systems of linear inequalities
systems of linear inequalities
solutions to systems of linear inequalities
linear inequalities
review
100
The school store must sell at least 18 notebooks to qualify for a special discount. Let n represent the number of notebooks sold.
Write an inequality that represents the situation.
n >= 18
100
Graph the system of inequalities and provide 2 possible solutions:
[y > x]
[y < 2x]
Answers vary
100
Without using Desmos
Show a graph for this system of inequalities:
[x >= 0]
[y <= 3]
Parris show capture 6
100
What is the inequality for:
a baker makes a $3 profit on each plain cookie (x) and $7 profit on each decorated cookie (y). She wants to make more than a $130 profit.
3x+7y > 130
100
In the sequence 2, 4, 6, 8, what is the common difference?
d = 2
200
A class has $120 available for a field trip. Transportation costs $30, and admission costs $15 per student. Let s represent the number of students.
Write and solve an inequality to find the maximum number of students who can attend.
30 + 15s ≤ 120 → 15s ≤ 90 → s ≤ 6. Maximum: 6 students.
200
A solid line is used when...?
less than or equal to OR greater than or equal to
200
Determine if the point (2, 2) is a solution to the system of inequalities:
[x + y < 4] [2x - y >= 0]
No
200
Carrie wants to spend at least 3 hours at the beach (X)
Write an inequality to show this situation
x >= 3
200
Given the sequence 3, 6, 9, 12,... what is the next term?
15
300
A school sells adult tickets for $8 and student tickets for $5. A group wants to buy at least 12 tickets while spending no more than $80. Let a represent adult tickets and s represent student tickets.
Write a system of inequalities representing both conditions.
a + s ≥ 12 and 8a + 5s ≤ 80. For a real-world ticket count, a ≥ 0 and s ≥ 0, with whole-number values.
300
Is (0,5) a possible solution to the system of inequalities? Why or why not?
Yes - it falls on a solid line in the double shaded area
300
Explain what it means for a point to be a solution in the context of a system of linear inequalities.
Answers vary: for it to be a solution, it needs to make both inequalities true and fall in the area where both sets of inequalities shading overlaps
300
What is the solution to a system of linear inequalities when both lines have the same slope with opposite shading?
no solution
300
In a relation, if each input is paired with exactly one output, it is called:
a. A function
b. A linear relation
c. An inverse function
d. A non-linear relation
a. a function
400
A student club sells bracelets for $4 each and keychains for $6 each. The club wants to earn at least $120, but members can make no more than 25 items. Let b represent bracelets and k represent keychains.
Write a system of inequalities representing both constraints.
4b + 6k ≥ 120 and b + k ≤ 25. Use nonnegative whole-number values for b and k.
400
Is the point (0,4) part of the solution set?
No - it falls on a dotted line
400
How could you determine if a point is a solution without graphing?
Answers vary: the x and y of the point should satisfy both equations and make them true
400
How can you verify the solution obtained from graphing a system of linear equations?
Answers vary: Plug in the x and the y values. Both equations should be made "true."
400
Which of the relations below is NOT a function?
a. {(1, 3), (2, 5), (3, 3)}
b. {(2, 4), (5, 7), (2, 6)}
c. {(1, 2), (2, 1), (3, 5)}
d. {(4, 7), (7, 5), (2, 9)}
b. {(2, 4), (5, 7), (2, 6)}
500
A food bank prepares fruit boxes for delivery. Each apple box weighs 3 pounds and each orange box weighs 5 pounds. A truck can carry no more than 40 pounds, and the food bank must deliver at least 10 boxes. Let a represent apple boxes and o represent orange boxes. A volunteer proposes delivering 4 apple boxes and 6 orange boxes.
Complete all four challenges below:
1. Write a system of inequalities for the situation.
2. Decide whether the proposed shipment satisfies BOTH constraints. Show your work.
3. Change as few boxes as possible to create a valid shipment.
4. Explain mathematically why your revised shipment works.
a + o ≥ 10 and 3a + 5o ≤ 40. Use nonnegative whole-number values for a and o.
$500 — Sample Reasoning
Original plan: 4 apple boxes and 6 orange boxes.
Box constraint: 4 + 6 = 10, so the minimum-box requirement is satisfied.
Weight constraint: 3(4) + 5(6) = 42 pounds, which exceeds the 40-pound limit.
Minimal revision: Replace one orange box with one apple box, giving 5 apple boxes and 5 orange boxes.
Check: 5 + 5 = 10 boxes, and 3(5) + 5(5) = 40 pounds. Both constraints are satisfied. One replacement is enough because an orange box weighs 2 pounds more than an apple box.
Teacher note: Other valid revised shipments may exist, but the one-box replacement is a minimal change to the proposed shipment.
500
List the two inequalities that make up this system of inequalities.
y <= 5
x > 3
500
What is a/the solution to this set of inequalities?
y >-2x +1
y - 1 > -2x
infinitely many solutions
500
What are the equations for this system of linear equations?
y = 4x + 2
y= -2x+3
500
What is the first term in an arithmetic sequence if the common difference is 5 and the 5th term is 23?
a. 5
b. 8
c. 10
d. 3
d. 3