Chapter 10
Graphing
Equations and
Inequalities
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10.7
Graphing Linear
Inequalities in Two
Variables
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Linear Inequalities in Two Variables
Linear inequality in two variables
• Written in the form Ax + By < C
• A, B, C are real numbers, A and B are not both 0
• Could use (>, ≥, ≤) in place of <
An ordered pair is a solution of the linear inequality
in x and y if replacing the variables with the
coordinates of the ordered pair results in a true
statement.
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Martin-Gay, Developmental Mathematics, 2e
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Linear Inequalities in Two Variables
To Graph a Linear Inequality
Step 1: Graph the boundary line found by replacing
the inequality sign with an equal sign. If the
inequality sign is > or <, graph a dashed
boundary line (indicating that the points on
the line are not solutions of the inequality).
If the inequality sign is ≥ or ≤, graph a solid
boundary line (indicating that the points on
the line are solution of the inequality).
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
continued
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Linear Inequalities in Two Variables
Step 2: Choose a point not on the boundary line as
a test point. Substitute the coordinates of
this test point into the original inequality.
Step 3: If a true statement is obtained in Step 2,
shade the half-plane that contains the test
point. If a false statement is obtained,
shade the half-plane that does not contain
the test point.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
continued
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Example
y
Graph: 7x + y > –14.
Step 1: First graph the boundary
line 7x + y = –14 as a dashed line
because the inequality symbol is >.
Step 2: Choose a test point not
on the boundary line. We
choose (0,0), and substitute the
coordinates into 7x + y > –14.
7(0) + 0 > –14 True
Step 3: Since the result is a true
statement, (0, 0) is a solution of
7x + y > –14, so shade the side containing (0,0).
(0, 0)
x
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Martin-Gay, Developmental Mathematics, 2e
6
Example
Graph: 3x + 5y ≤ –2.
Step 1: First graph the boundary
line 3x + 5y ≤ –2 as a solid line
because the inequality is ≤.
Step 2: Choose a test point not
on the boundary line. We
choose (0,0), and substitute the
coordinates into 3x + 5y ≤ –2.
3(0) + 5(0) ≤ –2 False
Step 3: Since the result is a
false statement, shade the side
that doesn’t contain (0,0).
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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Example
y
Graph 3x < 15.
Step 1: First graph the
boundary line 3x < 15 as a
dashed line.
Step 2: Choose a test point not
on the boundary line. We
choose (0,0), and substitute the
coordinates into 3x < 15.
3(0) < 15 True
Step 3: Since the result is a
true statement, shade the side
that contains (0,0).
(0, 0)
x
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Martin-Gay, Developmental Mathematics, 2e
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Helpful Hint
When graphing an inequality, make sure the
test point is substituted into the original
inequality.
Also, note that although all of our examples
allowed us to select (0, 0) as our test point,
that will not always be true. If the boundary
line contains (0,0), you must select another
point that is not contained on the line as your
test point.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Martin-Gay, Developmental Mathematics, 2e
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