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794 Chapter 8 Systems of Equations and Inequalities
Section
8.5
Objectives
� Graph a linear inequality in
�
�
�
two variables.
Graph a nonlinear inequality
in two variables.
Use mathematical models
involving linear inequalities.
Graph a system of
inequalities.
Systems of Inequalities
W
e opened the chapter noting that the modern
emphasis on thinness as the ideal body shape
has been suggested as a major cause of eating
disorders. In this section, as well as in the
exercise set, we use systems of linear
inequalities in two variables that will
enable you to establish a healthy weight
range for your height and age.
Linear Inequalities in Two
Variables and Their Solutions
We have seen that equations in the form
Ax + By = C are straight lines when graphed.
If we change the symbol = to 7, 6, Ú, or …,
we obtain a linear inequality in two variables.
Some examples of linear inequalities in
two variables are x + y 7 2, 3x - 5y … 15,
and 2x - y 6 4.
A solution of an inequality in two variables, x and y, is an ordered pair of real
numbers with the following property: When the x-coordinate is substituted for x and
the y-coordinate is substituted for y in the inequality, we obtain a true statement. For
example, (3, 2) is a solution of the inequality x + y 7 1. When 3 is substituted for x
and 2 is substituted for y, we obtain the true statement 3 + 2 7 1, or 5 7 1. Because
there are infinitely many pairs of numbers that have a sum greater than 1, the
inequality x + y 7 1 has infinitely many solutions. Each ordered-pair solution is
said to satisfy the inequality. Thus, (3, 2) satisfies the inequality x + y 7 1.
�
Graph a linear inequality in two
variables.
The Graph of a Linear Inequality
in Two Variables
We know that the graph of an equation in two variables is the set of all points whose
coordinates satisfy the equation. Similarly, the graph of an inequality in two variables
is the set of all points whose coordinates satisfy the inequality.
Let’s use Figure 8.14 to get an idea of what the
y
graph of a linear inequality in two variables looks
5
like. Part of the figure shows the graph of the linear
4
equation x + y = 2. The line divides the points in
Half-plane
3
x+y>2
the rectangular coordinate system into three sets.
2
First, there is the set of points along the line,
1
x
satisfying x + y = 2. Next, there is the set of points
−5 −4 −3 −2 −1−1 1 2 3 4 5
in the green region above the line. Points in the
−2
green region satisfy the linear inequality x + y 7 2.
Half-plane
−3
x
+
y
<
2
Finally, there is the set of points in the purple region
−4
Line x + y = 2
below the line. Points in the purple region satisfy the
−5
linear inequality x + y 6 2.
A half-plane is the set of all the points on one Figure 8.14
side of a line. In Figure 8.14, the green region is a
half-plane. The purple region is also a half-plane. A half-plane is the graph of a linear
inequality that involves 7 or 6. The graph of an inequality that involves Ú or … is a
half-plane and a line. A solid line is used to show that a line is part of a graph. A
dashed line is used to show that a line is not part of a graph.
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Section 8.5 Systems of Inequalities
795
Graphing a Linear Inequality in Two Variables
1. Replace the inequality symbol with an equal sign and graph the corresponding
linear equation. Draw a solid line if the original inequality contains a … or Ú
symbol. Draw a dashed line if the original inequality contains a 6 or 7
symbol.
2. Choose a test point from one of the half-planes. (Do not choose a point on
the line.) Substitute the coordinates of the test point into the inequality.
3. If a true statement results, shade the half-plane containing this test point. If a
false statement results, shade the half-plane not containing this test point.
Graphing a Linear Inequality in Two Variables
EXAMPLE 1
Graph:
2x - 3y Ú 6.
Solution
Step 1 Replace the inequality symbol by ⴝ and graph the linear equation. We
need to graph 2x - 3y = 6. We can use intercepts to graph this line.
y
We set y ⴝ 0 to find
the x-intercept.
5
4
3
2
1
1 2 3 4 5 6 7 8
−2
−3
−4
−5
2x - 3y = 6
3#0
2 # 0 - 3y = 6
-3y = 6
x
= 6
2x = 6
x = 3
y = -2
2x − 3y = 6
The x-intercept is 3, so the line passes through (3, 0). The y-intercept is -2, so the
line passes through 10, -22. Using the intercepts, the line is shown in Figure 8.15 as
a solid line. This is because the inequality 2x - 3y Ú 6 contains a Ú symbol, in
which equality is included.
Step 2 Choose a test point from one of the half-planes and not from the line.
