To solve the inequality:

\[
-1 \geq z + \frac{1}{3}
\]

### Step 1: Subtract \(\frac{1}{3}\) from both sides

\[
-1 - \frac{1}{3} \geq z
\]

### Step 2: Simplify the left-hand side

To subtract fractions, convert \(-1\) to \(-\frac{3}{3}\):

\[
-\frac{3}{3} - \frac{1}{3} = -\frac{4}{3}
\]

So now, the inequality becomes:

\[
-\frac{4}{3} \geq z
\]

### Final Answer:

\[
z \leq -\frac{4}{3}
\]

This is the solution for the inequality.

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Would you like a more detailed explanation of any part of the process, or have any other questions? 

Here are 5 related questions to explore:

1. What are the rules for solving inequalities involving fractions?
2. How do inequalities change if we multiply or divide by a negative number?
3. How can we graph \(z \leq -\frac{4}{3}\) on a number line?
4. What is the difference between strict inequalities (\(<\) and \(>\)) and non-strict inequalities (\(\leq\) and \(\geq\))?
5. How can inequalities be used to model real-world problems?

**Tip:** When solving inequalities, remember to reverse the inequality sign if you multiply or divide both sides by a negative number.

Master solving the inequality –1 ≥ z + 1/3 with a detailed breakdown of each step. This lesson covers key algebra concepts, including working with inequalities and fractions. Ideal for students in Grades 6-8.

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