Download Chapter 8 Section 4 - Canton Local Schools

Chapter 8
Systems of
Equations
and Inequalities
© 2010 Pearson Education, Inc.
All rights reserved
© 2010 Pearson Education, Inc. All rights reserved
1
SECTION 8.4 Systems of Inequalities
OBJECTIVES
1
2
3
4
Graph a linear inequality in two variables.
Graph systems of linear inequalities in two
variables.
Graph a nonlinear inequality in two variables.
Graph systems of nonlinear inequalities in two
variables.
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2
Definitions
The statements x + y > 4, 2x + 3y < 7, y ≥ x, and
x + y ≤ 9 are examples of linear inequalities in the
variables x and y.
A solution of an inequality in two variables x and
y is an ordered pair (a, b) that results in a true
statement when x is replaced by a, and y is
replaced by b in the inequality.
The set of all solutions of an inequality is called the
solution set of the inequality. The graph of an
inequality in two variables is the graph of the
solution set of the inequality.
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EXAMPLE 1
Graphing a Linear Inequality
Graph the inequality 2x + y > 6.
Solution
First, graph the equation 2x + y = 6. Notice that
if we solve this equation for y, we have
y = 6 − 2x.
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EXAMPLE 1
Graphing a Linear Inequality
Solution continued
Any point P(a, b) whose
y-coordinate, b, is
greater than 6 – 2a must
be above this line.
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EXAMPLE 1
Graphing a Linear Inequality
Solution continued
The graph of
y > −2x + 6, which is
equivalent to 2x + y > 6,
consists of all points in
the plane that lie above
the line 2x + y = 6.
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PROCEDURE FOR GRAPHING A LINEAR
INEQUALITY IN TWO VARIABLES
Step 1 Replace the inequality symbol by an
equals (=) sign.
Step 2 Sketch the graph of the corresponding
equation in Step 1. Use a dashed line
for the boundary if the given inequality
sign is < or >, and a solid line if the
inequality symbol is ≤ or ≥.
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PROCEDURE FOR GRAPHING A LINEAR
INEQUALITY IN TWO VARIABLES
Step 3 The graph in Step 2 divides the plane
into two regions. Select a test point in
the either region, but not on the graph
of the equation in Step 1.
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PROCEDURE FOR GRAPHING A LINEAR
INEQUALITY IN TWO VARIABLES
Step 4
(i) If the coordinates of the test point satisfy the
given inequality, then so do all the points in that
region. Shade that region.
(ii) If the test point’s coordinates do not satisfy
the given inequality, shade the region that does
not contain the test point.
The shaded region (including the boundary if it
is solid) is the graph of the given inequality.
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EXAMPLE 3
Graphing Inequalities
Sketch the graph of each of the following
inequalities.
a. x ≥ 2
b. y < 3
c. x + y < 4
Solution
a. Step 1 Change the ≥ to =: x = 2
Step 2 Graph x = 2
with a solid line.
Step 3 Test (0, 0). 0 ≥ 2 is
a false statement.
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EXAMPLE 3
Graphing Inequalities
Solution continued
Step 3 continued
The region not containing
(0, 0), together with the
vertical line, is the
solution set.
Step 4 Shade the
solution set.
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EXAMPLE 3
Graphing Inequalities
Solution continued
b. Step 1 Change the < to =: y = 3
Step 2 Graph y = 3 with a dashed line.
Step 3 Test (0, 0). 0 < 3 is
a true statement.
The region
containing (0, 0) is
the solution set.
Step 4 Shade the
solution set.
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EXAMPLE 3
Graphing Inequalities
Solution continued
c. Step 1 Change the < to = : x + y = 4
Step 2 Graph x + y = 4 with a dashed line.
Step 3 Test (0, 0).
0 + 0 < 4 is a true
statement. The
region containing
(0, 0) is the
solution set.
Step 4 Shade the
solution set.
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SYSTEMS OF LINEAR INEQUALITIES IN
TWO VARIABLES
An ordered pair (a, b) is a solution of a system
of inequalities involving two variables x and y
if and only if, when x is replaced by a and y is
replaced by b in each inequality of the system,
all resulting statements are true.
The solution set of a system of inequalities is
the intersection of the solution sets of all the
inequalities in the system.
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EXAMPLE 4
Graphing a System of Two Inequalities
Graph the solution set of the system of
2x  3y  6 (1)
inequalities: 
 y  x  0 (2)
Solution Graph each inequality separately.
Step 1 2x + 3y = 6
Step 2 Sketch as a dashed line by joining the
points (0, 2) and (3, 0).
Step 3 Test (0, 0). 2(0) + 3(0) > 6 is a false
statement.
Step 4 Shade the solution set.
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EXAMPLE 4
Graphing a System of Two Inequalities
Solution continued
Now graph the second inequality.
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EXAMPLE 4
Graphing a System of Two Inequalities
Solution continued
Step 1 y – x = 0
Step 2 Sketch as a solid line
by joining the points
(0, 0) and (1, 1).
Step 3 Test (1, 0).
0 – 1 ≤ 6 is a true
statement.
Step 4 Shade the solution set.
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EXAMPLE 4
Graphing a System of Two Inequalities
Solution continued
The graph of the solution set of inequalities
(1) and (2) is the region where the shading
overlaps.
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Sketch the graph and label the vertices of the
solution set of the system of linear inequalities.
Solution
On the same coordinate plane, sketch the
graphs of the three linear equations that
correspond to the three inequalities. All of the
equations are graphed as solid lines.
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Solution continued
Use (0, 0) as the test
point for each
inequality.
(i) The region on or
below 3x + 2y = 11 is
in the solution set of
inequality (1).
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Solution continued
(ii) The region on or
above x – y = 2
is in the solution set of
inequality (2).
(iii) The region
on or below
7x – 2y = –1 belongs to
the solution set of
inequality (3).
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Solution continued
The solution set is the
shaded region, including
the sides of the triangle.
The vertices are
obtained
by solving each
pair of equations
in the system.
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Solution continued
a. To find the vertex (3, 1) solve the system
Solve equation (2) for x and substitute x = 2 + y
in equation (1).
Back-substitute y = 1 in equation (2) to find the
vertex (3, 1).
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EXAMPLE 5
Solving a System of Three Linear
Inequalities
Solution continued
b. Solve the following system of equations
by the substitution method to find the vertex
(–1, –3):
c. Solve the following system of equations by
the elimination method to find the vertex (1, 4):
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Graphing a Nonlinear Inequality in Two
EXAMPLE 6 Variables
OBJECTIVE Graph a
EXAMPLE Graph y > x2 – 2.
nonlinear inequality in two
variables.
Step 1 Replace the inequality 1. y = x2 – 2
symbol with an equal (=) sign.
Step 2 Sketch the graph of the 2.
corresponding equation in
Step 1. Use a dashed curve if
the given inequality sign is <
or > and a solid curve if the
inequality symbol is ≤ or ≥.
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Graphing a Nonlinear Inequality in Two
EXAMPLE 6 Variables
OBJECTIVE Graph a
EXAMPLE Graph y > x2 – 2.
nonlinear inequality in two
variables.
Step 3 The graph in Step 2
divides the plane into two
regions. Select a test point in
either region, but not on the
graph.
3.
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Graphing a Nonlinear Inequality in Two
EXAMPLE 6 Variables
OBJECTIVE Graph a
EXAMPLE Graph y > x2 – 2.
nonlinear inequality in two
variables.
Step 4 (i) If the coordinates of 4. 0 > 02 – 2 = –2, so the test
the test point satisfy the
point (0, 0) satisfies the
inequality, then so do all of the
inequality .
points in that region. Shade
that region.
(ii) If the test point’s
coordinates do not satisfy the
inequality, shade the region
that does not contain the test
point.
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EXAMPLE 7
Solving a Nonlinear System of Inequalities
Graph the solution set of the following system of
y  4  x2
(1)