Substitute its coordinates into the inequality. The line 2x - 3y = 6 divides the plane
into three parts—the line itself and two half-planes. The points in one half-plane
satisfy 2x - 3y 7 6. The points in the other half-plane satisfy 2x - 3y 6 6. We need
to find which half-plane belongs to the solution of 2x - 3y Ú 6. To do so, we test a
point from either half-plane. The origin, (0, 0), is the easiest point to test.
Figure 8.15 Preparing to graph
2x - 3y Ú 6
y
2x - 3y Ú 6
5
4
3
2
1
−2 −1−1
2x - 3y = 6
2x -
−2 −1−1
We set x ⴝ 0 to find
the y-intercept.
?
2#0 - 3#0 Ú 6
?
0 - 0 Ú 6
1 2 3 4 5 6 7 8
−2
−3
−4
−5
Figure 8.16 The graph of
2x - 3y Ú 6
0 Ú 6
x
This is the given inequality.
Test (0, 0) by substituting 0 for x and 0 for y.
Multiply.
This statement is false.
Step 3 If a false statement results, shade the half-plane not containing the test
point. Because 0 is not greater than or equal to 6, the test point, (0, 0), is not part of
the solution set. Thus, the half-plane below the solid line 2x - 3y = 6 is part of the
solution set. The solution set is the line and the half-plane that does not contain the
point (0, 0), indicated by shading this half-plane. The graph is shown using green
shading and a blue line in Figure 8.16.
Check Point
1
Graph:
4x - 2y Ú 8.
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796 Chapter 8 Systems of Equations and Inequalities
When graphing a linear inequality, choose a test point that lies in one of the halfplanes and not on the line dividing the half-planes. The test point (0, 0) is convenient
because it is easy to calculate when 0 is substituted for each variable. However, if (0, 0)
lies on the dividing line and not in a half-plane, a different test point must be selected.
Graphing a Linear Inequality in Two Variables
EXAMPLE 2
Graph:
Solution
y
y=
− 23 x
5
4
3
2
1
−5 −4 −3 −2 −1−1
Rise = −2 −2
−3
−4
−5
Step 1 Replace the inequality symbol by ⴝ and graph the linear equation. Because
we are interested in graphing y 7 - 23 x, we begin by graphing y = - 23 x. We can use
the slope and the y-intercept to graph this linear function.
Test point: (1, 1)
1 2 3 4 5
y= –
x
Slope =
Run = 3
Figure 8.17 The graph of
y 7 - 23 x
2
y 7 - x.
3
2
x+0
3
−2
rise
=
3
run
y-intercept = 0
The y-intercept is 0, so the line passes through (0, 0). Using the y-intercept and the
slope, the line is shown in Figure 8.17 as a dashed line. This is because the inequality
y 7 - 23 x contains a 7 symbol, in which equality is not included.
Step 2 Choose a test point from one of the half-planes and not from the line.
Substitute its coordinates into the inequality. We cannot use (0, 0) as a test point
because it lies on the line and not in a half-plane. Let’s use (1, 1), which lies in the
half-plane above the line.
2
y 7 - x
3
This is the given inequality.
1 7 -
2#
1
3
Test (1, 1) by substituting 1 for x and 1 for y.
1 7 -
2
3
This statement is true.
?
Step 3 If a true statement results, shade the half-plane containing the test
point. Because 1 is greater than - 23 , the test point (1, 1) is part of the solution set.
All the points on the same side of the line y = - 23 x as the point (1, 1) are members
of the solution set. The solution set is the half-plane that contains the point (1, 1),
indicated by shading this half-plane. The graph is shown using green shading and a
dashed blue line in Figure 8.17.
Technology
Most graphing utilities can graph inequalities in two
variables with the 冷 SHADE 冷 feature. The procedure varies
by model, so consult your manual. For most graphing utilities, you must first solve for y if it is not already isolated.
The figure shows the graph of y 7 - 23 x. Most displays do
not distinguish between dashed and solid boundary lines.
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Section 8.5 Systems of Inequalities
2
Check Point
Graph:
797
3
y 7 - x.
4
Graphing Linear Inequalities without Using Test Points
You can graph inequalities in the form y 7 mx + b or y 6 mx + b without using
test points. The inequality symbol indicates which half-plane to shade.
• If y 7 mx + b, shade the half-plane above the line y = mx + b.
• If y 6 mx + b, shade the half-plane below the line y = mx + b.
Observe how this is illustrated in Figure 8.17 in the margin on the previous page. The
graph of y 7 - 23 x is the half-plane above the line y = - 23 x.
It is also not necessary to use test points when graphing inequalities involving
half-planes on one side of a vertical or a horizontal line.
