Lesson 26

Modeling with Systems of Inequalities in Two Variables

26.1: A Solution to Which Inequalities? (5 minutes)


This warm-up gives students a quick exposure to the inequalities \(x>0\), \(x\geq 0\), \(y>0\), and \(y\ge0\), so that they are prepared to deal with them later in this lesson. It also reinforces the idea of thinking carefully about whether the points on the boundary lines of a solution region are included in the solution set.

Student Facing

Is the ordered pair \((5.43,0)\) a solution to all, some, or none of these inequalities? Be prepared to explain your reasoning.





Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Invite students to share their responses. Then, display a blank four-quadrant coordinate plane for all to see.

Blank x y coordinate plane.

Ask students:

  • "If we were to graph the solutions to \(x>0\), what would the region look like?" (We would shade the right side of \(y\)-axis.) "Is the \(y\)-axis included in the solution region?" (No)
  • "What about the graph of the solutions to \(y>0\)?" (We would shade the upper side of the \(x\)-axis). "Is the \(x\)-axis included in the solution region?" (No)
  • "What about the graph of the solutions to the system \(x>0\) and \(y>0\)?" (The solution region would be the upper-right section of the graph, where the other two regions overlap.)

Remind students that this upper-right region of the coordinate plane is called the first quadrant.

26.2: Custom Trail Mix (20 minutes)


In this activity, students use their insights from the unit to analyze and interpret a set of mathematical models and a set of data in context. Each situation involves more than two constraints, and can therefore be represented with a system with more than two inequalities.

Interpreting and connecting the inequalities, the graphs, and the data set (which involves decimals) prompts students to make sense of problems and persevere in solving them (MP1), and to reason quantitatively and abstractly (MP2).


Give students a moment to skim through the task statement and familiarize themselves with the given information. Ask them to be prepared to share one thing they notice and one thing they wonder. Invite students to share their observations and questions.

Then, to help students interpret the variables in the given inequalities as representing the number of grams of each ingredient, ask them to use the table to write an expression to represent the total amount of fiber if they had \(a\) grams of almonds and \(b\) grams of raisins. Students should see that the expression is \(0.07a + 0.05b\).

Next, ask for an expression representing the total amount of sugar for the same amounts of almonds and raisins (\(0.21a+0.60b\)).

Arrange students in groups of 2. Ask them to analyze and answer the questions about one student's trail mix (either Tyler's or Jada's). If time permits, the groups could analyze the other trail mix.

Give students a few minutes of quiet work time and time to share their thinking with their partner. Follow with a whole-class discussion.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, highlight values from the table, equations, and graphs in corresponding colors as they discover the connections. Encourage students to label regions of the graph and label variables in addition to color coding to reinforce connections.
Supports accessibility for: Visual-spatial processing

Student Facing

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.


  • \(x+y > 50\)
  • \(4.76x + 6.67y \le 400\)
  • \(0.67x + 0.07y < 30\)
  • \(x>0\)
  • \(y>0\)
A graph of three intersecting inequalities.


  • \(w+z>50\)
  • \(0.14w + 0.03z > 4\)
  • \(0.61w + 0z \le 15\)
  • \(w>0\)
  • \(z>0\)
Graph of three intersecting inequalities.

Use the inequalities and graphs to answer these questions about each student's trail mix. Be prepared to explain your reasoning.

  1. Which two ingredients did they choose?
  2. What do their variables represent?
  3. What does each constraint mean?
  4. Which graph represents which constraint?
  5. Name one possible combination of ingredients for their trail mix.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Focus the discussion on the connections between the graphs and the inequalities, and on the inequalities \(x>0\) and \(y>0\). Ask questions such as:

  • “How did you know which ingredients each person used?” (By matching the coefficients in two of the inequalities to the nutritional values in the table.)
  • “The table shows the same values for some nutrients. How can you tell which one Tyler or Jada chose?” (The coefficients of \(x\) and \(y\) in one inequality and those in the other inequality must be for the same two ingredients.)
  • “Why do you think Jada and Tyler both included the inequalities \(x>0\) and \(y>0\)?” (There cannot be only one ingredient, so both \(x\) and \(y\) must be greater than 0.)
  • “How do those inequalities affect the graph of the solution region?” (They limit the solution region to the first quadrant.)
  • “Jada and Tyler each wrote five inequalities. Could all five form a single system?” (Yes) What does it mean to have a system with five inequalities?" (There are five constraints that must be met. The solutions to the system satisfy all five constraints simultaneously.)
Conversing: MLR2 Collect and Display. As students discuss their strategies, listen for and collect the language students use to identify and describe the boundary line or shading in their inequalities. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

26.3: Design Your Own Trail Mix (20 minutes)


This activity is designed to give students opportunities to use their understandings from this unit to perform mathematical modeling.

The trail mix context is familiar from the previous activity, but students are challenged to choose quantities, determine how to represent them, interpret and reason about them, and use the model they create to make choices. It also enables students to reflect on their model and revise it as needed (MP4).

Students are likely to want to use graphing technology, as the nutritional information involves decimals and the inequalities written would be inconvenient to graph by hand. This is an opportunity for students to choose tools strategically (MP5).

No time estimate is given here because the time would depend on decisions about the research students do and on the expectations for collaboration and presentation.


Arrange students in groups of 2–4. Provide access to Desmos or other graphing technology.

Explain the expectations for researching nutritional values, for collaboration with group members, and for presentation of student work. (If each group is presenting one response, provide each group with tools for creating a visual display. If each student is presenting a response, give each student tools for creating a visual display.)

Engagement: Develop Effort and Persistence. Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For each question, allot a specific amount of work time and display expectations. Display a countdown timer, along with a bulleted list of what teams or individuals are to produce to complete a given step. For instance, during question 1, display a 5 minute countdown timer and the list: “You’re finished if you. . . (1) Have your two ingredients (2) Have the nutrition information ready.” 
Supports accessibility for: Attention; Social-emotional skills

Student Facing

It's time to design your own trail mix!

  1. 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.)
  2. Think about the constraints for your trail mix. What do you want to be true about its calories, protein, sugar, fat, or fiber?
  3. Write inequalities to represent your constraints. Then, graph the inequalities.


    Blank coordinate plane with grid, origin O.
  4. 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.
  5. 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.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:

  • What constraints did every group use?
  • How do the graphs of the various mixes compare?
  • Did anyone have to revise or change their model in order to come up with a solution they could use?
  • How did you use the graph to choose a recipe for your mix?
Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite groups to create a visual display of their work. Students should consider what types of details (annotations, notes, diagrams, arrows, etc.) to include on their displays that will help communicate their reasoning. Begin the whole-class discussion by selecting and arranging 2–4 displays for all to see. Give students 2–3 minutes of quiet think time to interpret the displays before inviting the authors to present their work.
Design Principle(s): Optimize output; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Allow enough time for students to present their trail mix recipes. Consider a gallery walk as a way for students to share their display and to ask and answer questions.