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The given problem shows two inequalities involving the variable \( z \):

\[
\frac{z}{4} \geq 9 \quad \text{and} \quad \frac{z}{7} \leq 6
\]

### Solving each inequality:

1. **For** \( \frac{z}{4} \geq 9 \):
   \[
   z \geq 9 \times 4 = 36
   \]
   So, \( z \geq 36 \).

2. **For** \( \frac{z}{7} \leq 6 \):
   \[
   z \leq 6 \times 7 = 42
   \]
   So, \( z \leq 42 \).

### Combined Solution:
The solution must satisfy both inequalities, meaning \( z \) must be greater than or equal to 36 and less than or equal to 42. Thus, the solution is:

\[
36 \leq z \leq 42
\]

Would you like further explanation on any step or more details?

Here are five related questions to consider:
1. What is the solution set for \( z \) if the inequalities were reversed?
2. How would the solution change if \( \frac{z}{4} > 9 \) instead of \( \geq \)?
3. Can we represent the solution graphically on a number line?
4. What if the inequalities were combined using "or" instead of "and"?
5. How does solving a system of inequalities differ from solving a system of equations?

**Tip:** When solving compound inequalities, always consider if the condition is "and" (intersection) or "or" (union) for combining results.

Solve the compound inequality: z/4 ≥ 9 and z/7 ≤ 6.

Learn how to solve the compound inequality z/4 ≥ 9 and z/7 ≤ 6. This step-by-step guide covers key algebraic concepts, including solving inequalities and combining them to find the solution set. Suitable for students in Grades 8-10.

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