For the Vertical Line x ⴝ a:
For the Horizontal Line y ⴝ b:
• If x 7 a, shade the half-plane to the
right of x = a.
• If y 7 b, shade the half-plane above
y = b.
• If x 6 a, shade the half-plane to the
left of x = a.
• If y 6 b, shade the half-plane below
y = b.
Study Tip
Continue using test points to graph
inequalities in the form Ax + By 7 C
or Ax + By 6 C. The graph of
Ax + By 7 C can lie above or below
the line given by Ax + By = C,
depending on the value of B. The same
comment applies to the graph of
Ax + By 6 C.
x < a in the
yellow region.
x > a in the
green region.
y
y
y > b in the
green region.
y=b
x
x
y < b in the
yellow region.
x=a
EXAMPLE 3
Graphing Inequalities without Using Test Points
Graph each inequality in a rectangular coordinate system:
a. y … - 3
b. x 7 2.
Solution
b. x 7 2
a. y … - 3
Graph y = −3, a horizontal line with y-intercept
−3. The line is solid because equality is included
in y −3. Because of the less than part of ,
shade the half-plane below the horizontal line.
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
1 2 3 4 5
x
Graph x = 2, a vertical line with x-intercept 2.
The line is dashed because equality is not
included in x > 2. Because of >, the greater
than symbol, shade the half-plane to the right
of the vertical line.
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
1 2 3 4 5
x
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798 Chapter 8 Systems of Equations and Inequalities
3
Check Point
Graph each inequality in a rectangular coordinate system:
a. y 7 1
�
Graph a nonlinear inequality in
two variables.
b. x … - 2.
Graphing a Nonlinear Inequality in Two Variables
Example 4 illustrates that a nonlinear inequality in two variables is graphed in the
same way that we graph a linear inequality.
Graphing a Nonlinear Inequality in Two Variables
EXAMPLE 4
Graph:
x2 + y2 … 9.
Solution
Step 1 Replace the inequality symbol with ⴝ and graph the nonlinear equation. We
need to graph x2 + y2 = 9. The graph is a circle of radius 3 with its center at the origin.
The graph is shown in Figure 8.18 as a solid circle because equality is included in the
… symbol.
Step 2 Choose a test point from one of the regions and not from the circle. Substitute
its coordinates into the inequality. The circle divides the plane into three parts—the
circle itself, the region inside the circle, and the region outside the circle. We need to
determine whether the region inside or outside the circle is included in the solution. To
do so, we will use the test point (0, 0) from inside the circle.
x2 + y2 … 9
This is the given inequality.
?
02 + 02 … 9
Test (0, 0) by substituting 0 for x and 0 for y.
?
Square 0: 02 = 0.
0 + 0 … 9
0 … 9
Add. This statement is true.
Step 3 If a true statement results, shade the region containing the test point. The
true statement tells us that all the points inside the circle satisfy x2 + y2 … 9. The
graph is shown using green shading and a solid blue circle in Figure 8.19.
y
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
5
4
3
2
1
1 2 3 4 5
x
−2
−3
−4
−5
−5 −4 −3 −2 −1−1
1 2 3 4 5
−2
−3
−4
−5
Figure 8.18 Preparing to graph
Figure 8.19 The graph of
x2 + y2 … 9
x2 + y2 … 9
Check Point
4
Graph:
x 2 + y2 Ú 16.
x
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Section 8.5 Systems of Inequalities
Use mathematical models
involving linear inequalities.
Modeling with Systems of Linear Inequalities
Just as two or more linear equations make up a system of linear equations, two or
more linear inequalities make up a system of linear inequalities. A solution of a
system of linear inequalities in two variables is an ordered pair that satisfies each
inequality in the system.
Does Your Weight Fit You?
EXAMPLE 5
The latest guidelines, which apply to both men and women, give healthy weight
ranges, rather than specific weights, for your height. Figure 8.20 shows the healthy
weight region for various heights for people between the ages of 19 and 34,
inclusive.
Healthy Weight Region for
Men and Women, Ages 19 to 34
y
230
210
Weight (pounds)
�
799
4.9x − y = 165
190
A
170
Healthy Weight
Region
150
130
B
3.7x − y = 125
110
90
x
60
62
64
66
68
70
72
Height (inches)
74
76
78
Figure 8.20
Source: U.S. Department of Health and Human Services
If x represents height, in inches, and y represents weight, in pounds, the healthy
weight region in Figure 8.20 can be modeled by the following system of linear
inequalities:
b
4.9x - y Ú 165
3.7x - y … 125.
