MATH 141 501
Section 3.1 Lecture Notes
Graphing Systems of Linear Inequalities in Two Variables
A linear inequality is an inequality which can be written in one of the following forms:
Ax + By + C ≥ 0
Ax + By + C ≤ 0
Ax + By + C > 0
Ax + By + C < 0
Example 1: Graph 4x + 2y > 8
Step 1: Graph the corresponding linear equality 4x+2y = 8. (This is called
the boundary line.) Indicate whether it is part of the solution set (solid line =
“included”, dashed line = “not included”)
Step 2: Use a test point to determine which half-plane contains solutions
of the inequality.
Step 3: Shade the solution set.
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Systems of Linear Inequalities
If we have a system of linear inequalities, the solution set consists of all
points which make all of the inequalities true.
Example 2: a.) Sketch the solution set for the system
x+y ≤5
2x + y ≤ 10
b.) Is the solution set bounded or unbounded? (A set in the plane is
bounded if you can draw a circle around it, and unbounded if you can not.)
c.) A corner point is where two linear equalities in the system intersect.
Label all the corner points for this system. (A corner point is a point in the
solution set which comes from the intersection of the boundary lines from two
of the inequalities.)
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Example 3
Determine the graphical solution set for the following system of linear inequalities.
2x + y ≥ 40
x + 2y ≥ 40
x≥0
y≥0
Is the solution bounded or unbounded?
Label any corner points.
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Example 4
Determine the graphical solution set for the following system of linear inequalities.
3x + 2y ≤ 90
x + 4y ≤ 120
x≥0
y≥0
Is the solution set bounded or unbounded?
Label any corner points.
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