Modeling with Systems of Inequalities in Two Variables
- Let’s create mathematical models using systems of inequalities.
26.1: A Solution to Which Inequalities?
Is the ordered pair \((5.43,0)\) a solution to all, some, or none of these inequalities? Be prepared to explain your reasoning.
26.2: Custom Trail Mix
Here is the nutrition information for some trail mix ingredients:
|calories per gram (kcal)||protein per gram (g)||sugar per gram (g)||fat per gram (g)||fiber per gram (g)|
Tyler and Jada each designed their own custom trail mix using two of these ingredients. They wrote inequalities and created graphs to represent their constraints.
- \(x+y > 50\)
- \(4.76x + 6.67y \le 400\)
- \(0.67x + 0.07y < 30\)
- \(0.14w + 0.03z > 4\)
- \(0.61w + 0z \le 15\)
Use the inequalities and graphs to answer these questions about each student's trail mix. Be prepared to explain your reasoning.
- Which two ingredients did they choose?
- What do their variables represent?
- What does each constraint mean?
- Which graph represents which constraint?
- Name one possible combination of ingredients for their trail mix.
26.3: Design Your Own Trail Mix
It's time to design your own trail mix!
- Choose two ingredients that you like to eat. (You can choose from the ingredients in the previous activity, or you can look up nutrition information for other ingredients.)
- Think about the constraints for your trail mix. What do you want to be true about its calories, protein, sugar, fat, or fiber?
- Write inequalities to represent your constraints. Then, graph the inequalities.
- Is it possible to make trail mix that meets all your constraints using your ingredients? If not, make changes to your constraints or your ingredients and record them here.
- Write a possible combination of ingredients for your trail mix.
Pause here so your teacher can review your work and give further instructions for displaying your work.
- solutions to a system of inequalities
All pairs of values that make the inequalities in a system true are solutions to the system. The solutions to a system of inequalities can be represented by the points in the region where the graphs of the two inequalities overlap.
- system of inequalities
Two or more inequalities that represent the constraints in the same situation form a system of inequalities.