Download 2.8Graph Linear Inequalities in Two Variables

2.8
Before
Graph Linear Inequalities
in Two Variables
You solved linear inequalities in one variable.
Now
You will graph linear inequalities in two variables.
Why?
So you can model data encoding, as in Example 4.
Key Vocabulary
A linear inequality in two variables can be written in one of these forms:
• linear inequality in
two variables
• solution of a linear
inequality
• graph of a linear
inequality
• half-plane
Ax 1 By ≤ C
Ax 1 By < C
Ax 1 By ≥ C
Ax 1 By > C
An ordered pair (x, y) is a solution of a linear inequality in two variables if the
inequality is true when the values of x and y are substituted into the inequality.
EXAMPLE 1
★ Standardized Test Practice
Which ordered pair is a solution of 3x 1 4y > 8?
A (6, 23)
B (0, 2)
C (22, 21)
D (23, 5)
Solution
Ordered Pair
Substitute
Conclusion
(6, 23)
3(6) 1 4(23) 5 6 >/ 8
(6, 23) is not a solution.
(0, 2)
3(0) 1 4(2) 5 8 >/ 8
(0, 2) is not a solution.
(22, 21)
3(22) 1 4(21) 5 210 >/ 8
(22, 21) is not a solution.
(23, 5)
3(23) 1 4(5) 5 11 > 8
(23, 5) is a solution.
c The correct answer is D. A B C D
✓
GUIDED PRACTICE
for Example 1
Tell whether the given ordered pair is a solution of 5x 2 2y ≤ 6.
1. (0, 24)
2. (2, 2)
3. (23, 8)
4. (21, 27)
GRAPHING INEQUALITIES The graph of a linear inequality in two variables is the
set of all points in a coordinate plane that represent solutions of the inequality.
INTERPRET GRAPHS
A dashed boundary
line means that points
on the line are not
solutions. A solid
boundary line means
that points on the line
are solutions.
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y
All solutions of 3x 2 2y > 2
lie on one side of the
boundary line 3x 2 2y 5 2.
1
1
x
The boundary line divides the
plane into two half-planes.
The shaded half-plane is the
graph of 3x 2 2y > 2.
3x 2 2y > 2
Chapter 2 Linear Equations and Functions
10/20/05 10:17:13 AM
For Your Notebook
KEY CONCEPT
Graphing a Linear Inequality
To graph a linear inequality in two variables, follow these steps:
STEP 1
Graph the boundary line for the inequality. Use a dashed line for
< or > and a solid line for ≤ or ≥.
STEP 2 Test a point not on the boundary line to determine whether it is a
solution of the inequality. If it is a solution, shade the half-plane
containing the point. If it is not a solution, shade the other half-plane.
EXAMPLE 2
Graph linear inequalities with one variable
Graph (a) y ≤ 23 and (b) x < 2 in a coordinate plane.
a. Graph the boundary line y 5 23.
b. Graph the boundary line x 5 2.
Use a solid line because the
inequality symbol is ≤.
Use a dashed line because the
inequality symbol is <.
Test the point (0, 0). Because
(0, 0) is not a solution of the
inequality, shade the half-plane
that does not contain (0, 0).
Test the point (0, 0). Because
(0, 0) is a solution of the
inequality, shade the half-plane
that contains (0, 0).
y
y
(0, 0) 3
21
x
1
y ≤ 23
EXAMPLE 3
x<2
(0, 0)
3
x
Graph linear inequalities with two variables
Graph (a) y > 22x and (b) 5x 2 2y ≤ 24 in a coordinate plane.
a. Graph the boundary line y 5 22x.
AVOID ERRORS
It is often convenient to
use (0, 0) as a test point.
However, if (0, 0) lies on
the boundary line, you
must choose a different
test point.
b. Graph the boundary line
Use a dashed line because the
inequality symbol is >.
5x 2 2y 5 24. Use a solid line
because the inequality symbol is ≤.
Test the point (1, 1). Because
(1, 1) is a solution of the
inequality, shade the half-plane
that contains (1, 1).
