In a previous lesson, students learned to graphically represent the set of solutions to a linear inequality in two variables. They made a connection between the solutions to a linear inequality and the solutions to a related linear equation.
In this lesson, students deepen their understanding of the solutions to linear inequalities by studying them in context. They write inequalities in two variables to represent constraints, and interpret the points on a boundary line and on either side of it in terms of the situation.
The work here illustrates that the solution region represents the set of values that satisfy the constraint in a situation (MP2). Interpreting the solutions contextually also engages students in an aspect of mathematical modeling (MP4). It enables students to see that, while some values might make an inequality true, they might not be feasible or appropriate in the situation. The activity Rethinking Landscaping is an opportunity to make a generalization based on repeated reasoning (MP8), since students first find numbers that meet a constraint, and then use variables in place of those numbers to write an equation and an inequality.
Because reasoning about the solution region of an inequality is important here, graphing technology should not be used. Students will have opportunities to use graphing technology to solve inequalities in two variables in upcoming lessons.
Lesson overview
 22.1 Warmup: Landscaping Options (5 minutes)
 22.2 Activity: Rethinking Landscaping (20 minutes)

22.3 Activity: The Saturday Market (10 minutes)
 Includes "Are you Ready for More?" extension problem
 22.4 Optional Activity: Charity Concerts (15 minutes)
 Lesson Synthesis
 22.5 Cooldown: A Weekend of Games (5 minutes)
Learning goals:
 Find the solution to a twovariable inequality by graphing a related twovariable equation and determining the correct region for the solution.
 Interpret, in context, points on the graphs of equations and in the solution region of inequalities in two variables.
 Write inequalities in two variables to represent the constraints in a situation and identify possible solutions by reasoning.
Learning goals (student facing):
 Let’s write inequalities in two variables and make sense of the solutions by reasoning and by graphing.
Learning targets (student facing):
 Given a twovariable inequality that represents a situation, I can interpret points in the coordinate plane and decide if they are solutions to the inequality.
 I can find the solutions to a twovariable inequality by using the graph of a related twovariable equation.
 I can write inequalities to describe the constraints in a situation.
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.