To solve the expression \( -\frac{1}{2}(z + 4) - 3\left(\frac{1}{4}z + 1\right) \), we will first distribute the constants through the parentheses and then combine like terms.
- Start by distributing \( -\frac{1}{2} \) and \( -3 \):
\[ -\frac{1}{2}(z + 4) = -\frac{1}{2}z - 2 \]
\[ -3\left(\frac{1}{4}z + 1\right) = -3 \cdot \frac{1}{4}z - 3 \cdot 1 = -\frac{3}{4}z - 3 \]
- Now, combine these results:
\[ -\frac{1}{2}z - 2 - \frac{3}{4}z - 3 \]
- Combine the constant terms (-2 and -3):
\[ -2 - 3 = -5 \]
- Now, combine the \( z \) terms:
Convert \( -\frac{1}{2}z \) to a fraction with the same denominator as \( -\frac{3}{4}z \). The common denominator is 4:
\[ -\frac{1}{2}z = -\frac{2}{4}z \]
Now substitute this back into the expression:
\[ -\frac{2}{4}z - \frac{3}{4}z - 5 \]
- Combine the \( z \) terms:
\[ -\frac{2}{4}z - \frac{3}{4}z = -\frac{5}{4}z \]
So we have:
\[ -\frac{5}{4}z - 5 \]
Thus, the final answer is:
\[ -\frac{5}{4}z - 5 \]