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The equation given is \( |4x - 7| = 5 \). To solve for \( x \), we must consider two cases based on the definition of absolute value.

### Step 1: Break the absolute value equation into two cases:
1. \( 4x - 7 = 5 \)
2. \( 4x - 7 = -5 \)

### Case 1: \( 4x - 7 = 5 \)
- Add 7 to both sides:
  \[
  4x = 5 + 7 = 12
  \]
- Divide by 4:
  \[
  x = \frac{12}{4} = 3
  \]

### Case 2: \( 4x - 7 = -5 \)
- Add 7 to both sides:
  \[
  4x = -5 + 7 = 2
  \]
- Divide by 4:
  \[
  x = \frac{2}{4} = \frac{1}{2}
  \]

### Solutions:
The solutions are \( x = 3 \) and \( x = \frac{1}{2} \). Therefore, the solution set is \( \left\{ \frac{1}{2}, 3 \right\} \).

The correct answer is the second option: \( \left\{ \frac{1}{2}, 3 \right\} \).

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Solve \( |3x + 4| = 7 \).
2. Solve \( |2x - 9| = 11 \) for \( x \).
3. What is the solution to \( |5x + 1| = 4 \)?
4. Solve \( |6x - 8| = 10 \).
5. Solve \( |x - 2| = 3 \).

**Tip:** When solving absolute value equations, always break them into two cases—positive and negative.

Step-by-step solution for the absolute value equation |4x - 7| = 5, covering the breakdown into two linear equations, solving each, and finding x values of 3 and 1/2. Suitable for students in Grades 7-9.

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