Which point would not be a solution to the system of linear inequalities shown below?
y, is less than, minus, x, minus, 1, y, is less than, 5, x, minus, 7
y<−x−1y<5x−7
Answer
Multiple Choice Answers
left parenthesis, 1, comma, minus, 3, right parenthesis(1,−3)
left parenthesis, 5, comma, minus, 8, right parenthesis(5,−8)
left parenthesis, minus, 8, comma, minus, 2, right parenthesis(−8,−2)
left parenthesis, 4, comma, minus, 7, right parenthesis(4,−7)

To determine which point is **not** a solution to the system of linear inequalities:

1. \( y < -x - 1 \)
2. \( y < 5x - 7 \)

We need to check each given point by substituting its \(x\) and \(y\) values into both inequalities. If a point does not satisfy either of the inequalities, it is not a solution to the system.

### Checking each point:

1. **Point** \((1, -3)\)

   - Substitute \(x = 1\) and \(y = -3\):
     \[
     y < -x - 1 \implies -3 < -1 - 1 \implies -3 < -2 \quad \text{(True)}
     \]
     \[
     y < 5x - 7 \implies -3 < 5(1) - 7 \implies -3 < -2 \quad \text{(True)}
     \]
   - Both inequalities are satisfied, so \((1, -3)\) **is** a solution.

2. **Point** \((5, -8)\)

   - Substitute \(x = 5\) and \(y = -8\):
     \[
     y < -x - 1 \implies -8 < -5 - 1 \implies -8 < -6 \quad \text{(True)}
     \]
     \[
     y < 5x - 7 \implies -8 < 5(5) - 7 \implies -8 < 25 - 7 \implies -8 < 18 \quad \text{(True)}
     \]
   - Both inequalities are satisfied, so \((5, -8)\) **is** a solution.

3. **Point** \((-8, -2)\)

   - Substitute \(x = -8\) and \(y = -2\):
     \[
     y < -x - 1 \implies -2 < -(-8) - 1 \implies -2 < 8 - 1 \implies -2 < 7 \quad \text{(True)}
     \]
     \[
     y < 5x - 7 \implies -2 < 5(-8) - 7 \implies -2 < -40 - 7 \implies -2 < -47 \quad \text{(False)}
     \]
   - The second inequality is **not** satisfied, so \((-8, -2)\) **is not** a solution.

4. **Point** \((4, -7)\)

   - Substitute \(x = 4\) and \(y = -7\):
     \[
     y < -x - 1 \implies -7 < -4 - 1 \implies -7 < -5 \quad \text{(True)}
     \]
     \[
     y < 5x - 7 \implies -7 < 5(4) - 7 \implies -7 < 20 - 7 \implies -7 < 13 \quad \text{(True)}
     \]
   - Both inequalities are satisfied, so \((4, -7)\) **is** a solution.

### Conclusion

The point **\((-8, -2)\)** does not satisfy the system of linear inequalities because it does not satisfy the second inequality. 

**Correct answer:** \((-8, -2)\)

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. What are the general steps to solve a system of linear inequalities graphically?
2. How do you determine the solution region for a system of linear inequalities?
3. What is the significance of the boundary lines in linear inequalities?
4. How can you find the intersection points of two linear inequalities?
5. What are some real-world applications of solving systems of inequalities?

**Tip:** When solving systems of inequalities, always check each point against all inequalities to ensure it meets the conditions.

Learn how to identify non-solution points for a system of linear inequalities using specific examples. This problem covers key concepts in linear inequalities and graphing, including checking points against inequality conditions. Suitable for students in Grades 9-12.

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