Show that point A in Figure 8.20 is a solution of the system of inequalities that
describes healthy weight.
Solution Point A has coordinates (70, 170). This means that if a person is 70 inches
tall, or 5 feet 10 inches, and weighs 170 pounds, then that person’s weight is within the
healthy weight region. We can show that (70, 170) satisfies the system of inequalities
by substituting 70 for x and 170 for y in each inequality in the system.
4.9x - y Ú 165
3.7x - y … 125
?
?
4.91702 - 170 Ú 165
3.71702 - 170 … 125
?
343 - 170 Ú 165
173 Ú 165,
?
true
259 - 170 … 125
89 … 125,
true
The coordinates (70, 170) make each inequality true. Thus, (70, 170) satisfies the
system for the healthy weight region and is a solution of the system.
Check Point
5
Show that point B in Figure 8.20 is a solution of the system of
inequalities that describes healthy weight.
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800 Chapter 8 Systems of Equations and Inequalities
�
Graphing Systems of Linear Inequalities
Graph a system of inequalities.
The solution set of a system of linear inequalities in two variables is the set of all
ordered pairs that satisfy each inequality in the system. Thus, to graph a system of
inequalities in two variables, begin by graphing each individual inequality in the
same rectangular coordinate system. Then find the region, if there is one, that is
common to every graph in the system. This region of intersection gives a picture of
the system’s solution set.
Graphing a System of Linear Inequalities
EXAMPLE 6
Graph the solution set of the system:
b
x - y 6 1
2x + 3y Ú 12.
Solution Replacing each inequality symbol with an equal sign indicates that
we need to graph x - y = 1 and 2x + 3y = 12. We can use intercepts to graph
these lines.
xⴚyⴝ1
2x ⴙ 3y ⴝ 12
Set y = 0 in
each equation.
x-intercept:
x-0=1
x=1
The line passes through (1, 0).
Set x = 0 in
y-intercept:
0-y=1
each equation.
–y=1
y=–1
The line passes through (0, –1).
2x+3 ⴢ 0=12
2x=12
x=6
The line passes through (6, 0).
x-intercept:
2 ⴢ 0+3y=12
3y=12
y=4
The line passes through (0, 4).
y-intercept:
Now we are ready to graph the solution set of the system of linear inequalities.
Graph x − y < 1. The blue line, x − y = 1,
is dashed: Equality is not included in x − y < 1.
Because (0, 0) makes the inequality true
(0 − 0 < 1, or 0 < 1, is true), shade the halfplane containing (0, 0) in yellow.
Add the graph of 2x + 3y 12. The red line,
2x + 3y = 12, is solid: Equality is included in
2x + 3y 12. Because (0, 0) makes the inequality
false (2 ⴢ 0 + 3 ⴢ 0 12, or 0 12, is false),
shade the half-plane not containing (0, 0) using
green vertical shading.
y
−2
−3
−4
−5
y
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
The solution set of the system is graphed
as the intersection (the overlap) of the
two half-planes. This is the region in
which the yellow shading and the green
vertical shading overlap.
2x + 3y = 12:
passes through
(6, 0) and (0, 4)
1 2 3 4 5
x
x − y = 1: passes
through (1, 0)
and (0, −1)
The graph of x-y<1
5
4
3
2
1
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
1 2 3 4 5
x−y=1
Adding the graph of
2x+3y 12
x
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
1 2 3 4 5
x
This open dot shows
(3, 2) is not in
the solution set.
It does not satisfy
x − y < 1.
The graph of x-y<1
and 2x+3y 12
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Section 8.5 Systems of Inequalities
Check Point
6
801
Graph the solution set of the system:
b
x - 3y 6 6
2x + 3y Ú - 6.
Graphing a System of Inequalities
EXAMPLE 7
Graph the solution set of the system:
b
y Ú x2 - 4
x - y Ú 2.
Solution We begin by graphing y Ú x2 - 4. Because equality is included in Ú,
we graph y = x2 - 4 as a solid parabola. Because (0, 0) makes the inequality
y Ú x2 - 4 true (we obtain 0 Ú - 4), we shade the interior portion of the parabola
containing (0, 0), shown in yellow in Figure 8.21.
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
y
y
y = x2 − 4
1 2 3 4 5
x
−2
−3
−4
−5
Figure 8.21 The graph of y Ú x2 - 4
5
4
3
2
1
y = x2 − 4
−5 −4 −3 −2 −1−1
x−y=2
x
1 2 3 4 5
−2
−3
−4
−5
y = x2 − 4
5
4
3
2
1
−5 −4 −3 −2 −1−1
(−1, −3)
−2
x−y=2
1
3 4 5
x
(2, 0)
−4
−5
Figure 8.22 Adding the graph of x - y Ú 2
Figure 8.23 The graph of y Ú x2 - 4 and
x - y Ú 2
Now we graph x - y Ú 2 in the same rectangular coordinate system. First
we graph the line x - y = 2 using its x-intercept, 2, and its y-intercept, -2.
