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: $\left( -\frac{7}{2}, \frac{19}{6} \right)$

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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.