Linear Inequalities: An introduction

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Linear InequalitiesAn introduction
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A linear inequality is a mathematical statement
of one of the following forms
where a, b, and c are real numbers.
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Graph of a Linear inequality
The graph of a linear inequality of the form
or
is the half-plane that lies above or below the
line
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If the linear inequality is of the form
or
Then the line
is also part of the graph.
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Remark When the line
is included in the graph, we draw it as a solid
line. Otherwise, we draw it as a dotted line.
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Example 1 Graph the inequality
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The graph is the half plane above the line.
Use a test point
(0,6)
(-8,0)
We indicate which half plane with arrows
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Now we turn to systems of inequalities.
A system of inequalities is a collection of 2 or
more linear inequalities.
An example of a system of inequalities is
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Graphing a system of inequalities
To graph a system of linear inequalities

• graph each individual inequality separately.
• take the intersection of all the graphs.

So the graph is the overlapping region of the
graphs of the individual inequalities
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Example 2 Graph the system
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(0,6)
(6,0)
(4,0)
(0,- 3)
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A Word on Terminology
The graph of a system of linear inequalities
contains all (and only) the points whose
coordinates satisfy each of the linear
inequalities.
Hence we refer the graph of the system as the
solution of the system or the feasible region of
the system.
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Bounded and unbounded regions
A bounded feasible region is one that can be
enclosed in some circle with a sufficiently large
radius.
If a region is not bounded, it is unbounded.
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Example 3 Determine whether the feasible region
found in Example 2 is bounded.
No, the region is unbounded
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Corner Points
A point where two line segments form a boundary
of a feasible region is called a corner point.
To find the coordinates of a corner point, we
find the point of intersection of the lines that
meet at the corner point.
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Example 4 Determine the corner point of the
feasible region found in Example 2.
The corner point is the point of intersection of
the lines
and
Hence, the corner point is