This question has two parts. First, answer Part A. Then, answer Part B.
Part A
Solve the given inequality.

a.  ||3n−2||−2<1
 

Multiple choice question.
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A)
{n || n<−13 or n>53}
 

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B)
all real numbers

cross out

C)
{n || −13<n<53}
 

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D)
{n || −13≤n≤53}

Let's solve the inequality step by step.

### Given Inequality:
\[ |3n - 2| - 2 < 1 \]

### Step 1: Isolate the absolute value expression
Add 2 to both sides to isolate the absolute value:
\[
|3n - 2| < 3
\]

### Step 2: Break down the absolute value inequality
The inequality \( |3n - 2| < 3 \) implies:
\[
-3 < 3n - 2 < 3
\]

### Step 3: Solve the compound inequality
First, add 2 to all parts of the inequality:
\[
-3 + 2 < 3n < 3 + 2
\]
\[
-1 < 3n < 5
\]

Next, divide the entire inequality by 3 to solve for \( n \):
\[
-\frac{1}{3} < n < \frac{5}{3}
\]

### Step 4: Convert the inequality to set notation
This solution represents the interval:
\[
n \in \left(-\frac{1}{3}, \frac{5}{3}\right)
\]

### Step 5: Match the solution with the given choices
The correct answer is:
- **C) \(\{n \mid -\frac{1}{3} < n < \frac{5}{3}\}\)**

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

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Here are some related questions:

1. How would the inequality change if the absolute value had a different coefficient inside?
2. What happens if we reverse the inequality sign in the original problem?
3. How do you solve absolute value inequalities with a greater than sign?
4. What is the geometric interpretation of solving an absolute value inequality?
5. Can the inequality \( |3n - 2| - 2 \geq 1 \) have the same solution set?

**Tip:** When solving absolute value inequalities, always remember to consider both the positive and negative cases to capture the full solution set.

This question has two parts. First, answer Part A. Then, answer Part B. Part A Solve the given inequality. a. ||3n−2||−2<1. Multiple choice question. A) {n || n<−13 or n>53} B) all real numbers C) {n || −13<n<53} D) {n || −13≤n≤53}

Learn how to solve the absolute value inequality ||3n−2||−2<1 with a step-by-step breakdown. This solution covers essential algebraic concepts like inequalities and absolute value, providing clear explanations suitable for students in Grades 9-11.

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