Download Solving Systems of Linear Inequalities in Two Variables and more Exams Algebra in PDF only on Docsity!
7.9 Solve Systems of Linear Inequalities 433
"MHFCSB
Math and (^) BAND Ex. 31, p. 439
Solve Systems of
Linear Inequalities
Standards Alg. 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. Connect Before you graphed linear inequalities in two variables. Now you will solve systems of linear inequalities in two variables.
A system of linear inequalities in two variables, or simply a system of inequalities , consists of two or more linear inequalities in the same variables. An example is shown. x 2 y > 7 Inequality 1 2 x 1 y < 8 Inequality 2 A solution of a system of linear inequalities is an ordered pair that is a solution of each inequality in the system. For example, (6, 2 5) is a solution of the system above. The graph of a system of linear inequalities is the graph of all solutions of the system.
KEY CONCEPT For Your Notebook
Graphing a System of Linear Inequalities
STEP 1 Graph each inequality.
STEP 2 Find the intersection of the half-planes. The graph of the system
is this intersection.
E X A M P L E 1 Graph a system of two linear inequalities
Graph the system of inequalities. y > 2 x 2 2 Inequality 1 y 3 x 1 6 Inequality 2 Solution Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is the region shown in the darker shade of blue.
CHECK Choose a point in the dark blue region,
such as ( 0 , 1 ). To check this solution, substitute 0 for x and 1 for y into each inequality. 1 >?^ 0 2 2 1 ?^ 3( 0 ) 1 6 1 > 22 1 6
x
y
1 1
(0, 1)
x
y
1 1
(0, 1)
Key Vocabulary
- system of linear inequalities - solution of a system of linear inequalities - graph of a system of linear inequalities
For an interactive example of graphing a system of inequalities, go to classzone.com.
1-"/ "/% 13&1"3&
8BSN6Q&YFSDJTFT
Transparency Available
1. Graph y < }^2
x 2 1.
1 x
y
21
2. You are running one ad that costs
$6 per day and another that costs
$8 per day. You can spend no
more than $120. Name a possible
combination of days that you can
run the ads. Sample: 16 days of
the $6 ad and 3 days of the $8 ad
/PUFUBLJOH(VJEF
Transparency Available
Promotes interactive learning and
notetaking skills, pp. 174–177.
1BDJOH
Suggested Number of Days
Basic Average Advanced 2 Days 2 Days 2 Days Block: 1 Block
- (^) See Teaching Guide/Lesson Plan
in Chapter 7 Resource Book,
pp. 96–97.
'0$64 "/% .057"5&*
&TTFOUJBM2VFTUJPO
Big Idea 3, p. 372
How do you solve systems of linear
inequalities in two variables? Tell
students they will learn how to
answer this question by graphing
in the same coordinate plane.
.PUJWBUJOHUIF-FTTPO
You have a landscaping software
program to help you plan where to
place flowers and trees in a park. By
describing the outline of the park as
a system of inequalities, you can
"MH Students can use a system of inequalities to describe a region of show the park on a computer screen.
a plane.
- Students graph each of two inequalities, using a solid or dashed line for
each boundary and shading the appropriate half-plane for each inequality.
- They identify the region that is common to both shaded half-planes.
The solution to a system of inequalities is a region of the plane.
)PXUP5FBDIUIF$BMJGPSOJB4UBOEBSET
434 Chapter 7 Systems of Equations and Inequalities
E X A M P L E 2 Multiple Choice Practice
Which graph best represents the solution to this system of inequalities?
2 x y 1 1 2 x 1 2 y 2 2 A
2^ x
y
1
B
2^ x
y
1
C
2^ x
y
1
D
2^ x
y
1
Solution Notice that the boundary lines are the same for all graphs. Choose a point that lies in the shaded region of each graph. Test the point in both inequalities to determine whether the shaded region is the solution of the system. Choice A: The point (0, 1) lies in the shaded region. 2 x y 1 1 2 x 1 2 y 22 2( 0 ) ?^ 1 1 1 2 ( 0 ) 1 2( 1 ) ?^22 0 2 2 22 Choice B: The point (2, 1) lies in the shaded region. 2 x y 1 1 2 x 1 2 y 22 2( 2 ) ?^ 1 1 1 2 ( 2 ) 1 2( 1 ) ?^22 4 2 0 22 c The correct answer is B. A^ B^ C^ D
G UIDED P RACTICE for Examples 1 and 2
Graph the system of linear inequalities.
