It seems like you're trying to solve an inequality involving \( z \). Let's break it down and solve the inequality step by step.

The inequality given is:

\[
21 - 6z \geq 27 - 8z
\]

### Step 1: Add \( 8z \) to both sides of the inequality to eliminate the \( z \)-term on the right side.

\[
21 - 6z + 8z \geq 27 - 8z + 8z
\]

Simplifying this:

\[
21 + 2z \geq 27
\]

### Step 2: Subtract 21 from both sides to isolate the term involving \( z \):

\[
2z \geq 27 - 21
\]

Simplifying:

\[
2z \geq 6
\]

### Step 3: Divide both sides by 2 to solve for \( z \):

\[
z \geq \frac{6}{2}
\]

Simplifying:

\[
z \geq 3
\]

### Final Answer:
The solution to the inequality is:

\[
z \geq 3
\]

Would you like more details on how inequalities work, or do you have any other questions?

### Follow-up Questions
1. What happens if you switch the inequality direction?
2. How do you solve a similar inequality with fractions?
3. Can you graph the solution on a number line?
4. How does multiplying or dividing by a negative number affect the inequality?
5. What is the difference between strict inequalities and non-strict inequalities?
6. Can you set up and solve a word problem involving inequalities?
7. How do inequalities change when using absolute values?
8. What are compound inequalities and how are they solved?

### Tip:
Always remember, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign!

Learn how to solve the inequality 21 - 6z ≥ 27 - 8z with a detailed step-by-step solution. This problem covers key algebraic concepts, including inequalities and their solution methods. Suitable for students in Grades 7-9.

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