2x−y>52y+x>8x<8Which point lies in the area defined by the following system of inequalities?A.(5,−4)B.(−5,−4)C.(5,4)D.(−5,4)

We need to determine which of the given points satisfies all the inequalities in the following system:

1. \( 2x - y > 5 \)  
2. \( 2y + x > 8 \)  
3. \( x < 8 \)  

Let's test each point one-by-one to find the one that lies in the region defined by these inequalities.

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### Testing Points:

#### **Option A: (5, -4)**
1. \( 2(5) - (-4) = 10 + 4 = 14 > 5 \) ✅
2. \( 2(-4) + 5 = -8 + 5 = -3 \), but \(-3 \not> 8\) ❌
3. \( 5 < 8 \) ✅

Since the second inequality fails, **A is not in the solution region**.

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#### **Option B: (-5, -4)**
1. \( 2(-5) - (-4) = -10 + 4 = -6 \), but \(-6 \not> 5\) ❌
2. \( 2(-4) + (-5) = -8 - 5 = -13 \), but \(-13 \not> 8\) ❌
3. \( -5 < 8 \) ✅

Both the first and second inequalities fail, so **B is not in the solution region**.

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#### **Option C: (5, 4)**
1. \( 2(5) - 4 = 10 - 4 = 6 > 5 \) ✅
2. \( 2(4) + 5 = 8 + 5 = 13 > 8 \) ✅
3. \( 5 < 8 \) ✅

All inequalities are satisfied! **C is in the solution region**.

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#### **Option D: (-5, 4)**
1. \( 2(-5) - 4 = -10 - 4 = -14 \), but \(-14 \not> 5\) ❌
2. \( 2(4) + (-5) = 8 - 5 = 3 \), but \(3 \not> 8\) ❌
3. \( -5 < 8 \) ✅

Both the first and second inequalities fail, so **D is not in the solution region**.

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

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Do you need further details or clarification?

Here are five related questions to deepen your understanding:
1. How do we graph a system of linear inequalities?
2. What does it mean when inequalities have a strict inequality sign (like \(>\) or \(<\))?
3. How do we find the intersection region of multiple inequalities graphically?
4. Can systems of inequalities have no solution at all?
5. How would the solution change if the inequality \(x < 8\) became \(x \leq 8\)?

**Tip:** Always substitute given points carefully to avoid arithmetic mistakes when working with inequalities!

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 find the point that satisfies the system of inequalities 2x − y > 5, 2y + x > 8, and x < 8. Learn how to test each point in the system step-by-step and understand key algebraic concepts. Suitable for students in Grades 9-11.

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