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3-2 Solving Systems of Inequalities by Graphing
Solve each system of inequalities by graphing.
2. SOLUTION: The graph of the system of inequalities is
4. CCSS REASONING The most Kala can spend on hot dogs and buns for her cookout is $35. A package of 10 hot
dogs costs $3.50. A package of buns costs $2.50 and contains 8 buns. She needs to buy at least 40 hot dogs and 40
buns.
a. Graph the region that shows how many packages of each item she can purchase.
b. Give an example of three different purchases she can make.
SOLUTION: a. Let h be the number of package of hotdogs.
Let b be the number of package of buns.
The inequalities that represents the situation are
The graph of the region that shows the number of packages of each item Kala can purchase is
b. Sample answer: 4 packages of hotdogs, 5 packages of buns; 5 packages of hotdogs, 6 packages of buns; 6
packages of hotdogs, 5 packages of buns.
Find the coordinates of the vertices of the triangle formed by each system of inequalities.
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SOLUTION: The graph of the system of inequalities is
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b. Sample
3-2 Solving
Systems
by Graphing
answer:of4Inequalities
packages of hotdogs,
5 packages of buns; 5 packages of hotdogs, 6 packages of buns; 6
packages of hotdogs, 5 packages of buns.
Find the coordinates of the vertices of the triangle formed by each system of inequalities.
6. SOLUTION: The graph of the system of inequalities is
.
The coordinate (–4, 4) and (2, –8) can be determined from the graph.
Solve the system of equations
Substitute 13x – 34 for y in the equation
Substitute 3 for x in the equation
and solve for x.
and find y.
So, the coordinate is (3, 5).
The vertices of the triangle are at (–4, 4), (2, –8) and (3, 5).
Solve each system of inequalities by graphing.
8. SOLUTION: Graph the system of inequalities in a coordinate plane.
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.
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So, the coordinate is (3, 5).
3-2 Solving
Systems
Inequalities
by Graphing
The vertices
of theoftriangle
are at (–4,
4), (2, –8) and (3, 5).
Solve each system of inequalities by graphing.
8. SOLUTION: Graph the system of inequalities in a coordinate plane.
.
10. SOLUTION: Graph the system of inequalities in a coordinate plane.
.
12. SOLUTION: Graph the system of inequalities in a coordinate plane.
.
14. SOLUTION: Graph the system of inequalities in a coordinate plane.
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3-2 Solving Systems of Inequalities by Graphing
.
14. SOLUTION: Graph the system of inequalities in a coordinate plane.
.
16. RECORDING Jane’s band wants to spend no more than $575 recording their first CD. The studio charges at least
$35 an hour to record. Graph a system of inequalities to represent this situation.
SOLUTION: Let y be the cost to record the CD.
Jane’s band wants to spend no more than $575.
So:
Let x represent the time taken to record CD in hours.
The studio charges at least $35 an hour.
So:
.
Graph the system of inequalities.
Find the coordinates of the vertices of the triangle formed by each system of inequalities.
18. SOLUTION: Graph the system of inequalities.
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3-2 Solving Systems of Inequalities by Graphing
Find the coordinates of the vertices of the triangle formed by each system of inequalities.
18. SOLUTION: Graph the system of inequalities.
The coordinates of the vertices are (0, 2),
(4, 0), and (0, 0).
20. SOLUTION: Graph the system of inequalities.
The coordinates of the vertices are (1, 9),
(4, 0), and (4, 9).
22. SOLUTION: Graph the system of inequalities.
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The coordinates of the vertices are (–6, –5),
(–2, 4.5), and (7.5, –2).
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The coordinates
vertices areby
(1,Graphing
9),
3-2 Solving
Systemsofofthe
Inequalities
(4, 0), and (4, 9).
22. SOLUTION: Graph the system of inequalities.
The coordinates of the vertices are (–6, –5),
(–2, 4.5), and (7.5, –2).
24. BAKING Rebecca wants to bake cookies and cupcakes for a bake sale. She can bake 15 cookies at a time and 12
cupcakes at a time. She needs to make at least 120 baked goods, but no more than 360, and she wants to have at
least three times as many cookies as cupcakes. What combination of batches of each could Rebecca make?
SOLUTION: Let x be the number of cookies and y be the number of cupcakes.
So:
And:
Graph the inequalities.