1. y < x 2 4 2. 2 x 1 y > 21 3. y 2 x y 2 x 1 3 y < 3 x 1 2 x 22
THE SOLUTION REGION In Example 1, the half-plane for each inequality is shaded, and the solution region is the intersection of the half-planes. From this point on, only the solution region will be shaded.
1–3. See margin.
.BUIFNBUJDBM#BDLHSPVOE
For systems of equations, the
solution is the intersection of two
lines, which is a point. For systems
of inequalities, each graph is a
half-plane. The intersection of
half-planes is a region, so the solu-
tion of a system of inequalities is a
region of the coordinate plane.
5&"$)
&YUSB&YBNQMF
Graph the system of inequalities.
y < 3 x
y r 22 x 11
3
x
y
21
Algebra classzone.com
An Animated Algebra activity in
which students graph a system of
inequalities is available online for
Example 1. This activity is also avail-
able on the Power Presentations
CD-ROM.
&YUSB&YBNQMF
Using the four graphs in Example 2,
which graph best represents the
solution to the system x 2 2 y r 2
and 2 x 2 y b 1? D
1 x
y
21
x
y
2
1
6OJWFSTBM"DDFTT
"VEJUPSZ Tell students that the method used in Example 1
is sometimes called the “graph-and-check” method, because
students graph the inequalities and then check points in each
test region. Students should get in the habit of checking their
graphs by verifying that points in their solution satisfy each
inequality.
x
y
1 21
436 Chapter 7 Systems of Equations and Inequalities
BASEBALL The National Collegiate Athletic Association (NCAA) regulates the lengths of aluminum baseball bats used by college baseball teams. The NCAA states that the length (in inches) of the bat minus the weight (in ounces) of the bat cannot exceed 3. Bats can be purchased at lengths from 26 to 34 inches. a. Write and graph a system of linear inequalities that describes the information given above. b. A sporting goods store sells an aluminum bat that is 31 inches long and weighs 25 ounces. Use the graph to determine if this bat can be used by a player on an NCAA team.
Solution a. Let x be the length (in inches) of the bat, and let y be the weight (in ounces) of the bat. From the given information, you can write the following inequalities: x 2 y 3 The difference of the bat’s length and weight can be at most 3. x 26 The length of the bat must be at least 26 inches. x 34 The length of the bat can be at most 34 inches. y 0 The weight of the bat cannot be a negative number. Graph each inequality in the system. Then identify the region that is common to all of the graphs of the inequalities. This region is shaded in the graph shown.
b. Graph the point that represents a bat that is 31 inches long and weighs 25 ounces. c Because the point falls outside the solution region, the bat cannot be used by a player on an NCAA team.
x
y
(31, 25)
5 10
x
E X A M P L E 5 Write and solve a system of linear inequalities
G UIDED P RACTICE for Example 5
7. WHAT IF? In Example 5, suppose a Senior League (ages 10–14) player wants to buy the bat described in part (b). In Senior League, the length (in inches) of the bat minus the weight (in ounces) of the bat cannot exceed 8. Write and graph a system of inequalities to determine whether the described bat can be used by the Senior League player.
WRITE SYSTEMS OF INEQUALITIES Consider the values of the variables when writing a system of inequalities. In many real-world problems, the values cannot be negative.
See margin.
&YUSB&YBNQMF
A logo contest requires that the logo
width be between 3 and 5 inches, the
height no less than 2 inches, and the
sum of the width and height no more
than 9 inches.
a. Write and graph a system of
linear inequalities that describes
the information given above.
inequalities: x r 3, x b 5,
x 1 y b 9, y r 2
(^1) x
y
1
(4, 6)
b. You enter a logo 4 inches wide
and 6 inches high. Use the graph
to determine if it meets contest
requirements. Because the
point falls outside the solution
region, the logo does not meet
requirements.
$MPTJOHUIF-FTTPO
Students have learned to write,
graph, and solve systems of linear
inequalities. To bring closure, have
students answer these questions:
1. Essential Question: How do you
solve systems of linear inequali-
ties in two variables? Graph
each inequality in the same
coordinate plane. The graph of
the system is the intersection
of all of the graphs.