3

inequalities:  y  x  3 (2)
2

 y  6x  3 (3)
Solution
Graph each inequality separately in the same
coordinate plane. Since (0, 0) is not a solution of
any the corresponding equations, use (0, 0) as a
test point for each inequality.
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EXAMPLE 7
Solving a Nonlinear System of Inequalities
Solution continued
Step 1 y = 4 – x2
Step 2 Sketch as a
solid curve
with vertex
(0, 4).
Step 3 Test (0, 0).
0 ≤ 4 – 0 is a
true statement.
Step 4 Shade the region.
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EXAMPLE 7
Solving a Nonlinear System of Inequalities
Solution continued
3
Step 1 y  x  3
2
Step 2 Sketch as a
solid line
through (0, –3)
& (2, 0).
Step 3 Test (0, 0).
0 ≥ 0 – 3 is a
true statement.
Step 4 Shade the region.
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EXAMPLE 7
Solving a Nonlinear System of Inequalities
Solution continued
Step 1 y = –6x – 3
Step 2 Sketch as a
solid line
through (0, –3)
 1 
&   , 0 .
 2 
Step 3 Test (0, 0).
0 ≥ 0 – 3 is a
true statement.
Step 4 Shade the region.
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EXAMPLE 7
Solving a Nonlinear System of Inequalities
Solution continued
The region
common to all
three graphs is
the graph of the
solution set of
the given
system of
inequalities.
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