https://file.aiquickdraw.com/mathbot/file/1730576512622b4efa0em.jpg

The inequality given is:

\[
|3z + 5| > 2
\]

To solve this, we need to split it into two separate cases because of the absolute value.

### Case 1:
\[
3z + 5 > 2
\]
Subtracting 5 from both sides:
\[
3z > -3
\]
Dividing by 3:
\[
z > -1
\]

### Case 2:
\[
3z + 5 < -2
\]
Subtracting 5 from both sides:
\[
3z < -7
\]
Dividing by 3:
\[
z < -\frac{7}{3}
\]

### Solution:
The solution to the inequality is:
\[
z < -\frac{7}{3} \quad \text{or} \quad z > -1
\]

In interval notation, this is:
\[
(-\infty, -\frac{7}{3}) \cup (-1, \infty)
\]

Would you like me to provide a graphical sketch of this solution?

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**Further Questions:**

1. What are absolute value inequalities, and how are they solved?
2. How would the solution change if the inequality was \(\geq\) instead of \(>\)?
3. How do we represent solutions on a number line?
4. Can inequalities with absolute values have more than two cases?
5. How do absolute values affect the range of solutions?

**Tip:** When solving absolute value inequalities, always split them into positive and negative cases for accurate solutions.

Learn how to solve the absolute value inequality |3z + 5| > 2 and understand its solution. This guide provides step-by-step instructions, breaking down the inequality into cases, and explains how to represent the solution on a number line. Ideal for students in Grades 9-10.

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