2. Describe the solution of the
system of inequalities x r 22
and x b 5. The solution is all
points on the two vertical lines
x 5 2 2 and x 5 5 and all the
points in the plane between the
two lines.
7. x 2 y b 8, x r 26, x b 34, y r 0;
x
y
5 25
7.9 Solve Systems of Linear Inequalities 437
HOMEWORK KEY
SKILLS • P ROBLEM SOLVING • R EASONING
7.9 EXERCISES
1. VOCABULARY Copy and complete: The graph of a system of linear inequalities is the graph of all? of the system. 2. WRITING Describe the steps you would take to graph the system of inequalities shown.
CHECKING A SOLUTION Tell whether the ordered pair is a solution of the system of inequalities.
3. (1, 1) 4. (0, 6) 5. (3, 2 1)
x
y
1 1 x
y
1 1
x
y
1 1
6. MULTIPLE CHOICE Which ordered pair is a solution of the system 2 x 2 y 5 and x 1 2 y > 2? A (1, 2 1) B (4, 1) C (2, 0) D (3, 2)
MATCHING SYSTEMS AND GRAPHS Match the system of inequalities with its graph.
7. x 2 4 y 28 8. x 2 4 y 28 9. x 2 4 y 28 x 2 x 2 y 2 A.
x
y
1 1
B.
x
y
1 1
C.
x
y
1 1
GRAPHING A SYSTEM Graph the system of inequalities.
10. y < 22 x 1 3 11. y 0 12. y 2 x 1 1 y 4 y < 2.5 x 2 1 y < 2 x 1 4 13. x < 8 14. y 22 15. y 2 2 x < 7 x 2 4 y 28 2 x 1 3 y > 26 y 1 2 x > 21 16. x < 4 17. x 0 18. x 1 y 10 y > 1 y 0 x 2 y 2 y 2 x 1 1 6 x 2 y < 12 y 2 19. MULTIPLE CHOICE Which ordered pair is a solution of the system x 5, y 3, and x 2 y < 2? A (2, 2 2) B (6, 3) C (2, 0) D (2, 4)
x 2 y 7 Inequality 1 y 3 Inequality 2
x 2 y 7 Inequality 1 y 3 Inequality 2
(^) 5 MULTIPLE CHOICE PRACTICE Exs. 6, 19, 20, 34, and 39– 5 HINTS AND HOMEWORK HELP for Exs. 11, 23, and 33 at classzone.com
EXAMPLES 1, 2, and 3 on pp. 433– for Exs. 7 221 C A (^) B
10–18. See margin.
not a solution solution not a solution
D
solutions
2. Graph both inequalities in the same coordinate plane and shade the region that is the intersection of the two graphs. Use a test point to check the solution.
D
A
13"$5$& "/% "11-:*
"TTJHONFOU(VJEF
Answer Transparencies
available for all exercises
Basic:
Day 1: pp. 437–
Exs. 1–12, 19–
Day 2: pp. 437–
Exs. 22–24, 29–33, 39–
Average:
Day 1: MCP p. 371 Exs. 11, 12
pp. 437–
Exs. 1–9, 13–15, 19–
Day 2: CR p. 244 Exs. 27–
MCP p. 371 Exs. 13, 14
pp. 437–
Exs. 23–26, 28, 31–
Advanced:
Day 1: MRSPS p. 361 Exs. 3–
pp. 437–
Exs. 1–6, 13–
Day 2: MRSPS p. 361 Exs. 6, 7
CR p. 244 Exs. 10–
pp. 437–
Exs. 25–28, 32–38*
Block:
CR p. 244 Exs. 27–
MCP p. 371 Exs. 11–
pp. 437–
Exs. 1–9, 13–15, 19–21, 23–26, 28,
)PNFXPSL$IFDL
For a quick check of student under-
standing of key concepts, go over
the following exercises:
Basic: 8, 12, 22, 31, 32
Average: 12, 14, 24, 31, 33
Advanced: 16, 18, 26, 32, 33
&YUSB1SBDUJDF
- Student Edition, p. 816
- Chapter 7 Resource Book:
Practice Levels A, B, C, pp. 100–
1SBDUJDF8PSLTIFFU
An easily readable reduced
practice page (with answers)
for this lesson can be found
on pp. 372E–372H.
