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The problem shown is:

\[
\frac{1}{2} |x - 3| + 4 < 6
\]

### Step-by-step solution:

1. **Isolate the absolute value term**:
   \[
   \frac{1}{2} |x - 3| + 4 < 6
   \]
   Subtract 4 from both sides:
   \[
   \frac{1}{2} |x - 3| < 2
   \]

2. **Eliminate the fraction**:
   Multiply both sides of the inequality by 2 to get rid of the fraction:
   \[
   |x - 3| < 4
   \]

3. **Solve the absolute value inequality**:
   The inequality \( |x - 3| < 4 \) can be split into two cases:
   \[
   -4 < x - 3 < 4
   \]

4. **Solve the two inequalities**:
   Add 3 to all parts of the inequality:
   \[
   -4 + 3 < x < 4 + 3
   \]
   \[
   -1 < x < 7
   \]

### Final solution:
The solution is \( -1 < x < 7 \).

### Graphing on the number line:
On the number line, this inequality corresponds to an open interval between -1 and 7, meaning the values of \(x\) are between -1 and 7 but do not include -1 or 7.

Would you like further details or have any other questions?

Here are five related questions for further exploration:

1. How would the solution change if the inequality was \( \frac{1}{2} |x - 3| + 4 \geq 6 \)?
2. What is the process for solving a similar inequality with a greater absolute value, such as \( |x + 5| \)?
3. How do you graph inequalities that include "or equal to" conditions on a number line?
4. How would the solution be affected if the absolute value term was multiplied by a different fraction?
5. What happens if you have multiple absolute value terms in an inequality?

**Tip:** Always isolate the absolute value expression first when solving absolute value inequalities to make the process more straightforward.

Solve the following absolute value inequality and graph your solution on the number line below: \(\frac{1}{2} |x - 3| + 4 < 6\).

Learn how to solve the absolute value inequality \(\frac{1}{2} |x - 3| + 4 < 6\) and graph the solution on a number line. This step-by-step explanation is suitable for students in Grades 9-11, covering key algebra concepts such as absolute value and inequalities.

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