https://file.aiquickdraw.com/mathbot/file/1727312939920s4m38oel.jpg

Let's solve the inequality \(|3k - 2| > 7\) to determine which graph represents the solution.

### Step 1: Break down the absolute value inequality
We have \(|3k - 2| > 7\), which means two things:

1. \(3k - 2 > 7\)
2. \(3k - 2 < -7\)

### Step 2: Solve both inequalities

#### First Inequality: \(3k - 2 > 7\)
Add 2 to both sides:
\[
3k > 9
\]
Divide by 3:
\[
k > 3
\]

#### Second Inequality: \(3k - 2 < -7\)
Add 2 to both sides:
\[
3k < -5
\]
Divide by 3:
\[
k < -\frac{5}{3}
\]

### Step 3: Combine the results
The solution is \(k > 3\) or \(k < -\frac{5}{3}\), which means that \(k\) is either greater than 3 or less than \(-\frac{5}{3}\). This represents two separate intervals:

\[
k \in (-\infty, -\frac{5}{3}) \cup (3, \infty)
\]

### Step 4: Analyze the graphs
The correct graph should show two arrows going outward, one starting at \(k = -\frac{5}{3}\) (approximately \(-1.67\)) and going left, and the other starting at \(k = 3\) and going right. Both points should be open circles because the inequality is strict (\(>\)).

Looking at the provided graphs:
- The **first graph** shows arrows going outward, and the points are around \(-\frac{5}{3}\) and 3, with open circles. This graph matches the solution.
- The **second graph** shows the interval between two points, which doesn't fit our solution.
- The **third graph** shows the interval between two points, also not fitting.

### Conclusion:
The **first graph** represents the solution to \(|3k - 2| > 7\).

Would you like more details on this solution, or do you have any further questions?

Here are some related questions for further exploration:
1. How do we solve absolute value inequalities when the inequality symbol is \(<\)?
2. What happens if the inequality was \(|3k - 2| \geq 7\) instead of \(>\)?
3. Can you graph inequalities with compound conditions (e.g., \(3k - 2 \leq 7\))?
4. What are the real-world applications of absolute value inequalities?
5. How does the solution change if the inequality involves \(\leq\) or \(\geq\)?

**Tip**: Always check if the inequality is strict (\(>\) or \(<\)) or inclusive (\(\geq\) or \(\leq\)) to decide whether the graph uses open or closed circles at the boundary points.

Which graph represents the solution to |3k - 2| > 7?

Explore how to solve the absolute value inequality |3k - 2| > 7 with a detailed step-by-step solution. Understand how to break down absolute value inequalities into two separate conditions and graph the solution. Suitable for students in Grades 8-10.

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