13–18. See Additional Answers
beginning on p. AA1.
1 x
y
21
21
1 x
y 12.
2
x
y
1
7.9 Solve Systems of Linear Inequalities 439
31. COMPETITION SCORES In a marching band competition, scoring is based on a musical evaluation and a visual evaluation. The musical evaluation score cannot exceed 60 points, and the visual evaluation score cannot exceed 40 points. Write and graph a system of inequalities for the scores that a marching band can receive. California for problem solving help at classzone.com 32. NUTRITION For a hiking trip, you are making a mix of x ounces of peanuts and y ounces of dried fruit. You want the mix to have less than 60 grams of fiber and weigh less than 20 ounces. An ounce of peanuts has 14 grams of fiber, and an ounce of dried fruit has 2 grams of fiber. Write and graph a system of inequalities that models the situation. California for problem solving help at classzone.com 33. FISHING LIMITS You are fishing in a marina for surfperch and rockfish, which are two species of bottomfish. Gaming laws in the marina allow you to catch no more than 15 surfperch per day, no more than 10 rockfish per day, and no more than 15 total bottomfish per day.
a. Write and graph a system of inequalities that models the situation. b. Use the graph to determine whether you can catch 11 surfperch and 9 rockfish in one day.
34. MULTIPLE CHOICE A person’s maximum heart rate (in beats per minute) is given by 220 2 x where x is the person’s age in years (20 x 65). When exercising, a person should aim for a heart rate that is at least 70% of the maximum heart rate and at most 85% of the maximum heart rate. Which of the following heart rates is not in the suggested target range for a 40-year-old person who is exercising? A 120 beats per minute B 130 beats per minute C 140 beats per minute D 150 beats per minute 35. SHORT RESPONSE A self-service photo center allows you to make prints of pictures. Each sheet of printed pictures costs $8. The number of pictures that fit on each sheet is shown. a. You want at least 16 pictures of any size, and you are willing to spend up to $48. Write and graph a system of inequalities that models the situation. b. Will you be able to purchase 12 pictures that are 3 inches by 5 inches and 6 pictures that are 4 inches by 6 inches? Explain.
SurfperchSurfperch (^) RockfishRockfish
Four 3 inch by 5 inch pictures fit on one sheet.
Four 3 inch by 5 inch pictures fit on one sheet.
Two 4 inch by 6 inch pictures fit on one sheet.
EXAMPLE 5 on p. 436 for Exs. 31–
35. a. 8 x 1 8 y 48, 4 x 1 2 y 16, x 0, y **0, see margin for art.
- b. Yes, it will cost $48 and give you 18 pictures.**
x 60, y 40, x > 0, y > 0, see margin for art.
14 x 1 2 y < 60, x 1 y < 20, x 0, y 0, see margin for art.
s 15, r 10, s 1 r 15, s > 0, r > 0, see margin for art.
no
A
4UVEZ4USBUFHZ
Exercises 31–35 Remind students
that real-world situations often do
not include negative solutions. If the
situation warrants a restriction to a
certain quadrant, students should
include that inequality when they
write the system of inequalities.
"DBEFNJD7PDBCVMBSZ
Exercises 31, 33, 34 Encourage
students to pay close attention
to the phrases “cannot exceed,”
“no more than,” “at least,” and
“at most.” Point out that all of these
phrases share a common meaning
of “less than or equal to.”
Internet Reference
Exercise 33 For more information
about bottomfish and local fishing
laws, visit Hawaii’s Department of
Aquatic Resources at www.state.
hi.us/dlnr/dar/bottomfish/index.htm
Exercise 34 To learn more about
maximum heart rates, visit the
American Heart Association’s
website at www.americanheart.
org and do a search for “target
heart rates.”
35a.
21
1 x
y
10 x
y
210
x
y
4 24
33a.
25
5 s
r
440 Chapter 7 Systems of Equations and Inequalities
CHALLENGE Write a system of inequalities for the shaded region described.
