https://file.aiquickdraw.com/mathbot/file/1728403111642z2spksal.jpg

Let's solve the inequality step-by-step:

Given inequality:

\[
-8(z - 6) \geq 3(12 - 3z)
\]

### Step 1: Distribute
Distribute the coefficients on both sides:

\[
-8z + 48 \geq 36 - 9z
\]

### Step 2: Combine like terms
Add \( 9z \) to both sides to get all the \( z \)-terms on the left:

\[
-8z + 9z + 48 \geq 36
\]

This simplifies to:

\[
z + 48 \geq 36
\]

### Step 3: Solve for \( z \)
Subtract 48 from both sides:

\[
z \geq 36 - 48
\]

This results in:

\[
z \geq -12
\]

### Step 4: Express in interval notation
The solution set in interval notation is:

\[
[-12, \infty)
\]

So, the solution set is \( [-12, \infty) \).

If you have any questions or need further details, feel free to ask!

### Related Questions:
1. How would you solve a similar inequality with a different coefficient?
2. What is the effect of multiplying or dividing an inequality by a negative number?
3. How would the solution change if the inequality were \( -8(z - 6) > 3(12 - 3z) \)?
4. Can you express the solution graphically on a number line?
5. How does interval notation change when using strict inequalities (e.g., \( < \) or \( > \))?

**Tip:** When solving inequalities, always remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Solve the inequality symbolically. Express the solution set in interval notation: -8(z - 6) ≥ 3(12 - 3z).

Learn how to solve the inequality -8(z - 6) ≥ 3(12 - 3z) step-by-step and express the solution set in interval notation. This problem uses the distributive property and algebraic manipulation to solve. Ideal for students in Grades 8-10.

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