https://file.aiquickdraw.com/mathbot/file/1726781811056q2nkzhbk.jpg

Let's solve the inequality from the image step by step.

The inequality given is:

\[
-19z + 12 > -5z - 13z + 16
\]

### Step 1: Simplify both sides of the inequality.
On the right-hand side, combine like terms:

\[
-5z - 13z = -18z
\]

So the inequality becomes:

\[
-19z + 12 > -18z + 16
\]

### Step 2: Move all terms with \( z \) to one side.
Add \( 18z \) to both sides:

\[
-19z + 18z + 12 > 16
\]

Simplifying:

\[
-z + 12 > 16
\]

### Step 3: Move constant terms to the other side.
Subtract 12 from both sides:

\[
-z > 16 - 12
\]

\[
-z > 4
\]

### Step 4: Solve for \( z \).
To isolate \( z \), divide both sides by -1. **Remember** that when you divide or multiply an inequality by a negative number, the inequality sign flips:

\[
z < -4
\]

### Final Answer:
\[
z < -4
\]

Would you like more details on any step or have any questions?

Here are five related questions to explore:

1. How do we handle inequalities when multiplying or dividing by negative numbers?
2. What are the steps for solving more complex inequalities with fractions?
3. How would you graph the solution \( z < -4 \) on a number line?
4. What happens if we have multiple inequality symbols in one expression?
5. Can this process be applied to quadratic inequalities, and how does that differ?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the linear inequality -19z + 12 > -18z + 16 step by step. This problem covers key algebraic concepts and rules for handling inequalities, including flipping the inequality sign when dividing by negative numbers. Suitable for students in Grades 8-10.

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