Solve the inequality.
StartAbsoluteValue 3 x plus one half EndAbsoluteValue minus 4 less than 6
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Part 1
S

To solve the inequality \( |3x + \frac{1}{2}| - 4 < 6 \), follow these steps:

### Step 1: Isolate the absolute value expression
First, add 4 to both sides of the inequality to isolate the absolute value expression:
\[
|3x + \frac{1}{2}| - 4 + 4 < 6 + 4
\]
\[
|3x + \frac{1}{2}| < 10
\]

### Step 2: Remove the absolute value by considering two cases
The absolute value inequality \( |A| < B \) implies two inequalities:
\[
-10 < 3x + \frac{1}{2} < 10
\]

### Step 3: Solve the inequalities separately
**For the left side of the inequality:**
\[
-10 < 3x + \frac{1}{2}
\]
Subtract \(\frac{1}{2}\) from both sides:
\[
-10 - \frac{1}{2} < 3x
\]
\[
-\frac{20}{2} - \frac{1}{2} < 3x
\]
\[
-\frac{21}{2} < 3x
\]
Divide by 3:
\[
-\frac{21}{6} < x
\]
Simplify:
\[
-\frac{7}{2} < x
\]
or
\[
x > -\frac{7}{2}
\]

**For the right side of the inequality:**
\[
3x + \frac{1}{2} < 10
\]
Subtract \(\frac{1}{2}\) from both sides:
\[
3x < 10 - \frac{1}{2}
\]
\[
3x < \frac{20}{2} - \frac{1}{2}
\]
\[
3x < \frac{19}{2}
\]
Divide by 3:
\[
x < \frac{19}{6}
\]

### Step 4: Combine the two results
The solution to the inequality is:
\[
-\frac{7}{2} < x < \frac{19}{6}
\]
In interval notation, this is:
\[
\left( -\frac{7}{2}, \frac{19}{6} \right)
\]

Would you like more details or have any questions?

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**Related Questions:**
1. How do you solve absolute value inequalities involving "greater than"?
2. What is the difference between solving \( |A| < B \) and \( |A| > B \)?
3. How can absolute value inequalities be represented on a number line?
4. How do you verify solutions for absolute value inequalities?
5. What are some common mistakes when solving absolute value inequalities?

**Tip:** When solving absolute value inequalities, always consider both the positive and negative scenarios to avoid missing part of the solution.

Learn how to solve the absolute value inequality |3x + 1/2| - 4 < 6 step-by-step with detailed explanations. This problem covers absolute value inequalities and their solutions, suitable for high school students.

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