Because (0, 0) makes the inequality x - y Ú 2 false (we obtain 0 Ú 2), we shade
the half-plane below the line. This is shown in Figure 8.22 using green vertical
shading.
The solution of the system is shown in Figure 8.23 by the intersection (the
overlap) of the solid yellow and green vertical shadings. The graph of the system’s
solution set consists of the region enclosed by the parabola and the line. To find the
points of intersection of the parabola and the line, use the substitution method to
solve the nonlinear system
b
y = x2 - 4
x - y = 2.
Take a moment to show that the solutions are 1-1, -32 and (2, 0), as shown in
Figure 8.23.
Check Point
7
Graph the solution set of the system:
b
y Ú x2 - 4
x + y … 2.
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802 Chapter 8 Systems of Equations and Inequalities
A system of inequalities has no solution if there are no points in the rectangular
coordinate system that simultaneously satisfy each inequality in the system. For
example, the system
2x + 3y Ú 6
b
2x + 3y … 0,
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
2x + 3y ≥ 6
x
1 2 3 4 5
2x + 3y ≤ 0
whose separate graphs are shown in Figure 8.24, has no overlapping region. Thus,
the system has no solution. The solution set is , the empty set.
Graphing a System of Inequalities
EXAMPLE 8
Graph the solution set of the system:
Figure 8.24 A system of
x - y 6 2
c -2 … x 6 4
y 6 3.
inequalities with no solution
Solution We begin by graphing x - y 6 2, the first given inequality. The line
x - y = 2 has an x-intercept of 2 and a y-intercept of -2. The test point (0, 0)
makes the inequality x - y 6 2 true, and its graph is shown in Figure 8.25.
Now, let’s consider the second given inequality, - 2 … x 6 4. Replacing the
inequality symbols by =, we obtain x = - 2 and x = 4, graphed as red vertical lines
in Figure 8.26. The line of x = 4 is not included. Because x is between - 2 and 4, we
shade the region between the vertical lines. We must intersect this region with the
yellow region in Figure 8.25. The resulting region is shown in yellow and green
vertical shading in Figure 8.26.
Finally, let’s consider the third given inequality, y 6 3. Replacing the inequality
symbol by =, we obtain y = 3, which graphs as a horizontal line. Because of the less
than symbol in y 6 3, the graph consists of the half-plane below the line y = 3. We
must intersect this half-plane with the region in Figure 8.26. The resulting region is
shown in yellow and green vertical shading in Figure 8.27. This region represents the
graph of the solution set of the given system.
y
y
5
4
3
2
1
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
x = −2
1 2 3 4 5
x−y=2
x
−5 −4 −3
y
x=4
5
4
3
2
1
−1−1
−2
−3
−4
−5
x = −2
y=3
1 2 3
5
x
−5 −4 −3 −2 −1−1
−2
−3
−4
−5
x−y=2
x=4
5
4
3
2
1
1 2 3 4 5
x
x−y=2
Figure 8.25 The graph of
Figure 8.26 The graph of
Figure 8.27 The graph of x - y 6 2
x - y 6 2
x - y 6 2 and -2 … x 6 4
and - 2 … x 6 4 and y 6 3
In Figure 8.27 it may be difficult to tell where the graph of x - y = 2 intersects the
vertical line x = 4. Using the substitution method, it can be determined that this
intersection point is (4, 2). Take a moment to verify that the four intersection points
in Figure 8.27 are, clockwise from upper left, 1- 2, 32, (4, 3), (4, 2), and 1- 2, -42.
These points are shown as open dots because none satisfies all three of the system’s
inequalities.
Check Point
8
Graph the solution set of the system:
x + y 6 2
c
-2 … x 6 1
y 7 - 3.
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Section 8.5 Systems of Inequalities
803
Exercise Set 8.5
Practice Exercises
47. b
x2 + y2 … 16
x + y 7 2
48. b
x2 + y2 … 4
x + y 7 1
49. b
x2 + y2 7 1
x2 + y2 6 16
50. b
x2 + y2 7 1
x2 + y2 6 9
51. b
1x - 122 + 1y + 122 6 25
1x - 122 + 1y + 122 Ú 16
52. b
1x + 122 + 1y - 122 6 16
1x + 122 + 1y - 122 Ú 4
53. b
x2 + y2 … 1
y - x2 7 0
54. b
x2 + y2 6 4
y - x2 Ú 0
55. b
x2 + y2 6 16
y Ú 2x
56. b
x2 + y2 … 16
y 6 2x
In Exercises 1–26, graph each inequality.