The dark shaded region is the region where all the inequalities are true.
Pick a point in this region. Since we need to have whole numbers of batches, choose an x-coordinate which is a
multiple of 15 and a y-coordinate which is a multiple of 12.
Sample answer: (225, 72).
To make 225 cookies takes
batches, and to make 72 cupcakes takes batches.
26. TREES Trees are divided into four categories according to height and trunk circumference. In one forest, the trees
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are categorized by the heights and circumferences described in the table.
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Pick a point in this region. Since we need to have whole numbers of batches, choose an x-coordinate which is a
multiple of 15 and a y-coordinate which is a multiple of 12.
Sample answer: (225, 72).
3-2 Solving
of Inequalities
by batches, and to make 72 cupcakes takes Graphing
To makeSystems
225 cookies
takes
batches.
26. TREES Trees are divided into four categories according to height and trunk circumference. In one forest, the trees
are categorized by the heights and circumferences described in the table.
a. Write and graph the system of inequalities that represents the range of heights h and circumferences c for a codominant tree.
b. Determine the crown class of a basswood that is 48 feet tall. Find the expected trunk circumference
SOLUTION: a. The height of co-dominant tree ranges from 56 to 72.
So:
The trunk circumference of the co-dominant tree ranges from 48 to 60.
Therefore:
.
b. 48 lies between 40 and 55. So, the class is intermediate and the expected trunk circumference is 34 – 48 in.
Solve each system of inequalities by graphing.
28. SOLUTION: Graph the system of inequalities.
30. SOLUTION: Graph the system of inequalities.
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3-2 Solving Systems of Inequalities by Graphing
32. SOLUTION: Graph the system of inequalities.
34. SOLUTION: Graph the system of inequalities.
36. SOLUTION: Graph the system of inequalities.
Since the inequalities have no shaded region in common, the system has no solution.
38. JOBS Louie has two jobs and can work no more than 25 total hours per week. He wants to earn at least $150 per
week. Graph the inequalities to show possible combinations of hours worked at each job that will help him reach his
goal.
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3-2 Solving Systems of Inequalities by Graphing
Since the inequalities have no shaded region in common, the system has no solution.
38. JOBS Louie has two jobs and can work no more than 25 total hours per week. He wants to earn at least $150 per
week. Graph the inequalities to show possible combinations of hours worked at each job that will help him reach his
goal.
SOLUTION: Let x represent the number of hours that Louie works as a Busboy and y represent the number of hours he works as
a clerk.
So:
and Graph the system of inequalities.
Find the coordinates of the vertices of the figure formed by each system of inequalities.
40. SOLUTION: Graph the inequalities.
The Manual
coordinates
of by
theCognero
vertices
eSolutions
- Powered
are (0, 2),
,
, and (2.8, –6.4)
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3-2 Solving Systems of Inequalities by Graphing
Find the coordinates of the vertices of the figure formed by each system of inequalities.
40. SOLUTION: Graph the inequalities.
The coordinates of the vertices are (0, 2),
,
, and (2.8, –6.4)
42. SOLUTION: Graph the inequalities.
The coordinates of the vertices are (–4, 6),
(–3, 8), (4.8, –7.6), and
.
44. DODGEBALL A high school is selecting a dodgeball team to play in a fund-raising exhibition against their rival.
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There can be between 10 and 15 players on the team and there must be more girls than boys on the team.
a. Write and graph a system of inequalities to represent the situation.
b. List all of the possible combinations of boys and girls for the team.
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The coordinates of the vertices are (–4, 6),
(–3, 8), (4.8, –7.6), and
.
3-2 Solving Systems of Inequalities by Graphing
44. DODGEBALL A high school is selecting a dodgeball team to play in a fund-raising exhibition against their rival.
There can be between 10 and 15 players on the team and there must be more girls than boys on the team.
a. Write and graph a system of inequalities to represent the situation.
b. List all of the possible combinations of boys and girls for the team.
c. Explain why there is not an infinite number of possibilities.
SOLUTION: a. Let g be the number of girls and b be the number of boys in the team.
