Math 11 Foundations
Name: ________________________
Chapter 6 Review - Systems of Linear Inequalities
Short Answer
1. Graph the solution set for the linear inequality 2y – 6x < –10
.
2. Graph the system of linear inequalities:
{(x, y) | x + y ≤ 2, x > -1, x ∈ I, y ∈ I}
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3. The following model represents an optimization problem.
Restrictions:
x∈W
y∈W
Constraints:
x>0
y>0
2x ≥ y - 5
x + y ≤ 11
Objective function:
A = 2x + 3y
a)Graph the solution.
b) Determine the maximum solution.
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Problem
4. Gordon’s favourite activities are going to the movies and skating with friends. He budgets himself no more
than $160 a month for entertainment and transportation. Movie admission is $12 per movie, and skating costs
$10 each time. A student bus pass for the month costs $40.
a) Define the variables and write one linear inequality to represent the situation.
b) Graph the linear inequality.
c) Use your graph to determine:
i) a combination of activities that Gordon can afford with no money left over
ii) a combination of activities that will exceed his budget
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5. Science World has categorized its demonstrations as physics and chemistry.
• There are no more than 80 demonstrations altogether.
• No more than 60 of the demos are physics, and no less than 32 are chemistry.
• A ticket to any physics demo costs $10, and a ticket to any chemistry demo costs $12.
a) Create a model of this problem.
b) What combinations of physics and chemistry demos would maximize revenue?
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6. Extra graph paper (will not be marked).
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ID: A
Chapter 6 Review - Systems of Linear Inequalities
Answer Section
SHORT ANSWER
1. ANS:
PTS: 1
DIF: Grade 11
REF: Lesson 6.1
OBJ: 1.2 Graph the boundary line between two half planes for each inequality in a system of linear
inequalities, and justify the choice of solid or dashed lines. | 1.3 Determine and explain the solution region
that satisfies a linear inequality, using a test point when given a boundary line.
TOP: Graphing linear inequalities in two variables
KEY: linear inequality | solution set
2. ANS:
PTS: 1
DIF: Grade 11
REF: Lesson 6.2
OBJ: 1.4 Determine, graphically, the solution region for a system of linear inequalities, and verify the
solution. | 1.5 Explain, using examples, the significance of the shaded region in the graphical solution of a
system of linear inequalities.
TOP: Exploring graphs of systems of linear inequalities
KEY: systems of linear inequalities
1
ID: A
3. ANS:
(5, 20)
PTS:
OBJ:
TOP:
KEY:
1
DIF: Grade 11
REF: Lesson 6.6
1.6 Solve an optimization problem, using linear programming.
Optimization problems III: linear programming
optimization problem | linear programming | systems of linear inequalities | objective function
PROBLEM
4. ANS:
a) Let x represent the number of movies Gordon sees.
Let y represent the number of times Gordon goes skating.
{(x, y) | 12x + 10y + 50 ≤ 180, x ∈ W, y ∈ W}
b) Graph the line 12x + 10y = 130.
130
y-intercept: y = 13
x-intercept: x =
12
Since (0, 0) is in the solution set, the solution set is all points to the left of the line.
i) e.g., see 6 movies and go skating 7 times
12(5) + 10(7) + 50 = 180
ii) e.g., see 6 movies and go skating 6 times
12(6) + 10(6) + 50 = 182 > 180
PTS: 1
DIF: Grade 11
REF: Lesson 6.1
OBJ: 1.2 Graph the boundary line between two half planes for each inequality in a system of linear
inequalities, and justify the choice of solid or dashed lines. | 1.3 Determine and explain the solution region
that satisfies a linear inequality, using a test point when given a boundary line.
TOP: Graphing linear inequalities in two variables
KEY: linear inequality | solution set
2
ID: A
5. ANS:
Let x represent the number of herbivore exhibits.
Let y represent the number of carnivore exhibits.
Let R represent the revenue.
x ∈ W, y ∈ W
x ≤ 60
y ≤ 32
x + y ≤ 80
Objective function to maximize:
R = 10x + 12y
Graph the line x = 60 and shade the region between it and the y-axis.
Graph the line y = 32 and shade the region between it and the x-axis.
Graph the line x + y = 40 and shade the region below it and bounded by the axes.
The feasible region is all the whole number points in the overlapping area and its boundaries.
PTS:
OBJ:
TOP:
KEY:
1
DIF: Grade 11
REF: Lesson 6.4
1.1 Model a problem, using a system of linear inequalities in two variables.
Optimization problems I: creating the model
optimization problem | constraint | objective function
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ID: A
6. ANS:
Let x represent the number of women’s appointments.
Let y represent the number of men’s appointments.
Let T represent the total time.
Restrictions:
x ∈ W, y ∈ W
Constraints:
x ≥ 3y
x + y ≥ 120
Objective function to minimize:
E = 60x + 25y
Use technology to graph the lines and find the intersection points of the solution area.
The intersection points are (120, 0) and (90, 30).
The minimum occurs when x is minimized.
The minimum is at point (90, 30) and represents 90 women’s appointments and 30 men’s appointments.
E = 90(60) + 30(25)
E = 6150
The minimum amount of time is 6150 h.
PTS: 1
DIF: Grade 11
REF: Lesson 6.6
OBJ: 1.1 Model a problem, using a system of linear inequalities in two variables. | 1.6 Solve an optimization
problem, using linear programming.
TOP: Optimization problems III: linear programming
KEY: optimization problem | linear programming | systems of linear inequalities | objective function
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