This Math Talk focuses on substituting values from an ordered pair into an expression and comparing its value to a given number. It encourages students to think about the meaning of ordered pairs and inequalities and to rely on the structure of ordered pairs to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students test values in a region of the coordinate plane to determine if they are in the solution region of given inequality.

To correctly substitute into the expression, students need to look for and make use of structure (MP7).

To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to ’s strategy?”
• “Do you agree or disagree? Why?”
• “What connections to previous problems do you see?”

To help students recall the meaning of a solution to an inequality, ask: "Which pairs, if any, are solutions to the inequality ?" Make sure that students recognize that both and are solutions because they make the inequality true.

Earlier in the unit, students saw that a linear inequality in one variable has many solutions, which are represented by all the points on one side of a number on a number line. They also recalled that a linear equation in two variables has many solutions, which are represented by all the points on the graph of the equation. The work in this activity builds on those understandings.

Students see that the solutions to a linear inequality in two variables can be represented by many points on a coordinate plane, and that the set of points are in a region bounded by a line. To represent the set of all points that are solutions, we can shade that region. Students also see that the points that fall on the other side of the boundary line are not solutions to the inequality.

During the activity, students are not yet expected to recognize that the line is the graph of an equation related to the inequality (although some students may notice that). That insight is made explicit in the Activity Synthesis and reinforced in the next activity.

In the preceding activity, students looked at the regions that represent solutions and non-solutions to inequalities. They recognized that the boundary between the two regions is the graph of an equation that is related to the inequality. Students did not, however, look closely at whether the boundary line itself is a part of the solution. That investigation is the focus of this activity.

Students reason with algebraic and graphical representations of inequalities in two directions. They first graph the solutions to given inequalities, and later write inequalities whose solutions could be represented by given graphs.

If many students get stuck on graphing or writing inequalities, consider moving fairly quickly through the activity and using the discussion questions in the Lesson Synthesis to help students gain clarity and focus.