Solving Problems with Systems of Linear Inequalities in Two Variables
Integrated Math 1
8.1
Warm-up
Which three go together? Why do they go together?
A
Graph of two intersecting lines, origin O, no grid. Axes from negative 10 to 10 by 5’s. Blue horizontal line passes through 0 comma 7. Diagonal line passes through negative 7 comma 7, 0 comma 0, 5 comma negative 5.
B
Graph of two inequalities, no grid, origin O. Each axis from negative 10 to 10, by 5's. The first dashed line passes through negative 10 comma 10, 0 comma 0, and 5 comma negative 5. The region below the dashed line is shaded. Second dashed line passes through 0 comma 6, and 6 comma 0. The region above the dashed line is shaded.
C
Graph of two inequalities, no grid, origin O. Each axis from negative 10 to 10, by 5's. The first solid line passes through negative 10 comma 6, 0 comma 6, 5 comma 6. The region above the line is shaded. The second solid line passes through 6 comma 6, 6 comma 6, and 6 comma negative 10. The region to the right of the line is shaded.
D
8.2
Activity
Here are the graphs of the inequalities in this system:
Decide whether each point is a solution to the system. Be prepared to explain how you know.
A graph of two intersecting inequalities on a coordinate plane, origin O. Each axis from negative 10 to 5, by 2’s. The first dashed line starts below x axis and right of y axis, goes through negative 8 comma negative 8, 0 comma 0, and 8 comma 8. The region above the dashed line is shaded. Second line starts on negative 8 comma 10, goes through negative 4 comma 2, and ends on 2 comma negative 10. The region above the dashed line is shaded.
8.3
Activity
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card, and think about what information you need to answer the question.
Ask your partner for the specific information that you need. “Can you tell me ?”
Explain to your partner how you are using the information to solve the problem. “I need to know because . . . .”
Continue to ask questions until you have enough information to solve the problem.
Once you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card. Wait for your partner to ask for information.
Before telling your partner any information, ask, “Why do you need to know ?”
Listen to your partner’s reasoning and ask clarifying questions. Give only information that is on your card. Do not figure out anything for your partner!
These steps may be repeated.
Once your partner has enough information to solve the problem, read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
The blank coordinate planes are provided here in case they are useful.
Student Lesson Summary
A family has at most \$25 to spend on activities at Fun Zone. It costs \$10 an hour to use the trampolines and \$5 an hour to use the pool. The family can stay less than 4 hours.
What are some combinations of trampoline time and pool time that the family could choose given their constraints?
We could find some combinations by trial and error, but writing a system of inequalities and graphing the solution would allow us to see all the possible combinations.
Let represent the time, in hours, on the trampolines and represent the time, in hours, in the pool.
The constraints can be represented with the system of inequalities:
Here are graphs of the inequalities in the system.
The solution set to the system is represented by the region where shaded parts of the two graphs overlap. Any point in that region is a pair of times that meets both the time and budget constraints.
The graphs give us a complete picture of the possible solutions.
Two inequalities graphed on a coordinate plane, origin O, scale from 0 to 6 on both axes. Horizontal axis, hours on trampoline. Vertical axis, hours in pool. The dashed line starts on vertical axis at 4, goes through 1 comma 3 and ends on horizontal axis at 4. The region below the dashed line is shaded. The solid line starts on the vertical axis at 5, goes through 1 comma 3 and ends on the horizontal axis at 2 point 5. The region below the solid line is shaded.
Can the family spend 1 hour on the trampolines and 3 hours in the pool?
No. We can reason that it is because those times add up to 4 hours, and the family wants to spend less than 4 hours. But we can also see that the point lies on the dashed line of one graph, so it is not a solution.
Can the family spend 2 hours on the trampolines and 1.5 hours in the pool?
No. We know that these two times add up to less than 4 hours, but to find out the cost, we need to calculate , which is 27.5 and is more than the budget.
It may be easier to know that this combination is not an option by noticing that the point is in the region with line shading, but not in the region with solid shading. This means it meets one constraint but not the other.