Test the point (0, 0). Because
(0, 0) is not a solution of the
inequality, shade the half-plane
that does not contain (0, 0).
y
y
2
5x 2 2y ≤ 24
(1, 1)
1
y > 2 2x
(0, 0)
x
3x
23
"MHFCSB
at classzone.com
2.8 Graph Linear Inequalities in Two Variables
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133
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✓
GUIDED PRACTICE
for Examples 2 and 3
Graph the inequality in a coordinate plane.
5. y > 21
6. x ≥ 24
8. y < 2x 1 3
9. x 1 3y < 9
EXAMPLE 4
7. y ≥ 23x
10. 2x 2 6y > 12
Solve a multi-step problem
MOVIE RECORDING A film class is
recording a DVD of student-made
short films. Each student group is
allotted up to 300 megabytes (MB) of
video space. The films are encoded
on the DVD at two different rates: a
standard rate of 0.4 MB/sec for normal
scenes and a high-quality rate of
1.2 MB/sec for complex scenes.
• Write an inequality describing the
#LIPS
possible amounts of time available
for standard and high-quality video.
0HOTOS
!UDIO
4ITLES
4RANS
%FFECTS
$6$
$RAGCLIPSHERETOBUILDYOURPROJECT
'"FREE
• Graph the inequality.
• Identify three possible solutions of the inequality.
Solution
STEP 1
Write an inequality. First write a verbal model.
Standard
rate
p
Standard
time
(MB/sec)
0.4
1
(sec)
p
x
High-quality
rate
p
High-quality
time
(MB/sec)
1
1.2
≤
(sec)
p
y
Total
space
(MB)
≤
300
STEP 2 Graph the inequality. First graph the
boundary line 0.4x 1 1.2y 5 300. Use
a solid line because the inequality
symbol is ≤.
Test the point (0, 0). Because (0, 0) is
a solution of the inequality, shade the
half-plane that contains (0, 0). Because
x and y cannot be negative, shade only
points in the first quadrant.
High quality (sec)
An inequality is 0.4x 1 1.2y ≤ 300.
y
300
(150, 200)
200
100
0
0
(300, 120)
(600, 25)
200 400 600 800 x
Standard (sec)
STEP 3 Identify solutions. Three solutions are given below and on the graph.
(150, 200)
150 seconds of standard and 200 seconds of high quality
(300, 120)
300 seconds of standard and 120 seconds of high quality
(600, 25)
600 seconds of standard and 25 seconds of high quality
For the first solution, 0.4(150) 1 1.2(200) 5 300, so all of the available
space is used. For the other two solutions, not all of the space is used.
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Chapter 2 Linear Equations and Functions
10/20/05 10:17:20 AM
ABSOLUTE VALUE INEQUALITIES Graphing an absolute value inequality is similar
to graphing a linear inequality, but the boundary is an absolute value graph.
EXAMPLE 5
Graph an absolute value inequality
Graph y > 22x 2 3 1 4 in a coordinate plane.
Solution
STEP 1
Graph the equation of the boundary,
y
y > 22 z x 2 3 z 1 4
y 5 22x 2 3 1 4. Use a dashed line
because the inequality symbol is >.
2
STEP 2 Test the point (0, 0). Because (0, 0) is
a solution of the inequality, shade the
portion of the coordinate plane outside
the absolute value graph.
✓
GUIDED PRACTICE
(0, 0)
2
x
for Examples 4 and 5
11. WHAT IF? Repeat the steps of Example 4 if each student group is allotted up
to 420 MB of video space.
Graph the inequality in a coordinate plane.
12. y ≤ x 2 2 1 1
2.8
EXERCISES
13. y ≥ 2x 1 3 2 2
HOMEWORK
KEY
14. y < 3x 2 1 2 3
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 15, 25, and 45
★
5 STANDARDIZED TEST PRACTICE
Exs. 2, 21, 28, 39, 40, 41, 46, and 48
SKILL PRACTICE
1. VOCABULARY Copy and complete: The graph of a linear inequality in two
variables is a(n) ? .
2. ★ WRITING Compare the graph of a linear inequality in two variables with
the graph of a linear equation in two variables.
EXAMPLE 1
CHECKING SOLUTIONS Tell whether the given ordered pairs are solutions of the
on p. 132
for Exs. 3–6
inequality.
