3 \) through the respective expressions.
- Combine like terms.
Let's do it step by step.
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Distributing \( \frac{1}{2} \) to \( (z + 4) \): \[ \frac{1}{2}(z + 4) = \frac{1}{2}z + \frac{1}{2} \cdot 4 = \frac{1}{2}z + 2 \]
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Distributing \( -3 \) to \( \left(\frac{1}{4}z + 1\right) \): \[ -3\left(\frac{1}{4}z + 1\right) = -3 \cdot \frac{1}{4}z - 3 \cdot 1 = -\frac{3}{4}z - 3 \]
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Now combine the two results: \[ \frac{1}{2}z + 2 - \frac{3}{4}z - 3 \]
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Combine the \( z \) terms: \[ \frac{1}{2}z - \frac{3}{4}z \] To combine these, we can convert \( \frac{1}{2}z \) to a fraction with a denominator of 4: \[ \frac{1}{2}z = \frac{2}{4}z \] Now we can subtract: \[ \frac{2}{4}z - \frac{3}{4}z = -\frac{1}{4}z \]
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Now combine the constant terms: \[ 2 - 3 = -1 \]
Putting it all together, we have: \[ -\frac{1}{4}z - 1 \]
So, the final answer is: \[ -\frac{1}{4}z - 1 \]
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