In earlier lessons, students wrote and solved linear inequalities in one variable. In this lesson, they transition to linear inequalities in two variables.
Previously, students learned that the solutions to an equation in two variables are all pairs of values that make the equation true, and that, when graphed, the solutions are points on a line. Here, they learn that the solutions to inequalities in two variables also involve pairs of values. When graphed, the solutions are no longer points on a single line, but comprise a region that is bounded by a line. This region consists of all points in a plane on one side of a boundary line. That boundary line is the graph of an equation related to the inequality.
Students begin by noticing that the plots of solutions and nonsolutions occupy different parts of a coordinate plane. Next, they think about the boundary line between the two regions and whether it is a part of the solution. Finally, students write some inequalities given graphs that represent solution regions.
Lesson overview
 21.1 Warmup: Math Talk: Less Than, Equal to, or More Than 12? (5 minutes)
 21.2 Activity: Solutions and Not Solutions (20 minutes)

21.3 Activity: Sketching Solutions to Inequalities (10 minutes)
 Includes "Are you Ready for More?" extension problem
 Lesson Synthesis
 21.4 Cooldown: Pick a Graph (5 minutes)
Learning goals:
 Given the graph of a related equation, determine the solution region to an inequality in two variables by testing the points on the line and on either side of the line.
 Understand that the solutions to a linear inequality in two variables are represented graphically as a halfplane bounded by a line.
Learning goals (student facing):
 Let’s find out how to use graphs to represent solutions to inequalities in two variables.
Learning targets (student facing):
 Given a twovariable inequality and the graph of the related equation, I can determine which side of the line the solutions to the inequality will fall.
 I can describe the graph that represents the solutions to a linear inequality in two variables.
Required materials:
 Colored pencils
Required preparation:
 The colored pencils are optional.
 If used in the first activity, each student needs 2 different colors.
 As an alternative to two colors, students could just write two different symbols.
Standards:
 This lesson builds towards the standards: CCSS.HSAREI.D.12MS.AREI.12MO.A1.REI.C.7MO.A1.REI.C.8
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.