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We are tasked with solving the compound inequality:

\[
-4 < \frac{3x + 4}{2} \leq 3
\]

### Step-by-Step Solution:

#### 1. Break down the compound inequality:
The inequality can be written as two separate inequalities:
\[
-4 < \frac{3x + 4}{2} \quad \text{and} \quad \frac{3x + 4}{2} \leq 3
\]

We will solve each inequality separately and then combine the results.

---

#### 2. Solve the first inequality:
\[
-4 < \frac{3x + 4}{2}
\]
Multiply both sides of the inequality by 2 to eliminate the denominator:
\[
-8 < 3x + 4
\]
Subtract 4 from both sides:
\[
-8 - 4 < 3x
\]
\[
-12 < 3x
\]
Now, divide by 3:
\[
-4 < x
\]
This is the solution for the first part: \( x > -4 \).

---

#### 3. Solve the second inequality:
\[
\frac{3x + 4}{2} \leq 3
\]
Again, multiply both sides by 2:
\[
3x + 4 \leq 6
\]
Subtract 4 from both sides:
\[
3x \leq 2
\]
Now, divide by 3:
\[
x \leq \frac{2}{3}
\]

---

#### 4. Combine the results:
From both parts, we have:
\[
-4 < x \leq \frac{2}{3}
\]

### Final Solution:

- **Interval Notation**: The solution in interval notation is:
\[
(-4, \frac{2}{3}]
\]

- **Set-Builder Notation**: The solution in set-builder notation is:
\[
\{ x \, | \, -4 < x \leq \frac{2}{3} \}
\]

### Graph the solution:
On a number line, the solution would be a segment from \(-4\) (open circle) to \(\frac{2}{3}\) (closed circle).

---

Would you like further details or have any questions? Here are some related questions to explore further:

1. How do you solve compound inequalities involving absolute values?
2. What is the difference between open and closed intervals in interval notation?
3. How can we represent strict inequalities on a number line graph?
4. Can compound inequalities have no solutions? If yes, when does that happen?
5. How do we handle inequalities when multiplying or dividing by negative numbers?

**Tip:** Always pay attention to whether an inequality is strict (\(<\)) or inclusive (\(\leq\)), as it affects both the interval notation and the graph.

Solve and graph the solution set on a number line: -4 < (3x + 4) / 2 ≤ 3. Express your answer in interval notation and set-builder notation.

Learn how to solve the compound inequality -4 < (3x + 4) / 2 ≤ 3 and represent the solution on a number line. This problem involves key algebraic concepts like inequalities, and provides solutions in both interval and set-builder notation. Suitable for students in Grades 8-10.

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