EXAMPLES
2 and 3
on p. 133
for Exs. 7–20
3. x > 27; (0, 10), (28, 25)
4. y ≤ 25x; (3, 2), (22, 1)
5. y ≥ 22x 1 4; (0, 4), (21, 8)
6. 2x 2 y < 3; (0, 0), (2, 22)
GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.
7. x < 3
8. x ≥ 6
9. y > 22
10. 22y ≤ 8
11. y ≤ 22x 2 1
12. y < 3x 1 3
3x 1 1
13. y > }
4
2x 2 2
14. y ≥ 2}
3
15. 2x 1 y < 6
16. x 1 4y > 212
17. 3x 2 y ≥ 1
18. 2x 1 5y ≤ 210
2.8 Graph Linear Inequalities in Two Variables
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ERROR ANALYSIS Describe and correct the error in graphing the inequality.
20. y ≥ 23x 2 2
19. y < 2x 1 3
y
2
y
x
1
1
x
1
21. ★ MULTIPLE CHOICE Which ordered pair is not a solution of 3x 2 5y < 30?
A (0, 0)
B (21, 7)
C (1, 27)
D (25, 25)
EXAMPLE 5
ABSOLUTE VALUE INEQUALITIES Graph the inequality in a coordinate plane.
on p. 135
for Exs. 22–28
22. y > x 2 1
23. y < x 1 5
24. y > x 1 4 2 3
1 x22 11
25. y ≤ 2}


2
26. y < 3x 1 2
27. y ≥ 2x 2 1 2 4
28. ★ MULTIPLE CHOICE The graph of which inequality is shown?
y
A y ≤ 22x 1 1 1 3
B y ≥ 22x 2 1 1 3
C y > 22x 1 1 1 3
D y ≥ 22x 1 1 1 3
1
2x
CHECKING SOLUTIONS Tell whether the given ordered pairs are
solutions of the inequality.
2 x 1 1 ; (26, 8), (23, 23)
29. y ≥ 2}
}
3
2
30. 4.5 1 y < 1.6x; (0.5, 1), (3.8, 0)
31. 0.2x 1 0.7y > 21; (0.5, 21), (23, 21.5)
4 , 0 , 2 , 24
1 x 2 y > 1; }
32. }
}
3
3
4
1
21
2
GRAPHING INEQUALITIES Graph the inequality in a coordinate plane.
33. 3y < 4.5x 1 15
34. 21.5y 2 2x > 3
35. 2y 2 0.2 > 20.6x
2x 1 1y > 2
36. }
}
3
2
5 x23 2 3
37. y ≥ 2}

 }
2
2
38. 2y 2 4 ≤ 23x 1 2
39. ★ OPEN-ENDED MATH Write a linear inequality in two variables that has
(21, 3) and (1, 6) as solutions, but does not have (4, 0) as a solution.
40. ★ WRITING Explain why it is not helpful when graphing a linear inequality
in two variables to choose a test point that lies on the boundary line.
41. ★ SHORT RESPONSE Write an inequality for
the graph shown. Explain how you came up
with the inequality. Then describe a real-life
situation that the first-quadrant portion of
the graph could represent.
y
1
x
1
42. CHALLENGE Write an absolute value inequality that has exactly one solution
in common with y ≥ 2x 2 3 1 5. The common solution should not be the
vertex (3, 5) of the boundary. Explain how you found your inequality.
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5 WORKED-OUT SOLUTIONS
Chapter 2 Linear
and Functions
on p. Equations
WS1
★
5 STANDARDIZED
TEST PRACTICE
10/20/05 10:17:23 AM
PROBLEM SOLVING
EXAMPLE 4
on p. 134
for Exs. 43–48
43. CALLING CARDS You have a $20 phone card. Calls made using the card cost
$.03 per minute to destinations within the United States and $.06 per minute
to destinations in Brazil. Write an inequality describing the numbers of
minutes you can use for calls to U.S. destinations and to Brazil.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
44. RESTAURANT MANAGEMENT A pizza shop has 300 pounds (4800 ounces) of
dough. A small pizza uses 12 ounces of dough and a large pizza uses 18 ounces
of dough. Write and graph an inequality describing the possible numbers of
small and large pizzas that can be made. Then give three possible solutions.
