The solution to the inequality |2x-1| lt 6 is

Options

A. `x lt - 5/2` or `x gt 7/2`
B. `-5/2 lt x lt 7/2`
C. `x lt -7/2` or `x gt 5/2`
D. `-7/2 lt x lt 5/2`

Answer

To solve the inequality |2x - 1| < 6, we can break it down into two separate inequalities based on the property of absolute values. Here's the step-by-step solution:

### Step 1: Set up the inequalities
The inequality |2x - 1| < 6 can be rewritten as two separate inequalities:
1. 2x - 1 < 6
2. -(2x - 1) < 6

### Step 2: Solve the first inequality
1. Start with the first inequality:
\[ 2x - 1 < 6 \]
2. Add 1 to both sides:
\[ 2x < 6 + 1 \]
\[ 2x < 7 \]
3. Divide both sides by 2:
\[ x < \frac{7}{2} \]

### Step 3: Solve the second inequality
1. Now, solve the second inequality:
\[ -(2x - 1) < 6 \]
This simplifies to:
\[ -2x + 1 < 6 \]
2. Subtract 1 from both sides:
\[ -2x < 6 - 1 \]
\[ -2x < 5 \]
3. Divide both sides by -2 (remember to reverse the inequality sign):
\[ x > -\frac{5}{2} \]

### Step 4: Combine the results
From the two inequalities, we have:
1. \( x < \frac{7}{2} \)
2. \( x > -\frac{5}{2} \)

Combining these results gives us:
\[ -\frac{5}{2} < x < \frac{7}{2} \]

### Final Answer
The solution to the inequality |2x - 1| < 6 is:
\[ \boxed{\left(-\frac{5}{2}, \frac{7}{2}\right)} \]

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