The solution set of `|3-4x| gt 2` is:

Options

A. `(7/4, infty)`
B. `(-1/4, 7/4)`
C. `(-infty, -1/4) cup (7/4, infty)`
D. `(-infty, 1/4) cup (5/4, infty)`

Answer

To solve the inequality \( |3 - 4x| > 2 \), we need to consider the definition of absolute value, which leads us to two separate cases.

### Step 1: Set up the two cases from the absolute value inequality

The inequality \( |3 - 4x| > 2 \) can be split into two cases:
1. \( 3 - 4x > 2 \)
2. \( 3 - 4x < -2 \)

### Step 2: Solve the first case \( 3 - 4x > 2 \)

To solve \( 3 - 4x > 2 \):
- Subtract 3 from both sides:
\[ -4x > 2 - 3 \]
\[ -4x > -1 \]
- Divide by -4 (remember to flip the inequality sign):
\[ x < \frac{1}{4} \]

### Step 3: Solve the second case \( 3 - 4x < -2 \)

Now, solve \( 3 - 4x < -2 \):
- Subtract 3 from both sides:
\[ -4x < -2 - 3 \]
\[ -4x < -5 \]
- Divide by -4 (again, flip the inequality sign):
\[ x > \frac{5}{4} \]

### Step 4: Combine the solutions

From the two cases, we have:
1. \( x < \frac{1}{4} \)
2. \( x > \frac{5}{4} \)

Thus, the solution set is:
\[ x \in (-\infty, \frac{1}{4}) \cup (\frac{5}{4}, \infty) \]

### Final Answer

The solution set of \( |3 - 4x| > 2 \) is:
\[ (-\infty, \frac{1}{4}) \cup (\frac{5}{4}, \infty) \]

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