1. x + 2y … 8
2. 3x - 6y … 12
3. x - 2y 7 10
5. y …
4. 2x - y 7 4
1
x
3
6. y …
7. y 7 2x - 1
1
x
4
8. y 7 3x + 2
9. x … 1
10. x … - 3
11. y 7 1
12. y 7 - 3
13. x2 + y2 … 1
14. x2 + y2 … 4
15. x2 + y2 7 25
16. x2 + y2 7 36
17. 1x - 22 + 1y + 12 6 9
2
2
18. 1x + 222 + 1y - 122 6 16
19. y 6 x2 - 1
20. y 6 x2 - 9
21. y Ú x2 - 9
22. y Ú x2 - 1
23. y 7 2 x
24. y … 3x
25. y Ú log21x + 12
26. y Ú log31x - 12
In Exercises 27–62, graph the solution set of each system of
inequalities or indicate that the system has no solution.
27. b
3x + 6y … 6
2x + y … 8
28. b
x - y Ú 4
x + y … 6
29. b
2x - 5y … 10
3x - 2y 7 6
30. b
2x - y … 4
3x + 2y 7 - 6
y 7 2x - 3
31. b
y 6 -x + 6
y 6 - 2x + 4
32. b
y 6 x - 4
33. b
x + 2y … 4
y Ú x - 3
34. b
x + y … 4
y Ú 2x - 4
35. b
x … 2
y Ú -1
36. b
x … 3
y … -1
37. - 2 … x 6 5
38. - 2 6 y … 5
39. b
x - y … 1
x Ú 2
40. b
4x - 5y Ú - 20
x Ú -3
41. b
x + y 7 4
x + y 6 -1
42. b
x + y 7 3
x + y 6 -2
43. b
x + y 7 4
x + y 7 -1
44. b
x + y 7 3
x + y 7 -2
45. b
y Ú x2 - 1
x - y Ú -1
46. b
y Ú x2 - 4
x - y Ú 2
x - y … 2
57. c x 7 - 2
y … 3
3x + y … 6
58. c x 7 - 2
y … 4
x Ú 0
y Ú 0
59. d
2x + 5y 6 10
3x + 4y … 12
x Ú 0
y Ú 0
60. d
2x + y 6 4
2x - 3y … 6
3x + y … 6
2x - y … -1
61. d
x 7 -2
y 6 4
2x + y … 6
x + y 7 2
62. d
1 … x … 2
y 6 3
Practice Plus
In Exercises 63–64, write each sentence as an inequality in two
variables. Then graph the inequality.
63. The y-variable is at least 4 more than the product of -2 and
the x-variable.
64. The y-variable is at least 2 more than the product of - 3 and
the x-variable.
In Exercises 65–68, write the given sentences as a system of
inequalities in two variables. Then graph the system.
65. The sum of the x-variable and the y-variable is at most 4. The
y-variable added to the product of 3 and the x-variable does
not exceed 6.
66. The sum of the x-variable and the y-variable is at most 3. The
y-variable added to the product of 4 and the x-variable does
not exceed 6.
67. The sum of the x-variable and the y-variable is no more than
2. The y-variable is no less than the difference between the
square of the x-variable and 4.
68. The sum of the squares of the x-variable and the y-variable is
no more than 25. The sum of twice the y-variable and the
x-variable is no less than 5.
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804 Chapter 8 Systems of Equations and Inequalities
In Exercises 69–70, rewrite each inequality in the system without
absolute value bars. Then graph the rewritten system in rectangular
coordinates.
ƒxƒ … 2
69. b
ƒyƒ … 3
ƒxƒ … 1
70. b
ƒyƒ … 2
The graphs of solution sets of systems of inequalities involve
finding the intersection of the solution sets of two or more
inequalities. By contrast, in Exercises 71–72, you will be graphing
the union of the solution sets of two inequalities.
71. Graph the union of y 7 32 x - 2 and y 6 4.
77. Show that point A is a solution of the system of inequalities
that describes healthy weight for this age group.
78. Show that point B is a solution of the system of inequalities
that describes healthy weight for this age group.
79. Is a person in this age group who is 6 feet tall weighing
205 pounds within the healthy weight region?
80. Is a person in this age group who is 5 feet 8 inches tall
weighing 135 pounds within the healthy weight region?
