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The problem asks to determine the value of \( k \) for which the inequality

\[
-1 < 3x - k < 10 + k
\]

has the solution set \( \{ x | 0 < x < 4 \} \).

### Step-by-step solution:

We are given the inequality:
\[
-1 < 3x - k < 10 + k
\]
and we know the solution for \( x \) must be \( 0 < x < 4 \).

1. **Work on the inequality piece by piece**:

   First, split the compound inequality into two parts:
   \[
   -1 < 3x - k
   \]
   and
   \[
   3x - k < 10 + k.
   \]

2. **Solve the left inequality**:
   \[
   -1 < 3x - k.
   \]
   Add \( k \) to both sides:
   \[
   k - 1 < 3x.
   \]
   Divide by 3:
   \[
   \frac{k - 1}{3} < x.
   \]
   This inequality tells us that \( x \) must be greater than \( \frac{k - 1}{3} \).

3. **Solve the right inequality**:
   \[
   3x - k < 10 + k.
   \]
   Add \( k \) to both sides:
   \[
   3x < 10 + 2k.
   \]
   Divide by 3:
   \[
   x < \frac{10 + 2k}{3}.
   \]
   This inequality tells us that \( x \) must be less than \( \frac{10 + 2k}{3} \).

4. **Combine the solution**:
   From the problem statement, we know that the solution for \( x \) must be between 0 and 4, i.e., \( 0 < x < 4 \).

   We now have two inequalities:
   \[
   \frac{k - 1}{3} < x \quad \text{and} \quad x < \frac{10 + 2k}{3}.
   \]
   To match the desired solution \( 0 < x < 4 \), we set:
   \[
   \frac{k - 1}{3} = 0
   \]
   and
   \[
   \frac{10 + 2k}{3} = 4.
   \]

5. **Solve for \( k \)**:
   - From \( \frac{k - 1}{3} = 0 \), we get:
     \[
     k - 1 = 0 \quad \Rightarrow \quad k = 1.
     \]
   - From \( \frac{10 + 2k}{3} = 4 \), we get:
     \[
     10 + 2k = 12 \quad \Rightarrow \quad 2k = 2 \quad \Rightarrow \quad k = 1.
     \]

Therefore, the value of \( k \) is **1**.

Let me know if you need more details, or if you have any questions! Here are some related questions to practice:

1. How do you solve compound inequalities step by step?
2. What are the key steps to isolating \( k \) in an inequality?
3. How do you interpret solution sets in terms of intervals?
4. What happens if you change the boundaries of the solution set (e.g., \( -2 < x < 5 \))?
5. How do you graph inequalities involving variables on both sides?

**Tip:** When solving inequalities, remember to reverse the inequality sign if you multiply or divide by a negative number.

Determine the value of k for which the inequality -1 < 3x - k < 10 + k has the solution set {x | 0 < x < 4}.

Learn how to solve the compound inequality -1 < 3x - k < 10 + k and find the value of k such that the solution set is {x | 0 < x < 4}. This detailed solution involves working through algebraic inequalities and isolating the variable. Suitable for students in Grades 8-10.

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