Math 141: Business Mathematics I
Fall 2015
§3.1 Graphing Systems of Linear Inequalities in Two Variables
Instructor: Yeong-Chyuan Chung
Outline
• Graphing linear inequalities
• Graphing systems of linear inequalities
In many optimization problems, the constraints are given as inequalities. We will learn how
to solve some of these problems graphically, but before that, we need to know how to graph
systems of linear inequalities.
Graphing Linear Inequalities
Recall that a linear equation in two variables x and y can be written in the form ax+by +c =
0. Graphically, the equation represents a straight line. We may think of the line as consisting
of all the points (x, y) satisfying the equation. In other words, the line is the solution set of
the equation.
If we change the “=” sign to one of “<”, “>”, “≤”, or “≥”, then we get a linear inequality.
We need to be able to graphically determine the solution sets for such inequalities.
Steps for graphing linear inequalities:
1. Sketch the graph of the equation obtained by replacing the inequality sign with an
equal sign. Use a dotted or dashed line if the problem involves a strict inequality (<
or >). Use a solid line if the problem involves ≤ or ≥.
2. The line divides the plane into two parts, called half-planes. Pick a test point (x0 , y0 )
lying in one of the half-planes, and substitute the values into the given inequality (i.e.
plug in x = x0 and y = y0 ). The point (0, 0) is a convenient choice if it does not lie on
the line.
3. If the inequality is satisfied, then the solution set is the half-plane containing the test
point. If the inequality is not satisfied, then the solution set is the half-plane not
containing the test point.
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§3.1 Graphing Systems of Linear Inequalities in Two Variables
Example. Find the graphical solution of the inequality x > 1.
Example. Find the graphical solution of the inequality 5x + 2y ≤ −2.
Example. Determine the solution set for the inequality y − 2x ≥ 0.
§3.1 Graphing Systems of Linear Inequalities in Two Variables
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Graphing Systems of Linear Inequalities
Just as we had systems of linear equations, we also have systems of linear inequalities. The
solution set of a system of linear inequalities in two variables x and y is the set of all points
(x, y) satisfying every inequality in the system. The graphical solution of such a system
may be obtained by graphing the solution set for each inequality, and then determining the
common region.
The solution set of a system of linear inequalities is bounded if it can be enclosed by a
circle. Otherwise, it is unbounded.
Example (Exercise 24 in the text). Determine graphically the solution set for the following
system of linear inequalities and indicate whether the solution set is bounded or unbounded.
3x − 2y > −13
−x + 2y > 5
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§3.1 Graphing Systems of Linear Inequalities in Two Variables
Example (Exercise 30 in the text). Determine graphically the solution set for the following
system of linear inequalities and indicate whether the solution set is bounded or unbounded.
4x − 3y ≤ 12
5x + 2y ≤ 10
x≥0
y≥0
Example (Exercise 34 in the text). Determine graphically the solution set for the following
system of linear inequalities and indicate whether the solution set is bounded or unbounded.
3x + 4y ≥ 12
2x − y > −2
0≤y<3
x≥0