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Let's break down and solve each part of the problem step-by-step.

### 1. Solving the Equation \(|2x - 1| = 2\)

For an absolute value equation \(|A| = B\), we split it into two cases:

1. \( 2x - 1 = 2 \)
2. \( 2x - 1 = -2 \)

#### Case 1: \(2x - 1 = 2\)
1. Add 1 to both sides: \(2x = 3\)
2. Divide by 2: \(x = \frac{3}{2}\)

#### Case 2: \(2x - 1 = -2\)
1. Add 1 to both sides: \(2x = -1\)
2. Divide by 2: \(x = -\frac{1}{2}\)

So, the solutions to \(|2x - 1| = 2\) are:
\[
x = \frac{3}{2} \text{ and } x = -\frac{1}{2}
\]

### 2. Solving the Inequality \(|2x - 1| < 2\)

For an absolute value inequality \(|A| < B\), we can write it as a compound inequality:
\[
-2 < 2x - 1 < 2
\]

Now, solve this compound inequality step-by-step.

1. **Left Inequality**: \(-2 < 2x - 1\)
   - Add 1 to both sides: \(-1 < 2x\)
   - Divide by 2: \(-\frac{1}{2} < x\)

2. **Right Inequality**: \(2x - 1 < 2\)
   - Add 1 to both sides: \(2x < 3\)
   - Divide by 2: \(x < \frac{3}{2}\)

So, the solution to \(|2x - 1| < 2\) is:
\[
-\frac{1}{2} < x < \frac{3}{2}
\]

### Answers

- **For the equation \(|2x - 1| = 2\)**: The solutions are \(x = \frac{3}{2}\) and \(x = -\frac{1}{2}\).
- **For the inequality \(|2x - 1| < 2\)**: The solution set is \( -\frac{1}{2} < x < \frac{3}{2} \), which matches option B.

Would you like further details on these steps or have additional questions?

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#### Related Questions
1. How would the solution change if the inequality was \(|2x - 1| > 2\)?
2. How do we solve absolute value equations and inequalities with different constants, like \(|3x - 2| = 5\)?
3. Can we represent \(|2x - 1| = 2\) graphically and interpret the solution visually?
4. What if \(|2x - 1|\) had been less than or equal to 2 instead of strictly less?
5. How do we verify solutions for absolute value equations?

#### Tip
When solving absolute value inequalities, remember that \(|A| < B\) represents values of \(A\) between \(-B\) and \(B\), while \(|A| > B\) represents values of \(A\) outside this range.

Solve the equation |2x - 1| = 2 and the inequality |2x - 1| < 2.

Learn how to solve the absolute value equation |2x - 1| = 2 and inequality |2x - 1| < 2. This guide explains the solution process using properties of absolute values, suitable for students in Grades 8-10.

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