36. The shaded region is a rectangle with vertices at (2, 1), (2, 4), (6, 4), and (6, 1). 37. The shaded region is a triangle with vertices at ( 2 3, 0), (3, 2), and (0, 2 2). 38. CHALLENGE You make necklaces and keychains to sell at a craft fair. The table shows the time that it takes to make each necklace and keychain, the cost of materials for each necklace and keychain, and the time and money that you can devote to making necklaces and keychains.
Necklace Keychain Available Time to make (hours) 0.5 0.25 20 Cost to make (dollars) 2 3 120
a. Write and graph a system of inequalities for the number x of necklaces and the number y of keychains that you can make under the given constraints. b. Find the vertices (corner points) of the graph. c. You sell each necklace for $10 and each keychain for $8. The revenue R is given by the equation R 5 10 x 1 8 y. Find the revenue for each ordered pair in part (b). Which vertex results in the maximum revenue?
Q UIZ for Lessons 7.8–7.
Graph the inequality. (p. 425)
1. x 1 y 3 2. x < 14 3. 2 y 2 x 8
Graph the system of inequalities. (p. 433)
4. x > 23 5. y 2 6. 4 x y x < 7 y < 6 x 1 2 2 x 1 4 y < 4 7. x > 25 8. y 3 x 2 4 9. x 1 y < 2 x < 0 y x 2 x 1 y > 23 y 2 x 1 7 y 25 x 2 15 y 0
C ALIFORNIA STANDARDS SPIRAL R EVIEW
39. Which equation is equivalent to 2 (3 x 1 5) 1 4 x 5 0? (p. 139) A 2 x 5 2 10 B 2 x 5 10 C 10 x 5 2 10 D 10 x 5 10 40. What is the x -intercept of the graph of y 5 25? (p. 273) A 25 B 0 C 5 D Does not exist 41. The equation 2 x 1 5 y 5 20 models a purchase of $20 for x boxes of dog bones and y bags of dog food. Which ordered pair ( x , y ) does not give a possible combination of boxes of dog bones and bags of dog food? (p. 264) A (0, 4) B (2, 5) C (5, 2) D (10, 0)
Alg. 4.
Alg. 6.
Alg. 7.
x 2, x 6, y 1, y 4
37. y ^1 } 3 x 1 1,
y ^4 } 3 x 2 2,
y 2^2 } 3 x 2 2
0.5 x 1 0.25 y 20, 2 x 1 3 y 120, x 0, y 0, see margin for art. (0, 0), (40, 0), (0, 40), (30, 20)
B
D
B
4–9. See margin.
1–3. See margin.
C
"44&44 "/% 3&5&"$)
%BJMZ)PNFXPSL2VJ[
1. Write a system of inequalities
for the shaded region.
2
x
y
22
x < 2, y > x 1 1
2. A bibliography can refer to at most
8 articles, at most 4 books, and at
most 8 references in all. Write and
graph a system of inequalities that
models the situation. x 5 articles,
y 5 books; x b 8, y b 4, x 1 y b 8,
x r 0, and y r 0
(^1) x
y
1
Online Quiz classzone.com
An alternate quiz for
Lessons 7.8–7.9 is available
online in multiple choice format.
%JBHOPTJT3FNFEJBUJPO
- Practice A, B, C in Chapter 7
Resource Book, pp. 100–
Resource Book, pp. 106–
- Practice Workbook, pp. 108–
- California@HomeTutor
$IBMMFOHF
Additional challenge is available
in the Chapter 7 Resource Book,
p. 110.
2VJ[
An easily readable reduced
copy of the quiz on Lessons
7.8–7.9 (with answers) from
the Assessment Book can be
found on pp. 372I–372L.
38a, Quiz 1–9. See Additional
Answers beginning on p. AA1.
"MH U 8IJDIFRVBUJPOJTFRVJWBMFOUUP 2 2 Y 2 Y 5 D
6 Y 5 2 7 Y 5 8 Y 5 2 9 Y 5
"MH U 8IBUJTUIF Z JOUFSDFQUPGUIFHSBQIPG Y 5 2 D
9 %PFTOPUFYJTU
"MH U 8IJDIPSEFSFEQBJSEPFT OPU TBUJTGZUIFFRVBUJPO Y 2 Z 5 A
45"/%"3% 413"- 3&7&**
