Let's simplify the expression step by step:
Given expression:
\[ \frac{1}{2}(z + 4) - 3\left(\frac{1}{4}z + 1\right) \]
First, distribute \(\frac{1}{2}\) in the first term:
\[ \frac{1}{2}z + \frac{1}{2} \cdot 4 = \frac{1}{2}z + 2 \]
Now distribute \(-3\) in the second term:
\[ -3\left(\frac{1}{4}z + 1\right) = -3 \cdot \frac{1}{4}z - 3 \cdot 1 = -\frac{3}{4}z - 3 \]
Now combine these results:
\[ \frac{1}{2}z + 2 - \frac{3}{4}z - 3 \]
Next, combine like terms. Combine the \( z \) terms first:
\[ \frac{1}{2}z - \frac{3}{4}z \]
To combine these, convert \(\frac{1}{2}z\) to a fraction with a denominator of 4:
\[ \frac{1}{2}z = \frac{2}{4}z \]
Now we can combine:
\[ \frac{2}{4}z - \frac{3}{4}z = \frac{-1}{4}z \]
Now combine the constant terms:
\[ 2 - 3 = -1 \]
Putting it all together, the simplified expression is:
\[ -\frac{1}{4}z - 1 \]
Thus, the final result is:
\[ -\frac{1}{4}z - 1 \]