https://file.aiquickdraw.com/mathbot/file/1729526512017kkqzqxt6.jpg

Let's break down the system of inequalities and test which of the points lie in the defined area.

The inequalities are:
\[
\begin{cases}
2x - y > 5 \\
2y + x > 8 \\
x < 8
\end{cases}
\]

### Check each point:

**A. (5, -4)**:
1. \( 2(5) - (-4) = 10 + 4 = 14 \), and \( 14 > 5 \) (True)
2. \( 2(-4) + 5 = -8 + 5 = -3 \), and \( -3 > 8 \) (False)
   - Since this fails, point A is not a solution.

**B. (-5, -4)**:
1. \( 2(-5) - (-4) = -10 + 4 = -6 \), and \( -6 > 5 \) (False)
   - This point doesn't satisfy the first inequality, so it is not a solution.

**C. (5, 4)**:
1. \( 2(5) - 4 = 10 - 4 = 6 \), and \( 6 > 5 \) (True)
2. \( 2(4) + 5 = 8 + 5 = 13 \), and \( 13 > 8 \) (True)
3. \( x = 5 \), and \( 5 < 8 \) (True)
   - This point satisfies all three inequalities, so C is a solution.

**D. (-5, 4)**:
1. \( 2(-5) - 4 = -10 - 4 = -14 \), and \( -14 > 5 \) (False)
   - This point fails the first inequality, so it is not a solution.

### Conclusion:
The correct point that lies in the area defined by the system of inequalities is **C. (5, 4)**.

Would you like more details or have any questions about the solution?

Here are five related questions:
1. How do you graph inequalities to find the solution region?
2. Can inequalities have multiple solutions or areas?
3. How do systems of inequalities differ from systems of equations?
4. How do you determine the boundary line when graphing inequalities?
5. What is the significance of the direction of inequalities (greater than vs. less than)?

**Tip**: When solving inequalities, remember to check each inequality individually for each point, as failing one inequality is enough to exclude the point from the solution set.

Which point lies in the area defined by the following system of inequalities?
2x - y > 5
2y + x > 8
x < 8
A. (5, -4)
B. (-5, -4)
C. (5, 4)
D. (-5, 4)

Explore how to solve a system of inequalities step-by-step. Learn which points satisfy the conditions defined by 2x - y > 5, 2y + x > 8, and x < 8. This problem tests algebraic reasoning and is suitable for students in Grades 9-12.

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