To simplify the expression \(-\frac{1}{2}(z + 4) - 3\left(\frac{1}{4}z + 1\right)\), we will first distribute the terms.
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Distributing \(-\frac{1}{2}\) into \((z + 4)\): \[ -\frac{1}{2}(z + 4) = -\frac{1}{2}z - 2 \]
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Distributing \(-3\) into \(\left(\frac{1}{4}z + 1\right)\): \[ -3\left(\frac{1}{4}z + 1\right) = -\frac{3}{4}z - 3 \]
Now, we can combine the two results: \[ -\frac{1}{2}z - 2 - \frac{3}{4}z - 3 \]
Next, we need to combine the \(z\) terms and the constant terms:
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For the \(z\) terms: \[ -\frac{1}{2}z - \frac{3}{4}z \] To combine these, we should express \(-\frac{1}{2}\) in terms of quarters: \(-\frac{1}{2} = -\frac{2}{4}\), so: \[ -\frac{2}{4}z - \frac{3}{4}z = -\frac{5}{4}z \]
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For the constant terms: \[ -2 - 3 = -5 \]
Combining these gives us: \[ -\frac{5}{4}z - 5 \]
So, the simplified expression is: \[ -\frac{5}{4}z - 5 \]