81. Many elevators have a capacity of 2000 pounds.
72. Graph the union of x - y Ú - 1 and 5x - 2y … 10.
a. If a child averages 50 pounds and an adult 150 pounds,
write an inequality that describes when x children and
y adults will cause the elevator to be overloaded.
Without graphing, in Exercises 73–76, determine if each system has
no solution or infinitely many solutions.
b. Graph the inequality. Because x and y must be positive,
limit the graph to quadrant I only.
73. b
3x + y 6 9
3x + y 7 9
74. b
6x - y … 24
6x - y 7 24
75. b
1x + 422 + 1y - 322 … 9
1x + 422 + 1y - 322 Ú 9
76. b
c. Select an ordered pair satisfying the inequality. What
are its coordinates and what do they represent in this
situation?
82. A patient is not allowed to have more than 330 milligrams of
cholesterol per day from a diet of eggs and meat. Each egg
provides 165 milligrams of cholesterol. Each ounce of meat
provides 110 milligrams.
a. Write an inequality that describes the patient’s dietary
restrictions for x eggs and y ounces of meat.
1x - 422 + 1y + 322 … 24
1x - 422 + 1y + 322 Ú 24
b. Graph the inequality. Because x and y must be positive,
limit the graph to quadrant I only.
c. Select an ordered pair satisfying the inequality.What are its
coordinates and what do they represent in this situation?
Application Exercises
The figure shows the healthy weight region for various heights for
people ages 35 and older.
Healthy Weight Region for
Men and Women, Ages 35 and Older
y
a. Write a system of inequalities that models the following
conditions:
240
220
B
5.3x − y = 180
200
Weight (pounds)
83. On your next vacation, you will divide lodging between large
resorts and small inns. Let x represent the number of nights
spent in large resorts. Let y represent the number of nights
spent in small inns.
180
You want to stay at least 5 nights. At least one night
should be spent at a large resort. Large resorts
average $200 per night and small inns average $100
per night. Your budget permits no more than $700 for
lodging.
Healthy Weight
Region
160
A
140
120
b. Graph the solution set of the system of inequalities in
part (a).
4.1x − y = 140
100
x
60
62
64
66 68 70 72
Height (inches)
74
76
78
Source: U.S. Department of Health and Human Services
If x represents height, in inches, and y represents weight, in pounds,
the healthy weight region can be modeled by the following system
of linear inequalities:
b
5.3x - y Ú 180
4.1x - y … 140.
Use this information to solve Exercises 77–80.
c. Based on your graph in part (b), what is the greatest
number of nights you could spend at a large resort and
still stay within your budget?
84. A person with no more than $15,000 to invest plans to
place the money in two investments. One investment is
high risk, high yield; the other is low risk, low yield. At least
$2000 is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least
three times the amount invested at high risk. Find and
graph a system of inequalities that describes all
possibilities for placing the money in the high- and low-risk
investments.
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Section 8.5 Systems of Inequalities
805
The graph of an inequality in two variables is a region in the rectangular coordinate system. Regions in coordinate systems have numerous
applications. For example, the regions in the following two graphs indicate whether a person is obese, overweight, borderline overweight,
normal weight, or underweight.
Females
Males
35
35
Obese
30
Overweight
Borderline Overweight
25
Normal Range
20
Body-Mass Index (BMI)
Body-Mass Index (BMI)
Obese
30
Overweight
Borderline Overweight
25
Normal Range
20
Underweight
Underweight
15
10
20
30
40
50
Age
60
70
80
15
10
20
30
40
50
Age
60
70
80
Source: Centers for Disease Control and Prevention
The horizontal axis shows a person’s age. The vertical axis shows
that person’s body-mass index (BMI), computed using the following formula:
703W
.
BMI =
H2
The variable W represents weight, in pounds. The variable H
represents height, in inches. Use this information to solve
Exercises 85–86.
85. A man is 20 years old, 72 inches (6 feet) tall, and weighs
200 pounds.
a. Compute the man’s BMI. Round to the nearest tenth.
b. Use the man’s age and his BMI to locate this information
as a point in the coordinate system for males. Is this
person obese, overweight, borderline overweight, normal
weight, or underweight?
86. A woman is 25 years old, 66 inches (5 feet, 6 inches) tall, and
weighs 105 pounds.
a. Compute the woman’s BMI. Round to the nearest tenth.
b. Use the woman’s age and her BMI to locate this information as a point in the coordinate system for females. Is
this person obese, overweight, borderline overweight,
normal weight, or underweight?
95. Explain how to graph the solution set of a system of
inequalities.
