Question #6444294Fill in the Blank
Algebra-1
Question
Consider the system of inequalities: . Determine whether the point (0, 1) lies in the solution region of the system. _________(Yes or No).
Answer & Analysis
Analysis
Question Analysis
This question involves checking if the point lies within the solution region of a system of two linear inequalities.
This question involves checking if the point lies within the solution region of a system of two linear inequalities.
Key Concept Explanation
To analyze the solution region graphically:
Each linear inequality corresponds to a half-plane, bounded by a boundary line.A solid boundary line (for or ) includes the line in the solution, while a dashed line (for > or <) excludes it.
The solution region of the system is the intersection (overlap) of the half-planes defined by each inequality. A point is in the solution region if it lies within this overlap.
To analyze the solution region graphically:
Each linear inequality corresponds to a half-plane, bounded by a boundary line.A solid boundary line (for or ) includes the line in the solution, while a dashed line (for > or <) excludes it.
The solution region of the system is the intersection (overlap) of the half-planes defined by each inequality. A point is in the solution region if it lies within this overlap.
Step-by-step Solution
1. Graph the first inequality
Boundary line: (slope = 3, y-intercept = 2).
The inequality means the line is solid (included in the solution).
Shaded region: Test with the point : simplifies to (true).
Shade the half-plane below the line (including the line).
2. Graph the second inequality
Boundary line: (slope = , y-intercept = ).
The inequality > means the line is dashed (excluded from the solution).
Shaded region: Test with :
1. Graph the first inequality
Boundary line: (slope = 3, y-intercept = 2).
The inequality means the line is solid (included in the solution).
Shaded region: Test with the point : simplifies to (true).
Shade the half-plane below the line (including the line).
2. Graph the second inequality
Boundary line: (slope = , y-intercept = ).
The inequality > means the line is dashed (excluded from the solution).
Shaded region: Test with :
Click "Show Answer" to reveal the answer and analysis
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