Lesson 23

In this warm-up, students use graphing technology to graph simple linear inequalities in two variables. They practice adjusting the graphing window until the solution regions become visible and give useful information. Later in the lesson, students will write inequalities that represent constraints in different situations and find the solution sets. The exercises here prepare students to do the latter using graphing technology.

Give students access to graphing technology. If using Desmos:

• Explain to students that typing "\(< =\) " gives the \(\le\) symbol and typing "\(> =\)" gives the \(\ge\) symbol.
• Remind students that the \(+\) and \(-\) buttons can be used to zoom in and out of the graphing window, and that the wrench button in the upper right corner can be used to set the graphing window precisely.

If using other graphing technology available in your classroom:

• Demonstrate how to enter the \(\leq\) and \(\geq\) symbols.
• Remind students how to set a useful graphing window by zooming in or out, and how to set a precise graphing window by specifying the horizontal and vertical boundaries.
• (For technology that takes only equations or inequalities in slope-intercept form:) Remind students that some inequalities might need to be rewritten such that \(y\) is isolated before the inequality can be entered into the graphing tool.

Display the correct solution regions for all to see and ask students to check their graphs. Discuss any challenges students may have come across when trying to graph using technology. 

Explain to students that they will now use graphing technology to find solutions to some inequalities that represent constraints in situations.

This activity enables students to integrate several ideas and skills from the past few lessons. Students write inequalities in two variables to represent constraints in situations, use technology to graph the solutions, interpret points in the solution regions, and use the inequalities and the graphs to answer contextual questions. In doing so, they engage in aspects of mathematical modeling (MP4).

The questions in this activity are written in pairs. The same constraints and contexts will be used in an upcoming lesson on systems of linear inequalities in two variables.

Decide on the structure for the activity depending on the time available and the amount of practice each student needs. Here are some possibilities:

• Assigning all three pairs of questions to all students.
• Assigning each student a pair of questions, arranging students who work on different pairs in groups of 3, and asking them to explain their solutions to one another.
• Arranging students in groups of 3, assigning the same pair of questions to each group, and—for each pair of questions—asking one group to present the solutions to the class.

Continue to provide access to graphing technology. Explain that students will now write and graph inequalities to solve problems about some situations.

Assign at least one pair of questions about the same context to each student. See Activity Narrative for some possible ways to structure the activity.

Some students might not be familiar with terms such as "savings," "checking," or "premium." Explain any unfamiliar terms as needed.

One inequality in the bank account context involves only one variable. Some students might think that all inequalities they write must include two variables. Reassure them that this might not always be the case. Consider pointing to examples from earlier activities in which they wrote or graphed inequalities such as \(y>2\) or \(b<10\).

Select one student or one group to present their solutions for each pair of questions. Focus the discussion on how students used the inequalities and graphs to help answer the question about each situation.

Students who worked on the same questions might end up with different graphs and answers because they wrote different inequalities (which might not correctly represent the constraints in the situation). If this happens, analyze the different inequalities and look for the potential causes for the discrepancy. One possibility is that the wrong symbols or the wrong numbers were entered into the graphing tool.

Another possibility might be that students made different decisions about the quantities being assigned to the vertical and horizontal axes. In that case, both versions of the graphs might be correct, but the answers to questions might be different if one of the graphs is not interpreted correctly.

This optional activity allows students to practice interpreting inequalities in context and reasoning about their solutions graphically and numerically. A sorting and matching task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

As students explain their thinking to a partner, encourage them to use precise language and mathematical terms to refine their explanations (MP6).

Here are images of the sorted cards for reference and planning.

Arrange students in groups of 2. Distribute one set of pre-cut slips or cards to each group.

Ask students to take turns matching a group of 4 cards with representations of the same situation and explaining how they know the representations belong together. Emphasize that while one partner explains, the other should listen carefully, and the group should discuss any disagreements.

Select groups of students to share their results and explain their rationales. After a group explains why they believe a set of cards belong together, ask if other groups reasoned about the matches the same way or if they approached the matching differently.

Attend to the language that students use in their explanations by giving them opportunities to describe the inequalities, graphs, or solutions more precisely.