96. What does it mean if a system of linear inequalities has no
solution?
Technology Exercises
Graphing utilities can be used to shade regions in the rectangular
coordinate system, thereby graphing an inequality in two variables.
Read the section of the user’s manual for your graphing utility that
describes how to shade a region. Then use your graphing utility to
graph the inequalities in Exercises 97–102.
97. y … 4x + 4
98. y Ú
2
x - 2
3
99. y Ú x2 - 4
100. y Ú
1 2
x - 2
2
101. 2x + y … 6
102. 3x - 2y Ú 6
103. Does your graphing utility have any limitations in terms of
graphing inequalities? If so, what are they?
104. Use a graphing utility with a 冷 SHADE 冷 feature to verify
any five of the graphs that you drew by hand in
Exercises 1–26.
Writing in Mathematics
105. Use a graphing utility with a 冷 SHADE 冷 feature to verify
any five of the graphs that you drew by hand for the systems
in Exercises 27–62.
87. What is a linear inequality in two variables? Provide an
example with your description.
Critical Thinking Exercises
88. How do you determine if an ordered pair is a solution of an
inequality in two variables, x and y?
89. What is a half-plane?
90. What does a solid line mean in the graph of an inequality?
91. What does a dashed line mean in the graph of an inequality?
92. Compare the graphs of 3x - 2y 7 6 and 3x - 2y … 6.
Discuss similarities and differences between the graphs.
93. What is a system of linear inequalities?
94. What is a solution of a system of linear inequalities?
Make Sense? In Exercises 106–109, determine whether
each statement makes sense or does not make sense, and explain
your reasoning.
106. When graphing a linear inequality, I should always use (0, 0)
as a test point because it’s easy to perform the calculations
when 0 is substituted for each variable.
107. When graphing 3x - 4y 6 12, it’s not necessary for me to
graph the linear equation 3x - 4y = 12 because the
inequality contains a 6 symbol, in which equality is not
included.
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806 Chapter 8 Systems of Equations and Inequalities
108. The reason that systems of linear inequalities are appropriate
for modeling healthy weight is because guidelines give healthy
weight ranges, rather than specific weights, for various heights.
109. I graphed the solution set of y Ú x + 2 and x Ú 1 without
using test points.
Preview Exercises
Exercises 116–118 will help you prepare for the material covered
in the next section.
116. a. Graph the solution set of the system:
In Exercises 110–113, write a system of inequalities for each graph.
110.
111.
y
y
d
2
2
2
−2
x
4
−4 −2
−2
2
4
b. List the points that form the corners of the graphed
region in part (a).
x
c. Evaluate 3x + 2y at each of the points obtained in
part (b).
−4
−4
x … 8
y Ú 5.
4
4
−5 −3
x + y Ú 6
117. a. Graph the solution set of the system:
112.
113.
y
5
4
3
2
1
y = x2
−4 −3 −2 −1−1
1 2 3 4
x
−2
−3
x
y
d
3x - 2x
y
y
8
7
6
5
4
3
2
1
x
114. Write a system of inequalities whose solution set includes
every point in the rectangular coordinate system.
115. Sketch the graph of the solution set for the following system
of inequalities:
y Ú nx + b 1n 6 0, b 7 02
y … mx + b 1m 7 0, b 7 02.
Section
8.6
118. Bottled water and medical supplies are to be shipped to
survivors of an earthquake by plane.The bottled water weighs
20 pounds per container and medical kits weigh 10 pounds
per kit. Each plane can carry no more than 80,000 pounds. If x
represents the number of bottles of water to be shipped per
plane and y represents the number of medical kits per plane,
write an inequality that models each plane’s 80,000-pound
weight restriction.
West Berlin children at Tempelhof
airport watch fleets of U.S. airplanes
bringing in supplies to circumvent the
Soviet blockade. The airlift began
June 28, 1948 and continued for 15
months.
� Write an objective function
�
c. Evaluate 2x + 5y at each of the points obtained in
part (b).
Linear Programming
Objectives
�
0
0
6
- x + 7.
b. List the points that form the corners of the graphed
region in part (a).
1 2 3 4 5 6 7 8
b
Ú
Ú
…
…
describing a quantity that
must be maximized or
minimized.
Use inequalities to describe
limitations in a situation.
Use linear programming to
solve problems.
T
he Berlin Airlift (1948 – 1949) was an operation by the United States and Great
Britain in response to military action by the former Soviet Union: Soviet troops
closed all roads and rail lines between West Germany and Berlin, cutting off supply
routes to the city. The Allies used a mathematical technique developed during World