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Let's solve the system of linear inequalities step by step:

The given system is:

\[
\begin{aligned}
1. \quad & 3x + 4y \geq 12 \\
2. \quad & 2y > x + 8 \\
3. \quad & y < 5
\end{aligned}
\]

### Step 1: Analyze the inequalities
1. **Inequality 1:** \( 3x + 4y \geq 12 \)
   - This represents a region above or on the line \( 3x + 4y = 12 \).

2. **Inequality 2:** \( 2y > x + 8 \)
   - This represents a region strictly above the line \( 2y = x + 8 \) or equivalently \( y > \frac{x}{2} + 4 \).

3. **Inequality 3:** \( y < 5 \)
   - This represents a region strictly below the line \( y = 5 \).

### Step 2: Test integer coordinates
We are looking for integer values of \(x\) and \(y\) that satisfy all three inequalities. Let's test some integer points.

#### Testing \(x = 0\) and \(y = 4\):
1. For the first inequality: \( 3(0) + 4(4) = 16 \geq 12 \) (True)
2. For the second inequality: \( 2(4) = 8 > 0 + 8 \) (False)

#### Testing \(x = 0\) and \(y = 2\):
1. For the first inequality: \( 3(0) + 4(2) = 8 \geq 12 \) (False)

#### Testing \(x = 4\) and \(y = 3\):
1. For the first inequality: \( 3(4) + 4(3) = 12 + 12 = 24 \geq 12 \) (True)
2. For the second inequality: \( 2(3) = 6 > 4 + 8 = 12 \) (False)

#### Testing \(x = 4\) and \(y = 2\):
1. For the first inequality: \( 3(4) + 4(2) = 12 + 8 = 20 \geq 12 \) (True)
2. For the second inequality: \( 2(2) = 4 > 4 + 8 = 12 \) (True)
3. For the third inequality: \( 2 < 5 \) (True)

### Step 3: Conclusion
The integer point \( (4, 2) \) satisfies all three inequalities.

So, the solution is:
\[
(x, y) = (4, 2)
\]

Would you like more details or have any further questions?

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Here are five related questions for further exploration:
1. How do you graph systems of inequalities to visually find solutions?
2. What happens if one of the inequalities in the system has no solution?
3. Can we solve this problem by substitution or elimination?
4. How does the boundary of each inequality affect the solution set?
5. What if the inequalities had strict boundaries (e.g., \(3x + 4y > 12\))? How would the solution change?

**Tip:** When solving systems of inequalities, always check that each solution satisfies **all** conditions, not just one or two.

A system of linear inequalities is shown below:
3x + 4y ≥ 12
2y > x + 8
y < 5
There is only one set of integer coordinates that satisfies this system. Enter its x- and y-coordinates.

Learn how to solve a system of linear inequalities and find the integer coordinates that satisfy all three inequalities: 3x + 4y ≥ 12, 2y > x + 8, and y < 5. This detailed step-by-step solution is designed for students in Grades 9-12.

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