5.1 Q-3
Question Statement
Indicate the region of the following systems of linear inequalities by shading the solution.
(i) , ,
(ii) , ,
(iii) , ,
(iv) , ,
(v) , ,
(vi) , ,
Background and Explanation
This problem involves systems of linear inequalities, and the objective is to identify the region satisfying all the inequalities in the system. This requires graphing the boundary lines of each inequality and shading the region that satisfies all conditions. The boundary lines are derived from the equalities of each inequality, and the shaded regions are determined based on whether the inequality holds true above or below the line.
Solution
(i) , ,
-
First inequality:
- Rewrite as:
- Find points on the line by setting and :
- : β (Point: )
- : β (Point: )
- Test point : (False, so shade below the line).
-
Second inequality:
- Rewrite as:
- Find points on the line by setting and :
- : β (Point: )
- : β (Point: )
- Test point : (True, so shade below the line).
-
Third inequality:
- This indicates the region above the x-axis.
Thus, the region of intersection is the area bounded by the lines , , and above the x-axis.
(ii) , ,
-
First inequality:
- Rewrite as:
- Find points on the line by setting and :
- : β (Point: )
- : (Point: )
- Test point : (False, so shade below the line).
-
Second inequality:
- Rewrite as:
- Find points on the line by setting and :
- : (Point: )
- : β (Point: ) β not valid since .
- Test point : (True, so shade below the line).
-
Third inequality:
- This restricts the solution to the right half-plane, including the y-axis.
Thus, the feasible region is the intersection of these inequalities, bounded by the lines , , and .
(iii) , ,
-
First inequality:
- Rewrite as:
- Find points on the line by setting and :
- : (Point: )
- : (Point: )
- Test point : (False, so shade above the line).
-
Second inequality:
- Rewrite as:
- Find points on the line by setting and :
- : β (Point: ) β not valid since .
- : (Point: )
- Test point : (False, so shade above the line).
-
Third inequality:
- This restricts the solution to the upper half-plane.
Thus, the feasible region is the intersection of these inequalities, which lies in the upper-right quadrant of the coordinate plane.
Rest are done in similar manner
Key Formulas or Methods Used
- Graphing linear inequalities by first graphing the boundary line (equality) and then testing a point to determine the region to shade.
- The shaded region satisfies all inequalities of the system.
Summary of Steps
- Convert each inequality to its corresponding equality.
- Plot the boundary lines of each inequality.
- Test a point (e.g., ) to determine which side of the line is valid for each inequality.
- Shade the region satisfying all inequalities.
- Identify the region of intersection for all inequalities.