Graph the inequalities in the same coordinate plane.
b. (6, 4), (6, 5), (7, 3), (7, 4), (7, 5), (7, 6), (8, 2), (8, 3), (8, 4), (8, 5), (8, 6), (8, 7), (9, 1), (9, 2), (9, 3), (9, 4), (9,
5), (9, 6), (10, 0), (10, 1), (10, 2), (10, 3), (10, 4), (10, 5), (11, 0), (11, 1), (11, 2), (11, 3), (11, 4), (12, 0), (12, 1),
(12, 2), (12, 3), (13, 0), (13, 1), (13, 2), (14, 0), (14, 1), (15, 0)
c. Sample answer: You cannot have a fraction of a person.
46. OPEN ENDED Write a system of two inequalities in which the solution:
a. lies only in the third quadrant.
b. does not exist.
c. lies only on a line.
d. lies on exactly one point.
SOLUTION: a. Sample answer:
b. Sample answer:
y > 2, y < –2
c. Sample answer:
d. Sample answer:
The solution is (0, 0).
48. CCSS ARGUMENTS Determine whether the following statement is true or false . If false, give a counterexample.
A system of two linear inequalities has either no points or infinitely many points in its solution.
SOLUTION: True
50. WRITING IN MATH Explain how you would test to see whether (–4, 6) is a solution of a system of inequalities.
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SOLUTION: Page 11
Sample answer: Determine whether the point falls in the shaded area of the graphs and/or determine whether the
values satisfy each inequality.
48. CCSS ARGUMENTS Determine whether the following statement is true or false . If false, give a counterexample.
A system of two linear inequalities has either no points or infinitely many points in its solution.
SOLUTION: 3-2 Solving
Systems of Inequalities by Graphing
True
50. WRITING IN MATH Explain how you would test to see whether (–4, 6) is a solution of a system of inequalities.
SOLUTION: Sample answer: Determine whether the point falls in the shaded area of the graphs and/or determine whether the
values satisfy each inequality.
52. ACT/SAT The table at the right shows a relationship between x and y. Which equation represents this relationship?
F
G
H
J K
SOLUTION: Each output is two more than three times the input.
The correct choice is G.
54. GEOMETRY Look at the graph below. Which of
these statements describes the relationship between the two lines?
A They intersect at (6, 2).
B They intersect at (0, 2).
C They intersect at (3.5, 0).
D They intersect at (2, 6).
SOLUTION: The lines intersect at the point (2, 6).
The correct choice is D.
Graph each inequality.
56. SOLUTION: The boundary of the graph is the graph of x + y = 6. Since the inequality symbol is ≥, the boundary is solid.
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Test the inequality with the point (0, 0).
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D They intersect at (2, 6).
SOLUTION: The linesSystems
intersectofatInequalities
the point (2, 6).
3-2 Solving
by Graphing
The correct choice is D.
Graph each inequality.
56. SOLUTION: The boundary of the graph is the graph of x + y = 6. Since the inequality symbol is ≥, the boundary is solid.
Test the inequality with the point (0, 0).
The region that does not contain (0, 0) is shaded.
58. SOLUTION: The boundary of the graph is the graph of 5x + 7y = –20. Since the inequality symbol is ≥, the boundary is solid.
Test the inequality with the point (0, 0).
The region that contains (0, 0) is shaded.
Graph each function. Identify the domain and range.
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SOLUTION: Page 13
3-2 Solving Systems of Inequalities by Graphing
Graph each function. Identify the domain and range.
60. SOLUTION: D = {all real numbers},
R = {all real numbers},
62. BOOK CLUB For each meeting of the Putnam High School book club, $25 is taken from the activities account to
buy snacks and materials. After their sixth meeting, there will be $350 left in the activities account.
a. If no money is put back into the account, what equation can be used to show how much money is left in the
activities account after having x number of meetings?
b. How much money was originally in the account?
c. After how many meetings will there be no money left in the activities account?
SOLUTION: a. The amount spent for 6 meetings is $150.
The amount remaining in the account after 6 meetings is $350. Therefore, the initial amount in the account is $500.
Let y be the amount remaining in the account after x meetings.
So:
y = 500 – 25x.
b. $500.
c. Replace y with 0 and find x.
After 20 meetings, there will be no money left in the account.
Find each value if f(x) = 2x + 5 and g(x) = 3x – 4.
64. SOLUTION: Replace x with –2.
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SOLUTION: Page 14
Replace x with –2.
3-2 Solving Systems of Inequalities by Graphing
66. SOLUTION: Replace x with –0.5.
68. SOLUTION: Replace x with –0.75.
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