# Lesson 26

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.

\(x>0\)

\(y>0\)

\(x\ge0\)

\(y\ge0\)

### 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) | |
---|---|---|---|---|---|

peanuts | 5.36 | 0.21 | 0.04 | 0.46 | 0.07 |

almonds | 5.71 | 0.18 | 0.21 | 0.46 | 0.07 |

raisins | 3.00 | 0.03 | 0.60 | 0.00 | 0.05 |

chocolate pieces | 4.76 | 0.05 | 0.67 | 0.19 | 0.02 |

shredded coconut | 6.67 | 0.07 | 0.07 | 0.67 | 0.13 |

sunflower seeds | 5.50 | 0.20 | 0.03 | 0.47 | 0.10 |

dried cherries | 3.25 | 0.03 | 0.68 | 0.00 | 0.03 |

walnuts | 6.43 | 0.14 | 0.04 | 0.61 | 0.07 |

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.

Tyler

- \(x+y > 50\)
- \(4.76x + 6.67y \le 400\)
- \(0.67x + 0.07y < 30\)
- \(x>0\)
- \(y>0\)

Jada

- \(w+z>50\)
- \(0.14w + 0.03z > 4\)
- \(0.61w + 0z \le 15\)
- \(w>0\)
- \(z>0\)

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.

### Summary

### Glossary Entries

**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.