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
45. CRAFTS Cotton lace costs $1.50 per yard and linen lace costs $2.50 per yard.
You plan to order at most $75 of lace for crafts. Write and graph an inequality
describing how much of each type of lace you can order. If you buy 24 yards
of cotton lace, what are the amounts of linen lace you can buy?
46. ★ SHORT RESPONSE You sell T-shirts for $15 each and caps for $10 each.
Write and graph an inequality describing how many shirts and caps
you must sell to exceed $1800 in sales. Explain how you can modify this
inequality to describe how many shirts and caps you must sell to exceed
$600 in profit if you make a 40% profit on shirts and a 30% profit on caps.
47. MULTI-STEP PROBLEM On a two week vacation, you and your brother can
rent one canoe for $11 per day or rent two mountain bikes for $13 each per
day. Together, you have $120 to spend.
a. Write and graph an inequality describing the possible numbers of days
you and your brother can canoe or bicycle together.
b. Give three possible solutions of the inequality from part (a).
c. You decide that on one day you will canoe alone and your brother will
bicycle alone. Repeat parts (a) and (b) using this new condition.
48. ★ EXTENDED RESPONSE While camping, you and a friend filter river water
into two cylindrical containers with the radii and heights shown. You then
use these containers to fill the water cooler shown.
a. Find the volumes of the containers and the cooler in cubic inches.
b. Using your results from part (a), write and graph an inequality
describing how many times the containers can be filled and emptied
into the water cooler without the cooler overflowing.
c. Convert the volumes from part (a) to gallons (1 in.3 ø 0.00433 gal). Then
rewrite the inequality from part (b) in terms of these converted volumes.
d. Graph the inequality from part (c). Compare the graph with your graph
from part (b), and explain why the results make sense.
2.8 Graph Linear Inequalities in Two Variables
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49. CHALLENGE A widescreen television image has a width w and a height h
w > 4.
that satisfy the inequality }
}
h 3
a. Does the television screen shown at the right meet the
IN
requirements of a widescreen image?
b. Let d be the length of a diagonal of a television image.
IN
Write an inequality describing the possible values of d
and h for a widescreen image.
MIXED REVIEW
Look for a pattern in the table. Then write an equation that represents the table.
(p. 34)
50.
x
0
1
2
3
y
11
15
19
23
51.
x
0
1
2
3
y
60
45
30
15
PREVIEW
Graph the equation. (p. 89)
Prepare for
Lesson 3.1
in Exs. 52–57.
52. x 1 3y 5 26
53. 4x 2 3y 5 15
54. 8x 2 6y 5 18
55. 6x 1 9y 5 18
56. 22x 2 5y 5 20
57. 210x 1 4y 5 20
Write an equation of the line that satisfies the given conditions. (p. 98)
4 , passes through (10, 22)
58. m 5 }
5
59. m 5 23, passes through (3, 7)
60. passes through (0, 2) and (5, 8)
61. passes through (4, 21) and (7, 26)
QUIZ for Lessons 2.7–2.8
Graph the function. Compare the graph with the graph of y 5 x. (p. 123)
1. y 5 x 1 7 1 4
1 x21 25
3. f(x) 5 }


2
2. y 5 22x 1 10 2 1
Write an equation of the graph. (p. 123)
y
4.
21
5. y
1
6.
y
x
2
x
1
1
28
x
Graph the inequality in a coordinate plane. (p. 132)
7. y > 22
8. y ≤ 3x 1 1
9. 2x 2 5y ≥ 10
10. MINI-CARS You have a 20 credit gift pass to a mini-car raceway. It takes
2 credits to drive the cars on the Rally track and 3 credits to drive the cars on
the Grand Prix track. Write and graph an inequality describing how many
times you can race on the two tracks using your gift pass. Then give three
possible solutions. (p. 132)
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EXTRA PRACTICE for Lesson 2.8, p. 1011
ONLINE QUIZ at classzone.com
10/20/05 10:17:26 AM